Quaternion Fusion Packets 1470456656, 9781470456658

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765

Quaternion Fusion Packets Michael Aschbacher

Quaternion Fusion Packets Michael Aschbacher

765

Quaternion Fusion Packets Michael Aschbacher

EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss

Kailash Misra

Catherine Yan

2020 Mathematics Subject Classification. Primary 20D05; Secondary 20C20.

Library of Congress Cataloging-in-Publication Data Names: Aschbacher, Michael, 1944– author. Title: Quaternion fusion packets / Michael Aschbacher. Description: Providence, Rhode Island : American Mathematical Society, 2021. | Series: Contemporary mathematics, 0271-4132 ; volume 765 | Includes bibliographical references and index. Identifiers: LCCN 2020043155 | ISBN 9781470456658 (paperback) | ISBN 9781470464219 (ebook) Subjects: LCSH: Finite simple groups. | Sylow subgroups. | AMS: Group theory and generalizations – Abstract finite groups – Finite simple groups and their classification. | Group theory and generalizations – Abstract finite groups – Sylow subgroups, Sylow properties, π-groups, π-structure. Classification: LCC QA177 .A826 2021 | DDC 512/.23–dc23 LC record available at https://lccn.loc.gov/2020043155 Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online) DOI: https://doi.org/10.1090/conm/765

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established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

26 25 24 23 22 21

Contents Background and overview

1

Chapter 0. Introduction

3

Chapter 1. The major theorems and some background 1.1. Theorems 1 through 8 1.2. Background 1.3. An outline of the proof

7 7 10 13

Basics and examples

19

Chapter 2. Some basic results 2.1. Preliminary lemmas 2.2. Solvable components 2.3. Intrinsic SL2 [m]-components 2.4. A sufficient condition for quaternion fusion packets 2.5. Basic results on fusion packets 2.6. The case z ∈ Z(F). 2.7. F = SOτ 2.8. Modules for groups with a strongly embedded subgroup

21 21 27 34 39 39 41 45 48

Chapter 3. Results on τ 3.1. Δ(τ ), η(τ ), and μ(τ ) 3.2. The graph A 3.3. More basic lemmas 3.4. Generating F

53 54 64 66 72

Chapter 4. W (τ ) and M (τ ) 4.1. 3-transposition groups 4.2. The groups in M (τ ) 4.3. The groups ω ¯ (Φ, m)

79 79 86 90

Chapter 5. Some examples 5.1. AEn 5.2. The 2-share of the order of some groups 5.3. Orthogonal groups and packets 5.4. Linear, unitary, and symplectic groups and packets 5.5. Exceptional groups and packets 5.6. FS (G) is simple ¯ (Ad−1 , m) 5.7. Lπd [m] and ω 5.8. ω ¯ (Dn , m) v

95 96 102 105 109 113 117 120 123

vi

CONTENTS

5.9. ω ¯ (Cn , m) and 2¯ ω (Cn , m) 5.10. Some constrained examples 5.11. Summary of basics

127 128 132

Theorems 2 through 5

135

Chapter 6. Theorems 2 and 4 6.1. D(τ )c 6.2. Beginning the case z ∈ O2 (F) 6.3. The case E ≤ NG (K) 6.4. Subnormal closure 6.5. F ∗ (F) 6.6. z not in O2 (F) 6.7. The proof of Theorem 2

137 137 140 146 159 162 166 170

Chapter 7. Theorems 3 and 5 7.1. Packets of width 1 7.2. A(z) = ∅

173 173 184

Coconnectedness

193

Chapter 8. τ ◦ not coconnected 8.1. Dc disconnected

195 195

Theorem 6

203

Chapter 9. Ω = Ω(z) of order 2 9.1. |Ω(z)| = 2 9.2. Generation when |Ω(z)| = 2 9.3. |Ω(z)| = 2 and Z ∩ O(z) = {z} 9.4. |Ω(z)| = 2 and D∗ (z) = D(z) 9.5. |Ω(z)| = 2 and μ isomorphic to S4

205 205 211 215 223 240

Chapter 10. |Ω(z)| > 2 10.1. |Ω(z)| = 4 and μ isomorphic to Weyl(D4 ) 10.2. |Ω(z)| large

255 255 263

Chapter 11.1. 11.2. 11.3. 11.4. 11.5. 11.6.

11. Some results on generation |Ω(z)| = 2, μ isomorphic to Weyl(Dn ), n ≥ 4 Generation More generation Essentials and normal subsystems Generating Ωd [m] Generating AEk

269 269 273 289 292 295 300

Chapter 12.1. 12.2. 12.3. 12.4.

12. |Ω(z)| = 2 and the proof of Theorem 6 |Ω(z)| = 2, μ isomorphic to Weyl(D4 ) More |Ω(z)| = 2 Completing |Ω(z)| = 2 The proof of Theorem 6

305 305 312 325 338

CONTENTS

Theorems 7 and 8 Chapter 13.1. 13.2. 13.3. 13.4. 13.5. 13.6.

13. |Ω(z)| = 1 and μ abelian Systems with μ abelian Generic systems with μ abelian Symplectic groups and systems Linear and unitary groups and systems Generating symplectic and linear systems Finishing μ abelian

Chapter 14. More generation 14.1. A generation lemma 14.2. A generation lemma for E8 Chapter 15.1. 15.2. 15.3. 15.4. 15.5. 15.6. 15.7. 15.8.

15. |Ω(z)| = 1 and μ nonabelian |Ω(z)| = 1 The case r > 1 Φ = Dn Φ = D4 Φ = An Generating linear systems Wrapping up Φ = An Φ = En

vii

339 341 341 351 360 362 366 371 375 375 379 385 385 389 399 403 410 419 422 424

Theorem 1 and the Main Theorem

431

Chapter 16.1. 16.2. 16.3.

433 433 435 436

16. Proofs of four theorems The proof of Theorem 1 Proofs of the Main Theorem and Theorems 6, 7, and 8 Lie fusion packets

References and Index

439

Bibliography

441

Index

443

Background and overview

CHAPTER 0

Introduction The notion of fusion in groups first arose in the context of finite group theory. The fusion theorems of Alperin [Alp67] and Goldschmidt [Gol74] , and Glauberman’s Z ∗ -Theorem [Gla66] are highlights of the early work on fusion in finite groups. Saturated fusion systems were first defined and studied by L. Puig, although Puig calls these objects Frobenius categories rather than fusion systems; see for example [Pui09]. Our introduction to the subject was from [BLO03] and we adopt the notation and terminology found there and in [AKO11]. Let p be a prime and S a finite p-group. A fusion system on S is a category F whose objects are the subgroups of S and for P, Q ≤ S, the set homF (P, Q) of morphisms from P to Q consists of injective group homomorphisms from P into Q, and some weak axioms are satisfied. The standard example is a system FS (G), where G is a finite group, S ∈ Sylp (G), and the morphisms from P to Q are those induced via conjugation in G. The fusion system is saturated if it satisfies two additional axioms which are easily checked in FS (G) using Sylow’s Theorem. There are also exotic saturated fusion systems which are not realized in finite groups. See [BLO03] or [AKO11] for precise definitions, notation, terminology, and basic results on fusion systems. There is also some introductory discussion in section 1.2. Puig developed his theory as a tool in modular representation theory, while homotopy theorists use fusion systems to study the classifying spaces of finite groups, and more general objects called p-local finite groups. However our interest in the subject is motivated by local finite group theory. It is by now clear that techniques from the local theory of finite groups are effective in the investigation of fusion systems, and indeed some arguments are easier to carry out in the category of fusion systems than the category of groups. Thus we are interested in, first, using techniques from the local theory to prove theorems about saturated fusion systems, and perhaps to even classify the simple saturated 2-fusion systems, or the subclass of systems of component type (as defined in Definition II.14.1 of [AKO11]). Then, second, we hope to use the theorems about fusion systems to simplify the proof of the theorem classifying the finite simple groups, particularly that part of the proof on groups of component type. See section II.14 in [AKO11] and [Asc15] for discussion of such a program, begun by the author, in which this manuscript plays a major part. An obvious early step in a program to classify simple 2-fusion systems of component type is to prove an analogue of the so-called “classical involution theorem” on finite groups (found in [Asc77]) for 2-fusion systems. The hypotheses of that theorem are for the most part fusion theoretic, so a translation of the statement on groups into a statement about fusion systems is not difficult. This leads to the following definition.

3

4

0. INTRODUCTION

A quaternion fusion packet is a pair τ = (F, Ω) where F is a saturated fusion system on a finite 2-group S and Ω is an F-invariant set of subgroups of S such that (QFP1) There exist an integer m such that for all K ∈ Ω, K has a unique involution z(K) and is nonabelian of order m. (QFP2) For each pair of distinct K, J ∈ Ω, |K ∩ J| ≤ 2. (QFP3) If K, J ∈ Ω and v ∈ J − Z(J), then v F ∩ CS (z(K)) ⊆ NS (K). (QFP4) If K, J ∈ Ω with z = z(K) = z(J), v ∈ K, and φ ∈ homCF (z) (v, S) then either vφ ∈ J or vφ centralizes J. We next consider the generic class of examples of quaternion fusion packets. Let p be an odd prime and write Chev(p) for the collection of finite groups G of Lie type of characteristic p with an irreducible root system and generated by root groups. Let Chev∗ (p) consist of the members of Chev(p) which are not isomorphic to L2 (q) or 2 G2 (q) for any power q of p. Thus G ∈ Chev∗ (p) is quasisimple unless ˜ of type G, defined by the G∼ = SL2 (3), and G is an image of the universal group G Steinberg relations of type G. The fundamental subgroups of G are the conjugates of groups Z(Uα ), Z(U−α ) , for α a long root in a root system for G, and Uα the root subgroup of G corresponding to α. Each fundamental subgroup is isomorphic to SL2 (q) for a suitable power q of p. Let S ∈ Syl2 (G) and let Ω be the set of groups S ∩ L, as L varies over the fundamental subgroups L of G with S ∩ L ∈ Syl2 (L). Then the pair (FS (G), Ω) is a quaternion fusion packet; for example this follows from 14.5 in [Asc77] or 2.4.1 in this paper. We call such packets Lie fusion packets. Write SL2 [m] for the fusion system FS (G) of a group G isomorphic to SL2 (q), q odd, with m = (q 2 − 1)2 the 2-share of |G|. If (FS (G), Ω) is a Lie fusion packet, K ∈ Ω, and z = z(K), then the fundamental subgroup L of G containing K is subnormal in CG (z), so FK (L) ∼ = SL2 [m] is subnormal in CF (z). Given a saturated fusion system F on a finite 2-group S and a collection Σ of subgroups of S, the normal closure of Σ in F is the smallest normal subsystem [Σ]F of F such that a Sylow group of the subsystem contains Σ. We can now state our main theorem. Main Theorem. Assume F is a saturated fusion system on a finite 2-group S, z is an involution in S, and E is a subnormal subsystem of CF (z) containing z and isomorphic to SL2 [m] for some integer m. Let K be the Sylow group of E and set Ω = K F . Assume F is the normal closure of K. Then (F, Ω) is a Lie fusion packet. The Main Theorem is a corollary to Theorems 1 and 2, stated in section 1.1. The Main Theorem provides a characterization of the 2-fusion systems of the groups L in Chev∗ (p) for p an odd prime. By Theorem 5.6.18, if L is simple then its 2-fusion system is also simple. Section 1.2 contains a brief introduction to the theory of fusion systems, and then section 1.3 supplies an outline of the proof of Theorems 1 and 2, plus the proofs of Theorems 3-8, which are used in the proof of Theorem 1. We close with a historical remark. In January 1974, John Thompson gave a series of lectures at the winter meeting of the AMS in San Francisco. One of

0. INTRODUCTION

5

the talks was on a possible approach to proving the B-conjecture. That approach involved a finite group G and a normal subgroup Q of CG (z) for some involution z ∈ K ∈ Syl2 (Q) with K quaternion. It is easy to see that the 2-fusion system of G has an associated quaternion fusion packet. Thompson’s idea was to show that K G is a group of Lie type. Toward that end he constructed a 2-torus of G and part of the Weyl group of the 2-torus. This idea served as the foundation of the proof of the classical involution theorem, and is also the foundation for the study of quaternion fusion packets.

CHAPTER 1

The major theorems and some background 1.1. Theorems 1 through 8

To state the remainder of our major results we need more notation and terminology. Let τ = (F, Ω) be a quaternion fusion packet. We will show that τ comes equipped with a collection η(τ ) of sets η such that η consists of cyclic subgroups of S each of which is conjugate to a subgroup of index 2 in some member of Ω. For η ∈ η(τ ), W = η is abelian and should be viewed as a “2-torus” of F. For example if τ is the Lie packet of some group G of Lie type then W is a 2-torus of G. Each K ∈ Ω acts on W , K ∩ W ∈ η, and k ∈ K − W induces an automorphism dK∩W on W via conjugation that is independent of the choice of k. Indeed for V ∈ η, V is conjugate under AutF (W ) to K ∩ W for some K ∈ Ω, so we get a corresponding map dV , and hence a set Dη = {dV : V ∈ η} of involutions in AutF (W ). Set μ(τ ) = Dη ≤ AutF (W ), and call μ(τ ) the Thompson group of τ . It can be shown that usually η(τ ) contains a unique member, and in any event μ(τ ) is independent of the choice of η in η(τ ). One should think of the Thompson group as the Weyl group of its 2-torus W , and indeed if τ is the Lie packet of G then the Thompson group is the Weyl group of W , or at least a large subgroup of the Weyl group. Let Z = Z(τ ) be the union of the classes z(K)F as K ranges over Ω. We consider certain graphs D = D(τ ), D∗ = D ∗ (τ ), and A = A(τ ) on Z. Given such a graph B and z ∈ Z, write B(z) for the vertices adjacent to z in B. Write B c for the complementary graph. Write F ◦ for the normal closure [Ω]F of Ω in F and set τ ◦ = (F ◦ , Ω). For K ∈ Ω, Ω(z(K)) = {J ∈ Ω : z(K) = z(J)}. We are now ready to state our eight major theorems. The notation for describing the examples appearing as conclusions in the theorems can be found immediately after the statement of the eight theorems. Theorem 1. Assume τ = (F, Ω) is a quaternion fusion packet. Let K ∈ Ω, z = z(K), and assume that F is the normal closure of K and F is transitive on Ω. Then one of the following holds: (1) z ∈ O2 (F). (2) τ is a Lie fusion packet. (1) (3) F is L2 [2m](1) , L+ , or M12 . 3 [m] ˆ 6 (2) or Ω ˆ + (2). (4) F is a central factor system of Sp 8 7

8

1. THE MAJOR THEOREMS AND SOME BACKGROUND

Theorem 2. Assume τ = (F, Ω) is a quaternion fusion packet. Let K ∈ Ω, z = z(K), and assume that F is the normal closure of K in F, F is transitive on / Z(F). Then F is constrained, K ≤ O2 (CF (z)), and a Ω, and z ∈ O2 (F) but z ∈ model G for F satisfies one of the following: (1) G ∼ = L3 (2)/E8 . (2) G/Z(G) ∼ = L3 (2)/E64 or L3 (2)/23+6 . (3) G/Z(G) ∼ = AEn . (4) For η ∈ η(τ ), η  G and μ(τ ) is generated by 3-transpositions. Theorem 3. Assume τ = (F, Ω) is a quaternion fusion packet, K ∈ Ω, z = z(K), and A(z) = ∅. Set F1 = [Ω(z)]F ◦ , τ1 = (F1 , Ω(z)), and Ω2 = Ω − Ω(z). If Ω2 = ∅ set F2 = [Ω2 ]F ◦ and τ2 = (F2 , Ω2 ). Then (1) τ1 is a quaternion fusion packet with μ(τ1 ) ∼ = D12 . (2) If Ω2 = ∅ then τ2 is a quaternion fusion packet and F ◦ = F1 × F2 . (3) One of the following holds: (i) η  F1 for η ∈ η(τ1 ). (ii) F1 ∼ = L3 (2)/E8 . (1) (iii) F1 ∼ . = L+ 3 [m] ∼ (iv) F1 = G2 [m]. (v) F1 ∼ = M12 . Theorem 4. Assume τ = (F, Ω) is a quaternion fusion packet such that F = F ◦ . Then there exists a partition {Ωi : i ∈ I} of Ω such that, setting Fi = [Ωi ]F and τi = (Fi , Ωi ), each of the following holds: (1) τi is a quaternion fusion packet. (2) F is a central product of the Fi , for i ∈ I. (3) For i ∈ I, either Fi is transitive on Ωi or μ(τi ) ∼ = D12 . (4) For i ∈ I, either D(τi )c is a connected component of D(τ )c or Fi /Z(Fi ) ∼ = Spn [m] or L− n [m] for some n ≥ 6. In the situation described in Theorem 4, we say that τ is the central product of the subpackets τi and write τ = τ1 ∗ · · · ∗ τr . Moreover we call the subpackets τi , i ∈ I, the coconnected components of τ . For example from part (4) of Theorem 4, we have the graph D(τ ) on the set Z(τ ), and, aside from the exceptional cases appearing in (4), the sets Z(τi ) are the connected components of the complementary graph D(τ )c : the coconnected components of D(τ ). Alternatively from part (3) of Theorem 4, we can view coconnection as corresponding to orbits of F on Ω. Theorem 5. Assume τ = (F, Ω) is a quaternion fusion packet such that Ω = {K} is of order 1, F = F ◦ , and z(K) ∈ / Z(F). Then one of the following holds: (1) μ(τ ) ∼ = S3 and for η ∈ η(τ ), η  F. (2) F ∼ = L3 (2)/E8 . (3) F ∼ = L2 [2m](1) . (4) F ∼ = L3 [m]. (1) (5) F ∼ . = L+ 3 [m] (6) F ∼ = G2 [m]. (7) F ∼ = M12 .

1.1. THEOREMS 1 THROUGH 8

9

Theorem 6. Assume τ = (F, Ω) is a quaternion fusion packet such that F = F ◦ is transitive on Ω, |Ω(z(K))| > 1 for some K ∈ Ω, and A(τ ) = ∅. Then one of the following holds: (1) F ∼ = L3 (2)/E64 . (2) F/Z(F) ∼ = L3 (2)/23+6 . (3) F/Z(F) ∼ = AEn for some n ≥ 5. (4) For η ∈ η(τ ), η  F and μ(τ ) ∼ = Weyl(Dn ) for some n ≥ 3. (5) F/Z(F) ∼ (2). = Sp6 (2) or Ω+ 8 (6) F/Z(F) ∼ = P Ωn [m] for some n ≥ 5. Theorem 7. Assume τ = (F, Ω) is a quaternion fusion packet such that F = F ◦ is transitive on Ω, μ(τ ) is abelian, and for K ∈ Ω and z = z(K), Ω(z) = {K} = Ω. Then one of the following holds: ˆ 5. (1) F ∼ = AE (2) F/Z(F) ∼ = P Spn [m] for some n ≥ 4. (3) F/Z(F) ∼ = L− n [m] for some n ≥ 4. Theorem 8. Assume τ = (F, Ω) is a quaternion fusion packet such that F = F ◦ is transitive on Ω, μ = μ(τ ) is nonabelian, and for K ∈ Ω and z = z(K), Ω(z) = {K} = Ω. Then μ ∼ = Weyl(Φ) where Φ = An for some n ≥ 4; Φ = Dn for some n ≥ 3; or Φ is E6 , E7 , or E8 . Moreover one of the following holds: (1) For W ∈ W (τ ), W  F. (2) Φ = An for some n ≥ 4 and F is a central factor system of SL+ n+1 [m]. − (3) Φ = Dn for some n ≥ 3 and F ∼ [m], Spin [m], Spin = Spin+ 2n+1 2n 2n+2 [m], + ∼ ˆ ˆ AE2n , or AE2n+1 for some n ≥ 4; or n > 4 is even and F = HSpin2n [m] or ˆ 2n . H AE ˆ 3 (2)/E64 or Sp ˆ 6 (2). (4) m = 8, Φ = D3 , and F ∼ =L + 3+6 ∼ ˆ ˆ (5) Φ = D4 and F = Ω8 (2), L3 (2)/2 , F4 [m], or E6− [m]. (6) Φ = E6 and F ∼ = E6+ [m]. ˜7 [m]. (7) Φ = E7 and F ∼ = E7 [m] or E (8) Φ = E8 and F ∼ = E8 [m]. We next define the notation used to describe the fusion systems appearing in our theorems. We begin with a table summarizing our notation for the fusion system FS (G) of a Lie packet τ = (FS (G), Ω), where G is a group of Lie type over Fq . In the table, q is a prime power, π = ±1 ≡ q mod 4, and m = (q 2 − 1)2 is the 2-share of q 2 − 1. In the last two rows of the table, μ = μ(τ ) is the Thompson group of τ and W (Φ) is an abbreviation for the Weyl group Weyl(Φ) of type Φ.

group G2 (q), 2 D4 (q) system G2 [m] μ : π = 1 W (A2 ) μ : π = −1 W (A2 )

Ln (q) Lπ n [m] W (An−1 ) E2[n/2]

Spn (q) Ω+ Ω2n+1 (q) 4n (q) + Spn [m] Ω4n [m] Ω2n+1 [m] E2n/2 W (D2n ) W (Dn ) E2n/2 W (D2n ) W (Dn )

group Ω− Ω4n+2 (q) F4 (q) E6 (q) 4n+4 (q) − π system Ω4n+4 [m] Ω4n+2 [m] F4 [m] E6π [m] μ : π = 1 W (D2n+1 ) W (D2n+1 ) W (D4 ) W (E6 ) W (D2n ) W (D4 ) W (D4 ) μ : π = −1 W (D2n+1 )

E7 (q) E7 [m] W (E7 ) W (E7 )

E8 (q) E8 [m] W (E8 ) W (E8 )

10

1. THE MAJOR THEOREMS AND SOME BACKGROUND

Implicit in the table is the fact that the 2-fusion system of X(q) does not depend on q, but on X and (q 2 − 1)2 , and possibly also on π. This fact emerges in the proof of our major theorems, as a consequence of the uniqueness, up to isomorphism, of a quadratic fusion packet τ with suitable invariants, such as μ(τ ). For example the Thompson groups of the Lie packets are calculated in Chapter 5. The last section in the paper supplies an even stronger uniqueness condition for Lie packets. Next the notation for the exceptional examples. Write L2 [2m](1) for the 2-fusion system FS (G) of G, where G is the extension of L2 (q 2 ) of degree 2 with a semidihedral Sylow 2-subgroup S of order 2m. Write (1) L+ for FS (G) where G is the extension of L ∼ = L3 (q) by a graph automorphism 3 [m] and L has a wreathed Sylow 2-subgroup of order m2 /2. The 2-fusion system of L is − L+ 3 [m], while if L has semidihedral Sylow groups then its 2-fusion system is L3 [m]. Write M12 for the 2-fusion system of M12 . ˆ 6 (2), Ω ˆ + (2) for the universal covering group of Sp6 (2), Ω+ (2), respecWrite Sp 8 8 tively. We also use that notation for the 2-fusion system of the group. We write L3 (2)/E8 for both the nonsplit extension of E8 by its general linear group and for the 2-fusion system of that group. Similarly L3 (2)/E64 denotes both the maximal parabolic in Sp6 (2) with this structure and its 2-fusion system, and L3 (2)/23+6 denotes both the parabolic in Ω+ 8 (2) and its 2-fusion system. The 3+6 3+6 ˆ ˇ groups and systems L3 (2)/2 and L3 (2)/2 are defined in 5.10.10 and 5.10.12. ˆ n , H AE ˆ n , and AE ˇ n are defined and discussed in section The systems AEn , AE 5.1. For example AEn is the split extension of An by its natural module. A summary of basic concepts, notation, and terminology from the theory of quaternion fusion packets can be found in section 5.11.

1.2. Background For background involving fusion systems, we refer the reader for the most part to [BLO03] or [AKO11]. However in this section we recall some notation, terminology, and definitions involving fusion systems used frequently in this paper. Let C be a category and A, B objects in C. Write homC (A, B) for the set of morphisms in C from A to B, and AutC (A) for the group of automorphisms of A in C. Given an isomorphism α : A → B and C a subobject of A, write α∗ for the bijection α∗ : homC (C, A) → homC (Cα, B) defined by α∗ : φ → α−1 φα. ˜ A morphism α : F → F˜ of fusion Let F˜ be a fusion system on a p-group S. systems is a group homomorphism α : S → S˜ such that the induced map takes morphisms of F to morphisms of F˜ , as defined in II.2.2 in [AKO11]. For E a subsystem of F on T ≤ S and α ∈ homF (T, S), let Eα∗ be the fusion system on T α with homEα∗ (P α, Qα) = homE (P, Q)α∗ for P, Q ≤ T . Notice that α : E → Eα∗ is an isomorphism. Let S be a group. For x, y ∈ S, xy = y −1 xy is the conjugate of x by y, and cy : S → S defined by cy : x → xy is conjugation by y. A fusion system on S is a category F whose objects are the subgroups of S, and such that for P, Q ≤ S, homF (P, Q) is a set of injective group homomorphisms from P into Q, and such that:

1.2. BACKGROUND

11

(1) for each s ∈ S and P, Q ≤ S with P s ≤ Q, cs : P → Q is in homF (P, Q), and

(2) for each φ ∈ homF (P, Q), φ : P → P φ and φ−1 : P φ → P are morphisms in F. From now on let p be a prime, S a finite p-group, and F a fusion system on S. We call S the Sylow group of F. Given P ≤ S, let P F = {P φ : φ ∈ homF (P, S)} be the set of F-conjugates of P . Define P to be fully centralized, fully normalized if for all Q ∈ P F , |CS (P )| ≥ |CS (Q)|, NS (P )| ≥ |NS (Q)|, respectively. Write F f for the set of fully normalized subgroups of S. Given a subgroup T of S and a family (Δi : i ∈ I) of morphism in F between subgroups of T , define the fusion system generated by the family to be the intersection of all fusion systems on T containing the morphisms in each Δi , and write Δi : i ∈ I T for that fusion system. If the choice of T is clear, we often omit the subscript T . For U ≤ S, define NF (U ), CF (U ) to be the subsystems of F on T = NS (U ), CS (U ) such that morphisms from P ≤ T into T consist of those φ ∈ homF (P, T ) extending to ϕ ∈ homF (P U, T ) such that ϕ acts on U , centralizes U , respectively. Call NF (U ) the normalizer in F of U and CF (U ) the centralizer in F of U , respectively. Our fusion system F is saturated if: (I) For all P ∈ F f , P is fully centralized and AutS (P ) ∈ Sylp (AutF (P )), and (II) whenever P ≤ S and φ ∈ homF (P, S) with P φ fully centralized, then φ extends to a member of homF (Nφ , S), where Nφ = {g ∈ NS (P ) : c∗g ∈ AutS (P φ)}, using the ∗-notation defined above. In the remainder of the section assume F is a saturated fusion system on the finite p-group S. We have the following theorem of Puig; see I.5.5 in [AKO11] for a proof. Lemma 1.2.1. For U ∈ F f , NF (U ) and CF (U ) are saturated fusion systems. A subgroup P of S is centric if for each Q ∈ P F , CS (Q) ≤ Q. Further P is radical if Inn(P ) = Op (AutF (P )). Write F c , F r for the centric, radical subgroups of S, respectively. For Y ⊆ X = {f, r, c}, set  FY = Fy. y∈Y

Lemma 1.2.2. (Alperin’s Fusion Theorem) F = AutF (U ) : U ∈ F f rc . Proof. See for example I.3.5 in [AKO11].



Definition 1.2.3. The Alperin data for F consists of (1) S and (2) F f rc and (3) the subgroups AutF (U ) of Aut(U ), as U varies over F f rc . Lemma 1.2.4. The Alperin data determines F up to isomorphism; that is if F˜ is a saturated fusion systems on S˜ and α : S → S˜ is an isomorphism of groups

12

1. THE MAJOR THEOREMS AND SOME BACKGROUND

which induces a bijection α : F f rc → F˜ f rc such that for each U ∈ F f rc , α∗ : Aut(U ) → Aut(U α) maps AutF (U ) to AutF˜ (U α), then α induces an isomorphism of F with F˜ . Proof. This follows from Remark 9.11 in [Asc10]; it is almost immediate from Alperin’s Fusion Theorem.  For P ≤ S, define A(P ) = AF (P ) to consist of those α ∈ homF (NS (P ), S) such that P α ∈ F f . Lemma 1.2.5. Let P ≤ S and φ ∈ homF (P, S). (1) If P φ is fully centralized then φ extends to a member of homF (P CS (P ), S). (2) If P φ is fully normalized then there exists χ ∈ AutF (P ) such that χφ extends to a member of A(P ). (3) A(P ) = ∅. Proof. These are special cases of I.5.2.c in [AKO11].



A subgroup T of S is strongly closed in S with respect to F if for each t ∈ T , tF ⊆ T . The notions of an F-invariant subsystem of F, weakly normal subsystem and normal subsystem of F are defined in Definition I.6.1 of [AKO11]. We write E  F to indicate E is normal in F. If E is an F-invariant subsystem on T then T is strongly closed in S with respect to F and the invariant condition and Frattini condition of I.6.1 in [AKO11] are satisfied. An F-invariant system E is normal if E is saturated and the extension condition of I.6.1 in [AKO11] (called condition (N1) in [Asc08a]) is satisfied. Recall for Σ a collection of subgroups of S, [Σ]F is the smallest normal subsystem E of F such that Σ is contained in the Sylow group of E. Such a subsystem exists by Theorem 1 in [Asc11]. We call [Σ]F the normal closure of Σ in F.  Write O p (F) for the normal closure of S in F. Lemma 1.2.6. (Craven’s Theorem) If E is a saturated F-invariant subsystem  of F then O p (E)  F. Proof. See [Cra11].



The normal subsystem O p (F) of F is defined in I.7.4 in [AKO11]. If F0 is a normal subsystem of F on S0 and S0 ≤ T ≤ S with T strongly closed in S with respect to F, then there is a product subsystem T F0 of F on T defined in 8.19 of [Asc11]. If T is strongly closed in S with respect to F, then from section 8 in [Asc08a] we can form the factor system F/T and then form the surjective morphism Θ : F → F/T of fusion systems as in Definition 12.13 of [Asc11]. Lemma 1.2.7. Let T be strongly closed in S with respect to F, F + = F/T , and Θ : F → F + the natural surjection. Let E be a subsystem of F on E and E + = EΘ. (1) If E is saturated then so is E + . (2) If E  F then E +  F + . (3) If T ≤ Z(F) and E +  F then the preimage of E + in F is normal in F. Proof. See 8.5 in [Asc08a] or II.5.4 in [AKO11] for (1). See 8.9.2 in [Asc08a] for (2) and 8.10 in [Asc08a] for (3). 

1.3. AN OUTLINE OF THE PROOF

13

A subgroup Q of S is normal in F if F = NF (Q). Write Op (F) for the largest normal subgroup of F. Further F is constrained if there exists a normal centric subgroup of F. A model for F is a finite group G such that S ∈ Sylp (G), F ∗ (G) = Op (G), and F = FS (G). If G is a model for F then Op (G) = Op (F) is centric, so F is constrained. By a theorem in [BCG+ 05], the converse is true. Lemma 1.2.8. If F is constrained then (1) F possesses a model G. (2) G is unique up to an isomorphism which is the identity on S. ˜ is a model for F˜ (3) If α : F → F˜ is an isomorphism of fusion systems and G ˜ extending α, and α then there exists an isomorphism α ˇ:G→G ˇ is determined up an automorphism cz of G, for z ∈ Z(S). (4) The map H → FS∩H (H) is a bijection between the set of normal subgroups of G and the set of normal subsystems of F. Proof. See Theorem III.5.10 in [AKO11] for (1), and II.4.3 in [AKO11] for (3). Part (3) implies (2). See II.7.5 in [AKO11] for (4).  We recall Definition 2.7.2 from [Asc19]: Definition 1.2.9. For T ≤ S define sub0 (F, T ) = F, and, proceeding recursively, given subi (F, T ) define subi+1 (F, T ) = [T ]subi (F ,T ) . Write n(F, T ) for the first n such that subn (F, T ) = subn+1 (F, T ), and set sub(F, T ) = subn(F ,T ) (F, T ). We call sub(F, T ) the subnormal closure of T in F, and subi (F, T ), 1 ≤ i ≤ n(F, T ), the subnormal series for T in F. From 2.7.3 in [Asc19], the members of the subnormal series are indeed subnormal subsystems of F, so in particularly these systems are saturated. A subgroup X ∈ F f c is essential in F if OutF (X) has a strongly p-embedded subgroup. Write F e for the set of essential subgroups of F. Lemma 1.2.10. (Alperin-Goldschmidt Fusion Theorem) If F is saturated then F = AutF (R) : R ∈ F e ∪ {S} S . Proof. See for example I.3.5 in [AKO11].



Lemma 1.2.11. (1) F e ⊆ F f rc . (2) O2 (F) is contained in each member of F f rc . Proof. See I.3.3.a in [AKO11] for (1) and I.4.5 in [AKO11] for (2).



Lemma 1.2.12. Let U ∈ F f and E  F. Then NE (U )  NF (U ). Proof. This is 8.24 in [Asc11].



1.3. An outline of the proof In this section τ = (F, Ω) is a quaternion fusion packet. We begin with a preview of some of the concepts and notation involving τ that will be formally introduced later in the paper. Then, with this background in place, we give an outline of the proof of our main theorems appearing in section 1.1.

14

1. THE MAJOR THEOREMS AND SOME BACKGROUND

Let Z = Z(τ ) be the union of the classes z(K)F as K ranges over Ω, where z(K) is the involution in K. Define ZS = ZS (τ ) = {z(K) : K ∈ Ω}, and for z ∈ ZS set Ω(z) = {K ∈ Ω : z(K) = z}. Set WS = ZS . For K ∈ Ω we define a fusion system OK on K. If m = 8 set OK = FK (K). If m > 8 set OK = O 2 (AutF (Q)) : Q ∈ Q(K) K , where Q(K) is the set of Q8 -subgroups of K. Let z ∈ ZS ∩F f , and set O(z) = Ω(z) . By 2.6.6.1, O(z) is the central product of the members of Ω(z). Further by 2.6.8, F contains a central product O(z) of the subsystems OK , for K ∈ Ω(z), and by 2.6.11, O(z)  CF (z), so each OK is subnormal in CF (z). Similarly set O(τ ) = Ω . By 3.1.5, O(τ ) is a centralproduct of the members of Ω, and by 3.3.7, F contains a central product O(τ ) = K∈Ω OK that is normal in NF (WS ). Next for t ∈ Z = Z(τ ) and α ∈ A(t), Δ(t) = {(K ∩ CS (t)α)α−1 : K ∈ Ω(tα) and |K ∩ CS (t)α| > 2}. By 3.1.3, Δ(t) is independent of the choice of α ∈ A(t). Set ZΔ = ZΔ (τ ) = {t ∈ Z : Δ(t) = 1} and Δ = Δ(τ ) =



Δ(t).

t∈ZΔ

By 3.1.5, Δ(z) = Ω(z) for z ∈ ZS , and then by 3.1.11, each member of Δ − Ω is cyclic of order m/2. Let η(τ ) consist of the sets η of cyclic subgroups of S such that (i) η is η -invariant, and (ii) for each V ∈ η, V is contained in a unique D(V ) ∈ Δ, and either V = D(V ) or D(V ) ∈ Ω and |D(V ) : V | = 2, and (iii) the map V → D(V ) is a bijection of η with Δ. Let ηS (τ ) denote the set of S-invariant members of η(τ ). By 3.1.10, ηS (τ ) is nonempty, and by 3.1.11, for each η ∈ η(τ ), η is abelian. We write W (τ ) for {η : η ∈ ηS (τ ) . By 3.3.14: Lemma 1.3.1. Either η(τ ) = {η} is of order 1, or m = 8 and if F is transitive on Ω then Δ = Ω. Thus, the exceptional case of 1.3.1 aside, W (τ ) = {W } is of order 1 and W is weakly closed in S with respect to F. In any case pick η ∈ η(τ ) and set W = η . By 3.1.12, for each K ∈ Ω, K acts on W and W acts on K. Then k ∈ K − W induces an involutory automorphism dK∩W on W via conjugation. Then from 3.1.20, for each V ∈ η, V is conjugate in NF (W ) to K ∩ W for some K ∈ Ω, so we get a conjugate dV of dK∩W in AutF (W ). Set Dη = {dV : V ∈ η} and μη = Dη ≤ AutF (W ). If η(τ ) = {η} then μ = μη is uniquely determined. But even if |η(τ )| > 1, we find in 3.1.25 that μη ∼ = μη for all η, η  ∈ η(τ ). Thus we write μ = μ(τ ) for μη where η ∈ η(τ ), and we call μ the Thompson group of τ . If U, V ∈ η are distinct then by 3.1.18, dU , dV is isomorphic to E4 , S3 , or D12 . In the last case, by definition, z(U ) ∈ A(z(V )), so in particular the graph A(τ ) is nonempty. Moreover this shows that if A(τ ) = ∅ then Dη is a set of 3-transpositions of μ.

1.3. AN OUTLINE OF THE PROOF

15

Therefore in section 4.1 we consider groups generated by 3-transpositions. In 4.1.25 we determine all groups generated by a conjugacy class of 3-transpositions satisfying Hypothesis 4.1.12. Then in 4.2.14 we find that if Dη is a set of 3transpositions of μ then Hypothesis 4.1.12 is indeed satisfied. By 3.1.10 each W ∈ W (τ ) is normal in S. Therefore W ∈ F f and by 3.3.9, M = [Ω]NF (CS (W )) is constrained. Hence by 1.2.8, M has a model M . We write M (τ ) for the set of such groups M as W ranges over W (τ ). By 1.3.1, if m > 8 or F is transitive on Ω and μ is nonabelian, then W (τ ) = {W } and M (τ ) = {M } are of order 1. Assume Dη is a conjugacy class of 3-transpositions of μ. By an earlier remark, 4.1.25 supplies a list of possibilities for μ. Then, in 4.3.8, that list is refined to show that: Lemma 1.3.2. If Dη is a conjugacy class of 3-transpositions and μ is nonabelian, then μ is the Weyl group Weyl(Φ), where Φ is a Coxeter diagram of type An , Dn , or En . ¯=ω Moreover in 4.3.1 we define a universal group G ¯ (Φ, m) and show in 4.3.8 ¯ → M with ker(π) ≤ Z(G). ¯ that there is a surjective homomorphism π : G In particular a Sylow 2-group T of M is determined up to isomorphism by Φ, m, and ker(π); often in a Lie packet, we have T = S and FT (M ) = NF (W ), in which case S and NF (W ) are determined up to isomorphism. Further ξ = (FT (M ), Ω) is a quaternion fusion packet which decomposes as a central product ξ = ξ1 ∗ · · · ∗ ξr with ξi a coconnected component of ξ such that μ(ξi ) = μi and μ = μ1 × · · · × μr . The proof of Theorem 2 is completed in section 6.7, building on work in sections 6.2, 6.3, and 6.4. The proof quickly reduces to the case where F is constrained, so by 1.2.8, F has a model G. In particular F ∗ (G) = O2 (G), and the proof of Theorem 2 can now be carried out in the category of such groups, where things go relatively smoothly and some difficult machinery required to establish later theorems is unnecessary. Define the order of τ be the triple (|S|, |Ω|, |M or(F)|) and order such triples lexicographically. We prove many of our theorems by “induction on the order of τ ”. Hence, when necessary, we may assume the following Inductive Hypothesis : Theorem 1 holds in proper subpackets ρ = (Y, Γ) of τ with Y ≤ F, Γ ⊆ Ω, and ρ satisfying the hypothesis of Theorem 1. Sometimes we assume the stronger Extended Inductive + = Hypothesis: For each Q ≤ S centralizing F ◦ with Q ∩ Z(τ ) = ∅, we have τQ ◦+ + ◦ + ◦ (F , Ω ) satisfying the Inductive Hypothesis, where (QF ) = QF /Q. Observe that if τ is a counter example to Theorem 1 of minimal order, then τ satisfies the Extended Inductive Hypothesis. We handle the case A(τ ) = ∅ later when proving Theorem 3 in section 7.2; so assume A(τ ) = ∅. Then by 6.6.6, τ ◦ = τ1 ∗ · · · ∗ τr is a central product of subpackets τi = (Fi , Ωi ), where Fi = [Ωi ]Fi is transitive on Ωi . In particular the first three parts of Theorem 4 hold, and the fourth part holds by 6.6.6 if either τ = τi or τ = τi satisfies Theorem 1. Thus Theorems 1 and 3 imply Theorem 4. Similarly, with a bit of work, Theorem 1 (together with Theorem 2) implies Theorems 3-8, although in practice Theorem 1 is established by a series of steps which include Theorems 3-8.

16

1. THE MAJOR THEOREMS AND SOME BACKGROUND

Most of our theorems include the hypotheses that F = F ◦ and F is transitive on Ω. But it is difficult to use these hypotheses early in the game, so in effect we are looking at a system F with a normal subsystem F ◦ = F1 ∗ · · · ∗ Fr where τi = (Fi , Ωi ) for 1 ≤ i ≤ r are the coconnected components of τ ◦ . Thus often we produce a subpacket ρ = ρ1 ∗ · · · ∗ ρr with ρ = (Y, Ω), r > 1, and ρi = (Yi , Ωi ) the coconnected components of ρ. We then show that Y1 is subnormal in F with conjugates Yi , 1 ≤ i ≤ r, so as F = F ◦ we have F ◦ = Y, contradicting F transitive on Ω and r > 1. This is often accomplished by showing that Y1 is tightly embedded in F, and appealing to the theory of tightly embedded subsystems in [Asc19]. Such arguments appear in section 8.1. For example assume μ is nonabelian. Then from earlier remarks, ξ = ξ1 ∗· · ·∗ξr and as F is transitive on Ω, we have NF (W ) transitive on the ξi and the μi , 1 ≤ i ≤ r, with μi generated by a conjugacy class of 3-transpositions. Then by 1.3.2, μi ∼ = Weyl(Φ) for each i and some Coxeter diagram Φ of type An , Dn , or En . It turns out this forces μi = μ(ρi ), and if r > 1 and the Inductive Hypothesis holds then ρi is a known packet with Thompson group μi , facilitating the arguments described in the previous paragraph. So, via this approach and with some effort, we can reduce to the case where either μ is abelian; or A(τ ) = ∅ and μ ∼ = D12 ; or μ∼ = Weyl(Φ) with Φ of type An , Dn , or En . In order to work effectively with our system F and eventually identify it as one of our examples, it seems necessary to show that F is generated by some nice collection of subsystems. In Definition 3.4.2 we are led to define the following subsystem of F: E(τ ) = SO(τ ), NF (O(θ)), NF (W ) : W ∈ W (τ ) and ∅ = θ ⊆ ZS with O(θ) ∈ F f .  Here O(θ) = t∈θ O(t). Sometimes we can prove F = E(τ ) but sometimes F = E(τ ). In general, given a candidate B for F containing AutF (S), the AlperinGoldschmidt Fusion Theorem 1.2.10 says that if B is proper in F then their exists R ∈ F e such that AutB (R) is proper in AutF (R). This observation is exploited in section 3.4 and later sections where we prove F is generated by suitable subsystems: for example often F = E(τ ). As mentioned earlier, most of our theorems are proven under the hypothesis that F = F ◦ , but it can be difficult to make use of this condition early in our analysis. Therefore in many cases we need to identify a candidate E for F ◦ as a ˜ and prove F ◦ = E. Typically subsystem of F, prove E = F˜ for some example F, we begin by proving something like F = F1 , F2 is generated by a suitable pair of subsystems. Now F˜ satisfies our hypothesis, so also F˜ = F˜1 , F˜2 . Then we identify subsystems Ei of Fi , prove E2 ∼ = F˜2 , so that we may take E2 = F˜2 , and then prove that E1 = F˜1 . Thus E = E1 , E2 = F˜1 , F˜2 = F˜ ; in particular E = E ◦ is saturated. Then we also prove that Ei  Fi for i = 1, 2, and perhaps prove one or two other technical conditions. This makes possible an appeal to section 1.4 in [Asc10], which implies that E  F, so as E = E ◦ we have F˜ = E = F ◦ = F, completing our identification of F. The actual proof of Theorem 1 is begun with the case nS = |ZS | = 1. Most of our exceptional examples arise in this case, so it requires special treatment. Thus in the next few paragraphs we consider the case where nS = 1, so that ZS = {z}. In Theorem 5, |Ω| = 1, so nS = 1 and μ is Z2 or S3 = Weyl(A2 ). Theorem 5 is proved in section 7.1; see in particular Remark 7.1.30.

1.3. AN OUTLINE OF THE PROOF

17

Next in Theorem 3, A(τ ) = ∅. Then A(z) = ∅ for some z ∈ ZS , so by 7.2.2 we have |Ω(z)| = 2. Then, in the most important subcase where F = [O(z)]F , we have Ω = Ω(z), so indeed nS = 1, and μ ∼ = D12 . Theorem 3 is proved in section 7.2; see Remark 7.2.25. Given Theorem 3, from now on we may assume μ is generated by a set of 3transpositions. Then from 1.3.2, in each case to be considered we can eventually reduce to μ abelian or μ ∼ = Weyl(Φ) with Φ of type An , Dn , or En . We next consider the cases where |Ω(z)| = 2. Sections 9.1 and 9.2 contain basic results established under this hypothesis. Section 9.3 considers the case where Z ∩ O(z) = {z}, which turns out to force nS = 1 and μ ∼ = S4 = Weyl(A3 ) = Weyl(D3 ). Section 9.4 considers the case μ abelian; as |Ω(z)| = 2 this eventually forces μ ∼ = E4 . In addition to identifying F we prove that F = CF (z), AutF (Ri ) : 1 ≤ i ≤ k for suitable Ri ∈ F e and small k. Then in section 9.5 we treat the case where μ ∼ = S4 and Z ∩ O(z) = {z}. Here, in the process of identifying F, we show F = E(τ ) = CF (z), NF (W ) , except in one unusual case where F = E(τ ), CF (b) for a certain involution b in W . This completes the treatment of the case nS = 1 and |Ω(z)| = 2. In our examples, |Ω(z)| is 1, 2, or 4. In section 10.2, we prove this fact in general when F = F ◦ is transitive on Ω and τ satisfies the Extended Inductive Hypothesis , reducing to the case nS = 1 and μ ∼ = Weyl(D4 ), treated earlier in section 10.1, where we identify F as one of our examples, each of which is a subsystem of P Ω+ 8 [m]. This completes the treatment of the case nS = 1. We next turn to a proof of Theorem 6, where we assume F = F ◦ is transitive on Ω, |Ω(z)| > 1 for z ∈ ZS , and A(τ ) = ∅. We also assume the Extended Inductive Hypothesis. As discussed in the previous paragraph, from section 10.2 we have |Ω(z)| = 2 or 4, and in the latter case Theorem 6 holds. Therefore we may assume |Ω(z)| = 2. We interject a generational result proven in sections 11.2 and 11.3 under one of the following two hypotheses: Hypothesis 1.3.3. For z ∈ ZS , |Ω(z)| = 2 and μ ∼ = Weyl(Dn ) for some n ≥ 3.  Hypothesis 1.3.4. For z ∈ ZS , |Ω(z)| = 1 and either m(WS ) = nS or t∈ZS t = 1 and m(WS ) = nS − 1. In Theorems 11.3.10 and 11.3.11 it is shown that if 1.3.3 or 1.3.4 hold then, with known exceptions, F = B(τ ), where B(τ ) = E(τ ), CF (x) : x ∈ W(τ ) ∩ F f for a suitable set W(τ ) of involutions in W or WS . Note that Hypothesis 1.3.4 is satisfied if |Ω(z)| = 1 and μ ∼ = Weyl(Φ) with Φ of type An , Dn , or E6 . If Φ is E7 then nS = 7 is odd, so some member of Ω is normal in S; then 3.4.6 says F = E(τ ). If Φ is E8 then Theorem 14.2.24 supplies the required generational result. When Φ is Dn with n = 4 we appeal to Theorem 14.1.18. Next Theorem 11.3.11 is extended in Theorem 11.5.3 to show that if F ◦ = 2 O (F) is the system of an orthogonal group then F is generated by a small collection of nice subsystems. For example if m > 8 and F ◦ = P Ω+ 2n [m] then F = SO(τ ), FS (SM ) . Suppose μ ∼ = Weyl(Dn ) for some n ≥ 3. In section 12.1 we treat the case n = 4, while the case n ≥ 5 is handled in section 12.2. Note if n = 3 then μ ∼ = S4 with nS = 1, a case already dealt with.

18

1. THE MAJOR THEOREMS AND SOME BACKGROUND

Therefore to complete the proof of Theorem 6, modulo the Extended Inductive Hypothesis, it remains to show that, under the hypothesis of Theorem 6 and the Extended Inductive Hypothesis, we have μ ∼ = Weyl(Dn ) for some n ≥ 2. When μ is nonabelian this is accomplished in Theorem 12.3.50. The case μ abelian is handled in section 9.4. The proof of Theorem 6 is wrapped up in sections 12.4 and 16.2. Assuming the Extended Inductive Hypothesis, we have established Theorem 6 and treated the case |Ω(z)| > 1 for z ∈ ZS . Thus in the remainder of the discussion we may assume |Ω(z)| = 1. We turn next to Theorem 7, so we assume F = F ◦ is transitive on Ω, μ is abelian, and nS > 1. As usual, we also assume the Extended Inductive Hypothesis. We begin in section 13.1, where we reduce to the case where AutF (O(τ ))Ω = Sym(Ω). Set Θ = ZS . As |Ω(z)| = 1 for z ∈ Θ, also AutF (WS )Θ = Sym(Θ), so it follows that either WS is the natural module for Sym(Θ) or its image modulo eΘ . In particular Hypothesis 1.3.4 is satisfied, so F = B(τ ) by an earlier remark. Then this result is extended further in Theorem 13.5.3. Finally the proof of Theorem 7 (modulo the Extended Inductive Hypothesis) is completed in section 13.6; see in particular Remark 13.6.7. We’ve proven Theorems 2 through 5, established Theorems 6 and 7 under the Extended Inductive Hypothesis, and reduced the proof of Theorem 1 to the case where |Ω(z)| = 1 for z ∈ ZS and μ is nonabelian. This is the case covered by Theorem 8. After some basic lemmas are established in section 15.1, we prove in Theorem 15.2.2 that Dη is a conjugacy class of 3-transpositions of μ. Therefore by 1.3.2, μ is Weyl(Φ) where Φ is of type An , Dn , or En . As discussed earlier, Hypothesis 1.3.4 is satisfied unless Φ is Dn with n = 4, E7 or E8 , so, except possibly in these three cases, F = B(τ ). In the three exceptional cases, a similar result holds by other means. When Φ is of type An the result is extended further in Theorem 15.6.3. Finally the various possibilities for Φ are treated in sections 15.3-15.8. Formal proofs of Theorem 1 and Theorems 6, 7, and 8 appear in sections 16.1 and 16.2, respectively, although these proofs only gather together the various threads that we’ve been discussing here.

Basics and examples

CHAPTER 2

Some basic results Chapter 2 contains a variety of basic results, almost all on saturated fusion systems F on a p-group S, often with p = 2. For example Theorem 2.1.9 establishes an analogue for fusion systems of a theorem of D. Holt for groups, that determines the finite permutation groups containing an involution fixing a unique point. Asolvable component of F is a subnormal subsystem of F isomorphic to the 2fusion system of A4 or SL2 (3). Section 2.2 develops a theory of solvable components similar to the theory in [Asc11] on components. Section 2.3 studies intrinsic SL2 [m]-components and solvable components C of Fz = CF (z) for involutions z in S: that is z ∈ C. The results in this section are used in Theorem 2.4.1 to show that intrinsic SL2 [m] components and solvable components give rise to quaternion fusion packets. This makes possible the derivation of the Main Theorem from Theorems 1 and 2. Sections 2.5 and 2.6 establish basic results on quaternion fusion packets τ = (F, Ω). In particular, given K ∈ Ω and z = z(K), we find in Theorem 2.6.11 that K is Sylow in a subnormal subsystem OK of Fz , and that the normal closure O(z) of Ω(z) in Fz is the central product of the OJ for J ∈ Ω(z). In Theorem 2.7.3 we find that if F is the product SO 2 (F) of S with O 2 (F) the central product of SL2 [m]-components, then F is uniquely determined by S and the set Ω of Sylow groups of the components. Finally in section 2.8 we consider a finite group G with a strongly embedded subgroup, and an involution a in G such that G = aG . Then Theorem 2.8.15 determines all F2 G-modules V = [V, a]G with m([V, a]) = 2. This result is used in various places to study essential subgroups of S. 2.1. Preliminary lemmas In this section F is a fusion system on a finite p-group S. Definition 2.1.1. For Δ a collection of subgroups of S, define Φ(S, Δ) = O p (Aut(D)) : D ∈ Δ S to be the fusion system on S generated by these automizers. Set D(F) = {D ≤ S : O p (AutF (D)) = 1}. Lemma 2.1.2. Let F be saturated and U a set of representatives in F f for the orbits of F on F rc . Set Δ = D(F) ∩ U and assume for each D ∈ Δ that O p (AutF (D)) = O p (Aut(D)). Then F = Φ(S, Δ). Proof. As F is saturated, F = O p (AutF (D)) : D ∈ Δ by 1.3.1 in [Asc19]. Then the lemma follows.  21

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Example 2.1.3. Let S be semidihedral of order 2m. Let Δ(S) = {D ≤ S : O 2 (Aut(D)) = 1}. Then Δ(S) = Δ1 ∪ Δ2 , where Δ1 is the set of 4-subgroups of S and Δ2 is the set of Q8 -subgroups of S. Moreover S is transitive on Δi for i = 1, 2; for each D ∈ Δ1 , O 2 (Aut(D)) ∼ = Z3 ; for each D ∈ Δ2 , O 2 (Aut(D)) ∼ = A4 ; and 2 O (Aut(D) is the unique subgroup of Aut(D) containing Inn(D) which is not a 2-group. (1) Let L2 (q 2 )(1) be the extension of L2 (q 2 ) with semidihedral Sylow 2-groups. Let L2 [2m](1) be the fusion system of L ∼ = L2 (q 2 )(1) where q is an odd prime power 4 with (q − 1)2 = 2m. We may take S ∈ Syl2 (L) and observe that D(FS (L)) = Δ1 . Hence by 2.1.2, L2 [2m](1) = Φ(S, Δ1 ). (2) Next let SL2 (q)(2) be the extension of degree 2 of SL2 (q) with semidihedral Sylow 2-subgroups, where q is an odd prime power. If (q 2 − 1)2 = m we write SL2 [m](2) for FS (L) where L ∼ = SL2 (q)(2) and S ∈ Syl2 (L). Observe D(FS (L)) = (2) Δ2 , so by 2.1.2, SL2 [m] = Φ(S, Δ2 ). (3) Finally let L ∼ = L3 (q) for q an odd prime power with q ≡ −1 mod 4. If 2 (q − 1)2 = m we may choose S ∈ Syl2 (L) and we write L− 3 [m] for FS (L). Observe that D(FS (L)) = Δ(S), so by 2.1.2, L− 3 [m] = Φ(S, Δ(S)). Lemma 2.1.4. Assume F is saturated and T is strongly closed in S with respect to F. Then E = O p (AutF (D)) : D ≤ T T , regarded as a fusion system on T , is F-invariant. Proof. Observe first that, by definition of E, O p (AutE (D)) = O p (AutF (D)) for each D ≤ T . We next claim that AutF (T ) ≤ Aut(E). Namely for α ∈ AutF (T ), Eα∗

= O p (AutF (D)) : D ≤ T α∗ = O p (AutF (D))α∗ : D ≤ T = O p (AutF (Dα) : Dα ≤ T = E,

establishing the claim. By the claim and 3.3 in [Asc08a], to complete the proof it suffices to show that E is F-Frattini; that is for P ≤ T and α ∈ homF (P, S), α = ϕφ for some ϕ ∈ AutF (T ) and φ ∈ homE (P ϕ, T ). By Alperin’s Fusion Theorem and 5.5.3 in [Asc08a], F = AutF (U ) : U ∈ U , for any set U of representatives in F f for the orbits of F on F rc . We choose U so that for each U ∈ U, UT = U ∩ T is in F f . Now it suffices to take α to be the restriction to P of some β ∈ AutF (U ) for some U ∈ U with P ≤ U . Further we choose P and α to be a counter example with n = |T : P | minimal. If P = T then α ∈ AutF (T ), contrary to the choice of P ; therefore n > 1. Further by minimality of n, P = UT , so as T is strongly closed, P  NF (U ) and hence α ∈ AutF (P ). Let V = P CS (P ), G a model for NNF (V ) (P ), c : G → Aut(P ) the conjugation map, Q = NT (P ), and Y = O p (G)CG (P ). Thus Gc = AutF (P ), so Y c = O p (AutF (P )). As α ∈ AutF (P ), α = gc for some g ∈ G. As Q is strongly closed in NS (P ) with respect to G, QY  G and G = NG (Q)Y by a Frattini argument. Thus g = xy for some x ∈ NG (Q) and y ∈ Y . Let cx ∈ Aut(Q) be conjugation by x. As P < Q, cx = ϕγ for some ϕ ∈ AutF (T ) and γ ∈ homE (Qϕ, T ) by minimality of n. Also yc ∈ Y c = O p (AutF (P )) ≤ AutE (P ), so α = gc = ϕφ,  where φ = γ|P ϕ · (yc) ∈ homE (P ϕ, T ), contrary to the choice of α.

2.1.

PRELIMINARY LEMMAS

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Lemma 2.1.5. Assume F is saturated, T ≤ S, and the following hold: (a) There is a normal subgroup Q of F of index p in T . (b) Let G be a model for NF (QCS (Q)) and H = T G . Assume T ∈ Sylp (H) and O p (AutG (Q)) = O p (AutH (Q)) = 1. (c) There exists W ≤ T such that W is weakly closed in S with respect to F, W ≤ Q, and AutNF (W ) (T ) is a p-group. Then (1) Q is the greatest member of D(F) contained in T . (2) D = FT (H)  F. (3) D = O p (AutF (Q)) T as a fusion system on T . Proof. Let P ≤ T and φ ∈ homF (P, S). As Q  F by (a), φ extends to ϕ ∈ homF (P Q, S) acting on Q. If P ≤ Q then P φ = P ϕ ≤ Q. If P ≤ Q then as |T : Q| = p by (a), P Q = T . Similarly as W ≤ Q, T = W Q. Thus P φ ≤ (P Q)ϕ = T ϕ = W ϕQϕ = W Q = T as W is weakly closed by (c). Hence T is strongly closed in S. Next if P ∈ D(F) we may choose φ ∈ AutF (P ) to be a p -element. But then if P ≤ Q then ϕ ∈ AutF (T ) is a p-element as W ϕ = W since W is weakly closed, so ϕ ∈ AutNF (W ) (T ), which is a p-group by (c). Thus each member of D(F) contained in T is contained in Q. Further by (b), Q ∈ D(F), so (1) holds. Let E = O p (AutF (D)) : D ≤ T , regarded as a fusion system on T . By 2.1.4, E is F-invariant. By (1), E = B, where B = O p (AutF (Q)) T . On the other hand B = D by (b), so E = B = D and (3) holds. By definition in (2), D = FT (H), while T ∈ Sylp (H) by (b), so D is saturated. Finally by (c), AutF (T ) is a p-group, so F, D satisfies condition (N1) and hence (2) follows.  Lemma 2.1.6. Assume F is saturated and let I = {1, . . . , n}, let {Ci : i ∈ I} be an F-invariant set of components of F, let Qi ∈ Syl(Ci ), and let Q Sylow be in recursively, given 1 ≤ i < n C = C1 · · · Cn . Define F1 = NF (C1 ) and, proceeding  define Fi+1 = NFi (Ci+1 ). Set E = Fn and T = i∈I NS (Qi ). Then E is a normal subsystem of F on T . Proof. Let A = NF (Q), S = {Qi : i ∈ I}, and form B = A(S)∗ as in Theorem 2.8.11 in [Asc19]. By that theorem, B  A with T Sylow in B. Set Y = NE (Q). By 2.8.11.2 in [Asc19], for each V ≤ T , AutB (V ) = A(V ), and by construction of A(V ) in section 2.8 of [Asc19], each α ∈ A(V ) extends to α ˆ ∈ Aut(V Q) acting on each Qi , so A(V ) ≤ AutY (V ). Therefore B ≤ Y. Conversely Y ≤ B by 2.8.11.4 in [Asc19]. Thus Y = B  A. As {Ci : i ∈ I} is F-invariant, C  F by 7.4 in [Asc11]. Hence E  F by Theorem 1.5.1 in [Asc19].  Lemma 2.1.7. Assume F is saturated and constrained and let G be a model for F. Then the map H → FS∩H (H) is a bijection between the set of subnormal subgroups of G and the set of subnormal subsystems of F. Proof. Let S, T be the set of subnormal subgroups of G, subnormal subsystems of F, respectively, and for H ∈ S set Hϕ = FS∩H (H). Let H ∈ S. As F ∗ (G) = Op (G), also F ∗ (H) = Op (H). Let M be the normal closure of H in G; then either H = G or M is proper in G. We prove the lemma by induction on the order of G, so pick a counter example G of minimal order.

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If H = G then Hϕ = F ∈ T . If H = G then as F ∗ (M ) = Op (M ), Hϕ is subnormal in M ϕ by minimality of G. By 1.2.8.4, M ϕ ∈ T , so Hϕ ∈ T . Thus ϕ:S →T. Similarly if H ∈ T − {F} then H is subnormal in a proper normal subsystem N of F and N = N ϕ for some proper normal subgroup N of G by 1.2.8.4. Then by minimality of G, H = Hϕ for some H subnormal in N , so H ∈ S. That is ϕ is a surjection, so it remains to show that ϕ is an injection. Assume K ∈ S with Hϕ = Kϕ but H = K. Then some J ∈ {H, K} is proper in G, so its normal closure A is proper in G. As Hϕ = Kϕ, T = S ∩ K = S ∩ H ≤ A,   so B = T G ≤ A and hence also L = O p (H) and O p (K) are contained in A.   Now by 1.2.8.4, Lϕ = O p (Hϕ) = O p (Kϕ), so working in A, by minimality of G,  L = O p (K). Set Q = Op (L), Z = CG (Q), and P = Op (Z). As F ∗ (NG (Q)) = Op (NG (Q)), if Q is not normal in G, then working in NG (Q) we conclude H = K by minimality of G. Therefore Q  G, so also Z  G. As F ∗ (H) = Q = F ∗ (K), CH (Q) ≤ Q ≥ CK (Q) so p -elements of H and K are faithful on Q. As Hϕ = Kϕ, AutH (Q) = AutK (Q), so HZ = KZ. Hence by minimality of G, G = HZ = KZ. Let x be a p -element of H and W = [P, x]. Then also W = [W, x] and W ≤ M . Then by induction on the subnormal length of H, W ≤ H, so W ≤ Q. Therefore [P, x] ≤ P ∩ Q ≤ Z(Z), so as F ∗ (Z) = P we have [Z, x] ≤ CZ (P/Z(Z)) ≤ P , and then [Z, x] ≤ P ∩ Q ≤ Q. Therefore [Z, O p (H)] ≤ Q. Then as G = HZ, we conclude that O p (H)Q  G. By symmetry also O p (K)Q  G. By 7.8.2 in [Asc11], O p (H)ϕ = O p (Hϕ) = O p (K)ϕ, so if G = O p (H)O p (K)Q then O p (H) = O p (K) by minimality of G. But then H = O p (H)L = K, a contradiction. Therefore G = O p (H)O p (K)Q, so L  G. Hence H = LO p (H)Q  G and by symmetry also K  G. Therefore H = K by 1.2.8.4. This contradicts the choice of H and K and completes the proof.  Lemma 2.1.8. Assume F is saturated and E is a subsystem of F on S. Assume z ∈ Z(S) such that (a) CF (z) ≤ E and (b) for each s ∈ z F there is α ∈ homE (CS (s), S) with sα = z. Then E = F. Proof. By Alperin’s Fusion Theorem it suffices to show that if R ∈ F f rc then AutF (R) = AutE (R). As z ∈ Z(S), z ∈ Z(R). Let β ∈ AutF (R) and s = zβ. By (b), there is α ∈ homE (CS (s), S) with sα = z. As z ∈ Z(R), R ≤ CS (s) so ζ = βα ∈ homF (R, S) with zζ = z, so by (a), ζ ∈ homE (R, S). Then β = ζα−1 ∈  AutE (R), completing the proof. Theorem 2.1.9. (Holt’s Theorem for Fusion Systems) Assume F is saturated and E is a saturated subsystem of F on S. Assume z ∈ Z(S) such that (1) z F = z E and (2) CF (z) ≤ E. Then E = F. Proof. As z ∈ Z(S), z is fully centralized in E. Let s ∈ z F ; then by (1), s ∈ z E , so as z is fully centralized in E and E is saturated, there is α ∈ AE (s) with sα = z. Now the theorem follows from 2.1.8. 

2.1.

PRELIMINARY LEMMAS

25

Lemma 2.1.10. Assume F is saturated, p = 2, and S is dihedral or semidihedral. Then either (1) F has one class of involutions, or (2) F = O 2 (F). Proof. Assume F has more than one class of involutions. Then there is a subgroup T of S of index 2 that is strongly closed in S with respect to F. Then by I.7.5 in [AKO11], O 2 (F) = F.  Lemma 2.1.11. Assume F is saturated and let Q = Z(F) and F + = F/Q. Assume C + is a component of F + and let C be the preimage in F of C + . Then C = QO 2 (C) and O 2 (C) is a component of F. Proof. By 8.10 in [Asc08a], C is a subnormal subsystem of F. Let T be Sylow in C, B = O 2 (C), and R Sylow in B. By 7.7.3 in [Asc11], B  C, so B is subnormal in F. Then, replacing F by C, we may assume F = C. Hence F = QB by 7.15 in [Asc11]. Therefore F + = B + . Let Θ : F → F + be the natural map and ϕ = Θ|R : R → S + = R+ . Then ϕ : B → B + = B/(Q ∩ R) is the natural map with B + simple and Q ∩ R ≤ Z(B). Therefore ker(ϕ) = Z(B), so B is quasisimple, completing the proof.  Lemma 2.1.12. Assume F is a saturated and U ≤ Q ≤ S with U and Q normal in F. Define R = CF (Q/U ) to be the fusion system on R = CS (Q/U ) such that for P ≤ R, homR (P, R) consists of those φ ∈ homF (P, S) such that for some ˆ ≤ U . Then R is weakly normal in F, so φˆ ∈ homF (P Q, S) extending φ, [Q, φ]  O p (R)  F. Proof. First R is strongly closed in S with respect to F: if x ∈ R and α ∈ homF (x, S) then [Q, x] ≤ U so [Q, xα] = [Qα, xα] = [Q, x]α ≤ U α = U . Next by I.5.5 in [AKO11], R is a saturated fusion system on R. Claim R is F-invariant. For x ∈ R and φ ∈ homR (x, S) [x, φ] ∈ U , so [x, φ] = u for some u ∈ U . Then for α ∈ homF (X, S) with x ∈ X ≤ R and xφ ∈ X, we have (xα)(φα∗ ) = (xα)(α−1 φα) = (xφ)α = (xu)α = xαuα ∈ xαU, so φα∗ ∈ homR (x, R). This proves the claim. By the claim and paragraph two, R is weakly normal in F. Then by Craven’s  Theorem 1.2.6, O p (R)  F.  Definition 2.1.13. Assume F is saturated and p = 2. Following Definition 2.19 in [AO16], define the reduction of F to be red(F) = (CF (O2 (F))/Z(O2 (F)))∞ , where E ∞ is the last term in the Puig series of E, defined in Definition 2.18 in [AO16]: the series obtained by alternating applications of the operators O 2 and  O2 . Lemma 2.1.14. Assume F is saturated and p = 2. Assume (a) F0 is a quasisimple normal subsystem of F on S0 with Z(F0 ) = U ∼ = E4 . (b) AutF (U ) = GL(U ). (c) CS (S0 ) = CS (F0 ). (d) O 2 (CAutF (S0 ) (U )) = 1.

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(e) There exists T strongly closed in S with respect to F containing S0 , such that CT (S0 ) = U and [U, T ] = 1. Let E = [T ]F be the normal closure of T in F. Then (1) T is Sylow in E. (2) F ∗ (E) = F0 . (3) CS (S0 ) = U , so NE (S0 ) has a model G. (4) CG (U ) = CT (U ) and G/CT (U ) ∼ = S3 . (5) red(E) = F0 /U . Proof. Assume otherwise and choose F to be a counter example of minimal order. Observe that as [T, U ] = 1, it follows from (b) that AutE (U ) = GL(U ). As F0 is quasisimple, F0 is contained in the normal closure of S0 in F. Thus if D is a normal subsystem of F containing T , then as S0 ≤ T we have F0 ≤ D, and it follows that D satisfies conditions (a)-(e). Therefore as each of our conclusions is a statement about E, rather than F, we conclude that F = D by minimality of F. Then as E  F, we have F = E. Let C = E(CF (F0 )) and Sc Sylow in C. Then [T, Sc ] ≤ CT (S0 ) = U by (e), so by 9.5.2 in [Asc11], T centralizes C. Therefore E = [T ]F ≤ CF (C), so as F = E by paragraph one, we have F = CF (C), so F0 = E(F). Set Q = O2 (F) and R = CF (Q/U ). Now [T, Q] ≤ T ∩ Q ≤ CT (S0 ) = U by (e).   Therefore T ≤ CS (Q/U ), so E ≤ O 2 (R) by 2.1.12. Hence F = O 2 (R) as F = E by paragraph one. Let Q0 = S0 Q. Then S1 = CS (Q0 ) ≤ CS (S0 ) = CS (F0 ) by (c). As F0 = E(F) we have Q = F ∗ (CF (F0 )), so S1 = CS1 (Q) ≤ Q ≤ Q0 . That is CS (Q0 ) ≤ Q0 , so NF (Q0 ) has a model M . Set H = T M and L = O 2 (CM (U )).  As F = O 2 (R), L centralizes Q. By (d), L centralizes S0 . Thus L centralizes Q0 = QS0 , so as F ∗ (M ) = Q0 , we have L = 1. Therefore CM (U ) = O2 (M ) and M/O2 (M ) = GL(U ) ∼ = S3 by (b). Hence M = Y S with |S : O2 (M )| = 2 and Y ∼ = Z3 . Similarly as T is strongly closed in S with respect to M , we have H = Y T with H/CT (U ) ∼ = S3 . As CM (U ) = O2 (M ), we conclude that CF (F0 ) = Q and then that CF (Q) = CS (Q)F0 . Now by Lynd transfer [Lyn14], O 2 (CF (Q)) = F0 , so that red(F) = F0 /U . Hence (5) holds as F = E. As T is strongly closed in S, T O 2 (F)  F, so F = T O 2 (F) by paragraph one. Claim T = S. Assume otherwise; then T ≤ X < S with |S : X| = 2. Now S0 ≤ T is Sylow in F0  F, so NF (T ) ≤ NF (S0 Q) = NF (Q0 ), so NF (T ) = FS (NM (T )) = FS (S). Therefore F/T ∼ = NF (T )/T ∼ = S/T . Therefore hyp(F) ≤ T , 2 so as F = T O (F), we have T Sylow in F, establishing the claim. We’ve shown T = S; in particular (1) holds. As CS (S0 ) = CT (S0 ) = U , we conclude that Q = U , so F ∗ (F) = F0 Q = F0 , proving (2) and (3). Also Q0 = S0 , so H = M = G with CT (U ) = O2 (G), and (4) holds. Thus the proof is complete.  Lemma 2.1.15. Assume F is a saturated, Z ≤ Z(F), and set F + = F/Z. Assume {Fi : i ∈ I} is set of subsystems Fi of F on Si with Z ≤ Si such that F + = Fi+ : i ∈ I . Then F = Fi : i ∈ I . Proof. Let R ∈ F f rc and E = Fi : i ∈ I . As R ∈ F f rc , we have Z ≤ CS (R) ≤ R. By Alperin’s Fusion Theorem 1.2.2, it suffices to show that for each α ∈ AutF (R), α ∈ AutE (R). As F + = Fi+ : i ∈ I , there exists n and Fij -maps αj , 1 ≤ j ≤ n, with α+ = α1+ · · · αn+ . Now R+ ≤ S1+ and Z ≤ R ∩ S1 , so R ≤ S1 .

2.2.

SOLVABLE COMPONENTS

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Also β = α1 · · · αn is an E-map with α+ = β + . Next R+ β + = R+ α+ = R+ , so Rβ = R. As α+ = β + , φ = αβ −1 ∈ AutF (R) centralizes R/Z. Then as φ also centralizes Z, we have φ ∈ Op (AutF (R)) = Inn(R) as R ∈ F r . Then as R ∈ F c , we have φ = cr for some r ∈ R, so α = cr β is an E-map, completing the proof.  Lemma 2.1.16. Let X be a set of order n = 2a and for b ≤ a let Ub = {Π1 , . . . , Πb } be a set of partitions of X such that Πk as 2k blocks of size n/2k b and Πk+1 refines Πk . Let Gb be the stabilizer in Sym(X) of Ub and GU b the inb duced action of Gb on Ub . Then GU b is a 2-group. b Proof. Indeed GU b is an iterated wreath product of Z2 ; this follows by induction on b. 

2.2. Solvable components In this section we assume F is a saturated fusion system on a finite 2-group S. Define a solvable component of F to be a subnormal subsystem C of F on T ≤ S, such that C ∼ = FT (G) for some group G isomorphic to SL2 (3) or L2 (3). Write Sol(F) for the set of solvable components of F, Comp(F) for the set of components of F, and set Comp+ (F) = Comp(F) ∪ Sol(F). Lemma 2.2.1. Assume C is a solvable component of F on T . Then (1) T ≤ O2 (F). (2) Let Γ = {Cφ∗ : φ ∈ homF (T, S)} be the set of conjugates of C in F. Then a central product X of the members of Γ is a normal subsystem of F. (3) Suppose T ∈ F f and let G be a model for NNF (VT ) (T ), where VT = T CS (T ). Then there is a normal subgroup H of G that is a model for C. Further [VT , H] = T and if T ∼ = Q8 then H centralizes CS (T ). (4) If T ∈ F f then C  NF (T ). Proof. As T  C   F, T is subnormal in F, so (1) follows from 9.10 in [Asc11]. Assume T ∈ F f and let C = F1  · · ·  Fn = F be a subnormal series. If n = 1 then (2) is trivial, so assume n > 1, and let E = Fn−1 be a system on E ≤ S. Let Δ = {Cψ ∗ : ψ ∈ homE (T, E)}. By induction on n, there is a central product Y of the members of Δ normal in E; let Y be Sylow in Y. For α ∈ AutF (E), α ∈ Aut(E) by 3.3 in [Asc08a], so by 7.3.4 in [Asc11], Y α  E. As AutE (Y ) is irreducible on Y /Φ(Y ), either Y = Y α or Y ∩ Y α ≤ Φ(Y ) ∩ Φ(Y α) ≤ Z(Y) ∩ Z(Yα). In the first case Y = Yα as AutY (Y ) is an AutF (Y )-invariant normal subgroup of AutE (Y ), and in the second Yα ≤ CE (Y) and YYα is the central product of Y and Yα by 9.2 in [Asc11]. Set  Ω= Δα. α∈AutF (E)

Then by repeated applications of 9.2 in [Asc11], there is a central product Z of the members of Ω that is normal in E by Theorem 3 in [Asc11]; let Z be Sylow in Z. By 7.4 in [Asc11], Z  F. Let φ ∈ homF (T, S). By 3.3 in [Asc08a], φ = βγ, where β ∈ AutF (Z) and γ ∈ homZ (T β, Z). Then β ∈ Aut(Z), so Cβ ∗ ∈ Ω

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and γ fixes Cβ ∗ . Thus T φ = T β and Cφ∗ = Cβ ∗ with β ∈ AutF (Z), so Z = X , completing the proof of (2). Let X be Sylow in X . Recall VX = XCS (X). Let G(X) be a model for NF (VX ). As X  F, there is a model L for X normal in G(X) by Theorem 2 in [Asc08a]. Choose G as in (3). As VX ≤ VT and X is strongly closed in S, X and VX are normal in N = NNF (VT ) (T ), so we may take G = NG(X) (VT ) ∩ NG(X) (T ). As C  X , a similar argument shows there is a model H for C normal in L, so H is subnormal in G(X). Indeed by (2), L is the central product of copies of H permuted by G(X), so as the Sylow group T of H is normal in G it follows that G ≤ NG(X) (H). Hence [VT , H] ≤ VT ∩ H = T and H acts on CVT (T ) = CS (T ). Further if T ∼ = Q8 then [H, CS (T )] ≤ CH (T ) = Z(H), so as H = O 2 (H), we conclude that H centralizes CS (T ), completing the proof of (3). Finally as C is the fusion system of a group, C is saturated, and as AutC (T ) = O 2 (Aut(T ))  Aut(T ), C is NF (T )-invariant. By (3), C satisfies (N1) in NF (T ), so (4) holds.  Lemma 2.2.2. (1) F contains a normal subsystem E which is a central product of the solvable components of F. (2) E ≤ CF (E(F)). (3) CF (E(F)) is constrained, and if G is a model for CF (E(F)) then the normal model for E in G is the central product of copies of SL2 (3) and L2 (3). Proof. Let C be a solvable component of F on T ∈ F f . By 2.2.1.1, T ≤ O2 (F), so T ≤ D = CF (E(F)). Set N = NF (T ). By 2.2.1.4, C  N , so as T centralizes E(F), it follows from 10.3 in [Asc11] that E(F) = E(N ). Then as C is normal in N and constrained, CE(F) is the central product of C and E(F) by 9.14 in [Asc11]. Thus C ≤ D, establishing (2). Define X as in 2.2.1.2 and let X, E be Sylow in X , E(F), respectively. As C ≤ D  F, X ≤ D. By 3.6.2 in [Asc08a], X is D-invariant, and by 2.2.1.2, X is saturated. By 2.2.1.3, α ∈ AutC (T ) of order 3 lifts to β ∈ AutF (T ECS (T E)) with [β, ECS (T E)] ≤ Z(T ). As E is strongly closed in S, it follows that [E, β] ≤ E ∩ Z(T ) ≤ Z(E(F)), so [E, β] = 1 by 9.5.1 in [Asc11]. Then β|T CD (T ) ∈ AutD (T CD (T )), so X satisfies (N1) in D. Therefore X  D, so that C   D. By 9.12.3 in [Asc11], D is constrained, so there exists a model G for D. As X  D, there is a model M for X normal in G by Theorem 1 in [Asc08a]. If C1 ∈ Sol(F) is not a conjugate of C then arguing as in the proof of 2.2.1.2, C1 centralizes X , so (3) follows. Then (3) implies (1).  Lemma 2.2.3. Let U ∈ F f and C ∈ Comp+ (F) be contained in NF (U ). Then C ∈ Comp+ (NF (U )). Proof. By 2.2.1.2 there is a normal subsystem X of F which is a central product of the conjugates of C. By Theorem 5 in [Asc11] we can form the system U X . Let T be Sylow in C. Suppose C is a component of F. Claim U acts on T . If not [T, T u ] = 1 for some u ∈ U , so [T, u, T ] ≤ Z(C), impossible as C acts on U and [T, u] ≤ U . This establishes the claim, and by the claim, U ≤ NF (C). By 8.24 in [Asc11], C ≤ NX (U )  NF (U ). Then applying 1.1.10 in [Asc19] to C, NX (U ), NF (C) in the role of E, D, F, we conclude C  NX (U ). But now the lemma holds in this case, so we may assume that C is solvable.

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As C is solvable, U X is constrained, so it has a model G. By Theorem 1 in [Asc08a], there is a normal model L for X in G, and a normal model H for C in L. Then NL (U ) is a model for NX (U ). As C normalizes U , so does H, so H  NL (U ),  and hence C  NX (U ). Finally NX (U )  NF (U ) by 8.24 in [Asc11]. Lemma 2.2.4. Let C be a solvable component of F on T ≤ S, and U ≤ S. Let D ≤ F be the central product of the subsystems C u , u ∈ U and set E = U D. Then (1) E is a saturated constrained fusion system on some E ≤ S, D = O 2 (E), and C is a solvable component of E. Moreover E = FE (G) for some model G of E, and there exists H  K  G with D = FS∩K (K), C = FT (H), and G = KU . (2) Assume U is strongly closed in E with respect to E. Then either E is the central product of C and U , or T ≤ U . (3) Assume U is strongly closed in S with respect to F and W  U such that (*) for each R ≤ W and φ ∈ homF (R, U ) with R ∩ Rφ = 1, we have Rφ ≤ W . Assume CW (H) = 1. Then either T ≤ W or W  E and C centralizes W . (4) O2 (G) = T CO2 (G) (H), so O2 (F) = T CO2 (F ) (C). Proof. Let B be the product of all F-conjugates of C. By 2.2.1.2, B  F. Then by Theorem 5 in [Asc11], U B is a saturated fusion system with B = O 2 (U B), so replacing F by U B, we may assume F = U B. Then by 7.4 in [Asc11], D  F, so again by Theorem 5 in [Asc11], U D is saturated with D = O 2 (U D). As C is a solvable component, E is constrained, and as C  D  E, C is a solvable component of E. Thus (1) holds. We next prove (4), where we take U = O2 (F), so that also U = O2 (G). If u ∈ U − NU (T ) then [H, H u ] = 1, so [H, u] is not a 2-group, contradicting [H, u] ≤ U . Therefore T  H, U = G, so H  G by 2.2.1.4. Thus [H, U ] = T and as H is irreducible on T /Φ(T ), U = T CU (T ). Finally for A ∈ Syl3 (H), [CU (T ), A] ≤ Z(T ), so CU (T ) = Z(T )CU (H), and hence (4) holds. Thus it remains to prove (2) and (3), where we may assume F = E and U is strongly closed in S with respect to F. Set V = O2 (K) ∩ U ; as F = E and U is strongly closed in S, we have V  G. Then [H, V ] ≤ T ∩ V and as H is irreducible on T /Φ(T ), either [H, V ] = 1 or T ≤ V . We first prove (2), so assume F is a counter example to (2). Then T ≤ U , so [H, V ] = 1. Hence K = H G centralizes V . If [T, U ] = 1 then as H  NG (T ) by 2.2.1.4 and U is strongly closed in S with respect to G, either G = HU is a central product of H with U , or T = [H, U ] ≤ U , contrary to the choice of F as a counter example. Therefore [T, U ] = 1. On the other hand, [T, U ] ≤ O2 (K) ∩ U = V . Set G∗ = G/V , so that G∗ , K ∗ , U ∗ satisfy the hypothesis of G, K, V with [T ∗ , U ∗ ] = 1 and T ∗ ≤ U ∗ . Thus by an earlier case, G∗ is a central product of H ∗ with U ∗ , so U  G. But then as K = O 2 (K) centralizes V and U ∗ , K centralizes U , completing the proof of (2). So assume the hypothesis of (3). Thus CW (H) = 1, so by (*) applied to φ = ch for h ∈ H and W ∩ Q in the role of R: (**) H acts on W ∩ Q for each H-invariant 2-subgroup Q of S containing CW (H). Thus if W centralizes T then [H, W ] ≤ CH (T ) = Z(T ), so applying (**) to Q = W T we conclude that H acts on W . Then as [H, W ] ≤ Z(T ), either T ≤ W or W centralizes H, and in the latter case as W  U , also W  H, U = G. Therefore we may assume [T, W ] = 1. Therefore [H, U ] = 1, so that T ≤ U ≤ NG (W ) by (2). Thus [T, W ] ≤ O2 (K)∩W = WV . Further WV acts on T and hence

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also on H, so [H, WV ] ≤ T . Hence by (**) applied to Q = T CW (H)WV , H acts on CW (H)WV , and hence also on WV . Therefore either T ≤ WV or T ∩ WV ≤ Φ(T ) and [H, WV ] = 1, and we may assume the latter. As W  U , WV  U , so WV  H, U = G. ¯ ∈ Sol(G), ¯ so as W ¯ centralizes T¯, W ¯ acts on H. ¯ Hence ¯ = G/WV . Then H Set G ¯ W ¯ ] ≤ CH¯ (T¯ ) ≤ T¯, so H acts on T W . Then by (**) applied to Q = T W , H [H, acts on W , so [H, W ] ≤ T ∩ W ≤ Φ(T ), and hence [H, W ] = 1, completing the proof of (3).  Lemma 2.2.5. Assume z ∈ F f is an involution and C is a solvable component of Fz = CF (z) on T ∈ Fzf . Let α ∈ A(T ) and set T¯ = T α, F¯ = NF (T¯ ), etc. Let G be a model for NNF (VT¯ ) (T¯). Then ¯ z ) that is a model for C. (1) There is a normal subgroup H of CG (¯ ¯ z ), H]. (2) T = [CVT¯ (¯ (3) If T ∼ = Q8 then H centralizes CS (T¯) and C¯  F¯ . (4) Suppose R ≤ CS (T¯ ) is H¯ z -invariant. Then either H centralizes R or ¯ T ≤ R. Proof. As T ∈ Fzf , it follows from 2.2 in [Asc10] that z¯ = zα ∈ F¯ f and z ) is an isomorphism. Hence as C is a solvable component of Fz , α : NFz (T ) → CF¯ (¯ z ). Let Sz be Sylow in Fz and G0 a model we have C¯ is a solvable component of CF¯ (¯ ˇ : G0 → CG (z) is an isomorphism, for NNFz (T CSz (T )) (T ). Then by 2.2 in [Asc10], α z ), C¯ in the role of F, C. Indeed if so (1) and (2) follow from 2.2.1.3 applied to CF¯ (¯ ∼ ¯ z ) by 2.2.1.3, so by the Thompson A×B-Lemma, T = Q8 then H centralizes CS (T ¯ ¯ establishing (3). H centralizes CS (T¯ ). Then the proof of 2.2.1.4 shows C¯  F, ¯ Assume the setup of (4). Then as H is irreducible on T /Φ(T¯), it follows from z ) or T¯ ≤ R. In the latter case (4) holds, and in (2) that either H centralizes CR (¯ the former H centralizes R by the A × B-Lemma, completing the proof of (4).  Lemma 2.2.6. Let z ∈ F f be an involution, C ∈ Comp+ (CF (z)), T ∈ Syl(C), L ∈ Comp+ (F), and L ∈ Syl(L). Let E be the product of the F-conjugates of L and F Sylow in E. Then (1) T acts on L. (2) If L = Lz then either C centralizes L or C ≤ LLz is a morphic image of L. (3) If L = Lz then one of the following hold: (i) T centralizes L, or (ii) T ≤ L, or (iii) C ∼ = FT (L2 (3)) centralizes Z(E) and, setting E¯ = E/Z(E), there exist three conjugates L = L1 , L2 , L3 of L under AutF (F ) such that z acts on each Li , Li is dihedral or quaternion, CL¯ i (z) = t¯i ∼ = Z2 for some ti ∈ Li , and each t ∈ T # is of the form t = ti tj for some 1 ≤ i < j ≤ 3. In particular each involution in T centralizes some F conjugate of L, and some involution in T centralizes L. (4) If z ∈ C then either C and L commute or T ≤ L. (5) If z acts on L and C ∼ = SL2 [m] then either C and L commute or T ≤ L. Proof. We begin a series of reductions which will eventually establish (1)-(3). Then at the end of the proof we prove (4) and (5). (a) The lemma holds if C is a component, so we may assume C is a solvable component.

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For assume C is a component. Then by Theorem 7 in [Asc11], C ≤ E(F). Thus if L is a solvable component then C centralizes L by parts (1) and (2) of 2.2.2, so the lemma holds trivially. Hence we may assume L is a component. Therefore as T ≤ E ∈ Syl(E(F)) and E acts on L, (1) holds. Next by 10.11 in [Asc11], there exists a component K of F, such that either K = Kz and C ≤ KKz , or K = Kz and C ≤ K. Hence if L ∈ / {K, Kz } then C centralizes L, so we may take L = K. But then the lemma holds, so (a) is established. (b) We may assume z does not centralize L. For if z centralizes L then L ∈ Comp+ (CF (z)) by 2.2.3, so L = C or C centralizes L by 2.2.2. In either case the lemma holds, so (b) is established. Let E be the product of the F-conjugates of L, so that E is a normal subsystem of F, which is the central product of the conjugates of L. (c) If L = Lz then CLLz (z) = WCZ(LLz ) (z), where W ∈ Comp+ (CF (z)) is a morphic image of L centralized by C. Further NT z (L) is of index 2 in T z and centralizes L. By 10.11 and 8.24 in [Asc11], CLLz (z) = WCZ(LLz ) (z), where W is a morphic image of L. In particular if L is quasisimple then so is W, while if L is solvable then W is L2 (3) or SL2 (3). In any event as L is subnormal in F, W is subnormal in CF (z) by 8.24 in [Asc11]. Therefore W ∈ Comp+ (CF (z)). Then by 2.2.2, either C centralizes W or C = W, and the lemma holds in the latter case, so C centralizes W. Now 2.3.3.2 in [Asc19] completes the proof of (c). (d) T does not centralize L. Assume otherwise; then (1) and (3i) hold, so it remains to prove (2), and hence we may assume L = Lz , and adopt the notation in (c). We may assume T ∈ Fzf ¯L ¯ z¯ ≤ CS (T¯ ). Let Y be the product and adopt the notation in 2.2.5. Then R = L ¯ ¯ of the set Γ of F -conjugates of L and Y Sylow in Y. Thus Y ≤ CS (T¯ ) ≤ VT¯ and Y is strongly closed in CS (T¯ ) with respect to F¯ . Pick h of order 3 in H and set ¯ so γ = ch ∈ AutF¯ (Y ). Then γ permutes Γ and centralizes a Sylow group W of W, z¯ ¯ ¯ ¯ γ acts on {L, L }. Then as γ is of order 3, γ acts on L. Next as γ centralizes W , ¯ γ] ≤ Z(L), ¯ so γ centralizes L¯ by 9.5.1 in [Asc11]. But then C centralizes L, so [L, (2) holds, completing the proof of (d). (e) We may take Z(E) = 1. Let γ ∈ AutC (T ) be of order 3, and set E = Z(E) and U = CE (z). Let X be the product of the CF (z)-conjugates of C, so that X is a normal subsystem of CF (z) on some X ≤ S. Then U X  Fz , so γ extends to a 3-element β ∈ AutFz (XU ) with [XU, β] = T . Then [U, β] = [U, β, β] ≤ [XU, β] ∩ U = T ∩ U , so as β is irreducible on T /Φ(T ), either [U, β] = 1 or T = [U, β] ≤ U . However in the latter case, E ≤ CF (T ), contrary to (d), so we may assume [U, β] = 1. But β extends to a 3-element δ ∈ AutF (XE) as E  F, and then by the Thompson A × B-Lemma, [E, δ] = 1. Therefore T = [T, δ] also centralizes E, so C ≤ CF (E). By 2.2.3, C ∈ Sol(CF (U z )) and L ∈ Comp+ (CF (U )). Then by induction on the order of F, U ≤ Z(F). Now we form the factor system F + = F/U as in 8.10 of [Asc08a]. By 8.10 in [Asc08a], the image C + of C is a solvable component of

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CF + (z + ), and L+ ∈ Comp+ (F + ). Thus if E = 1, then by induction on the order of F, the lemma holds for C + and L+ , and hence also for C and L. Note that, adopting the notation of 2.2.6.3.iii, T = t1 t2 , t1 t3 , so T is abelian, and hence is E4 rather than Q8 . This completes the proof of (e). (f) We may assume z acts on L. Assume otherwise; then we are in the setup of (c), and we write W for the Sylow group of W. If T acts on L then T centralizes L by (c), contrary to (d). Therefore by (c), T0 = NT (L) is of index 2 in T . Set Z = Ω1 (Z(L)) and let W0 be the inverse image in W of Z under the projection map, which is an isomorphism by (e). Set V = T W0 and V¯ = V /Φ(T ). ¯ 0 = V¯0 Then Z acts faithfully on V¯ as the group of transvections with axis T¯0 W ¯ and centers in W0 . Let γ ∈ AutC (T ) be of order 3. By 2.2.1.3 applied in CF (z), γ lifts to a 3-element γˆ on VT with [CS (T ), γˆ ] ≤ Z(T ), so γˆ induces β ∈ AutF (V ). As γ is irreducible on T /Φ(T ) and β acts on the strongly closed subgroup V1 = V ∩ F , where F is Sylow in the product E, either T ≤ V1 or T ∩ V1 ≤ Φ(T ), and as T0 = T , it must be the latter. Thus V1 = W0 or W0 Φ(T ), and as V1 is strongly closed in V , V1 is β-invariant. Thus [β, W0 ] ≤ V1 ∩ T ≤ Φ(T ), so β centralizes W0 . Let Y = AutZ (V ), β ≤ AutF (V ). Now Z centralizes V1 and V /V1 , so AutZ (V ) ≤ O2 (Y ), and hence β lifts to δ ∈ AutF (Q) with ZV ≤ Q ≤ S. Next ZδW0 = (ZW0 )δ ≤ Q ∩ F with Zδ ≤ (Q ∩ L)δ ≤ K ∈ LF and (Zδ)z ≤ (Q ∩ Lz )δ ≤ Kz . Then W0 ≤ (ZZ z )δ ≤ KKz , so {K, Kz } = {L, Lz }, and hence {Q ∩ L, Q ∩ Lz }γ = {Q ∩ L, Q ∩ Lz }. Therefore as δ is of odd order, δ acts on Q ∩ Lz , so T = [T, δ] also acts on Q ∩ Lz , contradicting T = T0 . This completes the proof of (f), and establishes (2). (g) We may assume T ≤ F , so that T acts on L. Let P consist of those P ≤ Lz with z ∈ P ∈ F f , CS (z) ≤ NS (P ), and C a solvable component of FP = NF (P ). For example z ∈ P. As the Sylow group F of E is strongly closed in S with respect to F, U = NF (P ) is strongly closed in NS (P ) with respect to FP . Applying 2.2.4.2 to FP , C, and U , we conclude that either T ≤ U or U centralizes C. In the former case (g) holds, so we may assume the latter. When P = z , this shows that Lz = CL (z) centralizes C. ¯ f . Here ¯ ∈ F f and z ∈ NF (L) We can choose a conjugate L¯ of L such that L ¯ if L is quasisimple then NF (L) is defined in Definition 2.2.1 of [Asc19], while if ¯ = NF (L). ¯ We may assume T ≤ F , so applying the L is solvable we define NF (L) ¯ ¯ z . Then, using argument above to L in place of L, we conclude that C centralizes L ¯ ¯ 2.1.5 in [Asc19], C ≤ NF (Lz ) ≤ NF (L). Then by induction on the order of F, either L¯  F, or our lemma holds in NF (L), and hence also in F. Therefore we may assume that L¯  F, so E = L¯ = L. Hence U = NL (P ), and as CS (z) ≤ NS (P ), also CS (z) ≤ NS (U ). Let V = U z and α ∈ A(V ); as CS (z) ≤ NS (V ) and z ∈ F f , also zα ∈ F f , so replacing z, V by zα, V α, we may assume that V ∈ F f . As CF (V ) ≤ Fz and C is a solvable component of Fz , C is also a solvable component of CF (V ), and hence also of FV . But now choosing P maximal in P, we conclude that P = V , so L = U centralizes C, contrary to (d). This finally completes the proof of (g). By (g), part (1) of our lemma holds, so it remains to prove (3).

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(h) We may assume that CL (C) = 1. Suppose not, and set W = CL (z). Then CW (C) = 1. Further suppose R ≤ L and γ ∈ homF (R, F ) with L ∩ Rγ = 1. By 3.3 in [Asc08a], γ = ϕφ with ϕ ∈ AutF (F ) and φ ∈ homE (Rϕ, F ). Thus as 1 = L ∩ Rγ and (Rγ ∩ L)φ−1 ≤ L, we have 1 = Rϕ ∩ L, so as ϕ permutes LF , Rϕ ≤ L, and then Rγ = Rϕφ ≤ L. Thus condition (*) of 2.2.4.3 is satisfied by Fz , CF (z), W in the roles of F, U , W , so applying 2.2.4.3, we conclude either T ≤ W or C centralizes W . In the former case (3ii) holds, so we may assume the latter. Let π : T → L be the projection map and TL = T π. If TL = 1 then T centralizes L, contrary to (d), so TL = 1. As π is z-equivariant, TL ≤ W , so C centralizes TL . But then for α ∈ AutC (T ), π is α-equivariant, contradicting CT (α) = Φ(T ) and C centralizing TL . This establishes (h). Form CE (z) in F as in 8.24 in [Asc11]; by that lemma, CE (z)  Fz . By (f), L  z E, so we can form CL (z) in z E as in 8.24 in [Asc11], and CL (z)  z E. Therefore CL (z) is subnormal in Fz , so either CL (z) contains a component D, or CL (z) is constrained. In the former case D centralizes C by 2.2.2, contrary to (h). Thus R = O2 (CL (z)) = F ∗ (CL (z)) ≤ O2 (Fz ) = U as CL (z) is subnormal in Fz . Now the hypothesis of 2.2.4 are satisfied with CF (z) in the role of F, so we can form the group G of 2.2.4.1, and the normal model H for C in G. By 2.2.4.4, U = T CU (H), so by (h), |R| ≤ |T : CT (H)| = 4. Indeed as T ≤ F ≤ NS (L), we can pick an involution r ∈ Z(T R) in R. Now [H, R] ≤ T , so if H ∼ = SL2 (3), then H acts on Z(r, T ) = r, Z(T ) , and hence H = O 2 (H) centralizes r, contrary to (h). Hence (i) C ∼ = FT (L2 (3)), so T ∼ = E4 . Let a be of order 3 in H. Then Ra = Rϕφ, where ϕ ∈ AutF (F ) and φ ∈ homE (Rϕ, F ). Thus Ra ≤ Lϕ. By (i), U = T × CU (H), so by (h) we have 1 = CR (H) = CR (a). Also [R, a] ≤ T . Hence as we may assume (3ii) does not hold, a does not act on R, so L = Lϕ∗ . Therefore RT ≥ RRa = R × Ra . 2 Suppose |R| = 4. Then R ∼ = E4 and Ra ≤ R × Ra = RT ≤ F . This is a i contradiction as Ra is contained in a component Ji ∈ LF and if Ji is a system 2 2 on Ji ≤ S, then as Ra ≤ LJ1 , we have J2 ∈ {L, J1 }, so Ra ≤ L ∩ RT = R or J1 ∩ RT = Ra . Therefore: (j) CL (z) = R = r ∼ = Z2 . It follows from (j) that Lz is dihedral or semidihedral, and hence as Z(L) = 1, that L is dihedral. Further RH ∼ = Z2 × A4 and for each involution t ∈ T , t = r1 r2 ∈ 2 L1 L2 for some ri ∈ {r, r a , r a }, with ri ∈ Li ∈ LF a component on Li ≤ F . In particular t centralizes L3 . Therefore (3iii) holds, completing the proof of (3). Finally we prove (4) and (5), so assume the hypothesis of (4) or (5). In the first case, z ∈ C, so z acts on L by (1), while in the second z acts on L by hypothesis. Thus in either case, z acts on L, so by (3), either T centralizes L or T ≤ L. In the latter case we are done, so assume the former. If z ∈ T then z centralizes L, so L ∈ Comp+ (Fz ) by 2.2.3, so as C ∈ Comp+ (Fz ), C and L commute by 2.2.2. Therefore (4) is established, so we may assume C ∼ = SL2 [m].

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If m = 8 then conjugating in Fz we may assume T ∈ Fzf , and adopt the ¯ so L and C commute. Finally take notation of 2.2.5. By 2.2.5.3, C¯ centralizes L, m > 8; then C is a component of Fz , so C ≤ E(F) by E-balance, and then as T  centralizes L, we have C = [T ]E(F ) ≤ CE(F ) (L), completing the proof of (5). Lemma 2.2.7. Let z ∈ F f be an involution, C ∈ Comp+ (CF (z)), and T ∈ Syl(C). Assume z ∈ T , C does not centralize E(F), and C is not a component of F. Then (1) there exists a unique component L of F such that C does not centralize L. Moreover T ≤ L ∈ Syl(L). (2) Let F0 = CF ∗ (F ) (L) and Y = CNF (L) (F0 ). Then C ≤ Y. (3) F ∗ (Y) = LZ(Y). (4) Set Y + = Y/Z(Y). Then Y + is almost simple with F ∗ (Y + ) = L+ . (5) C + ∈ Comp+ (CY + (z + )) and z + = 1. Proof. Let L be the set of components of F not centralized by z. As C does not centralize E(F) and z ∈ T it follows from 2.2.3 applied to z in the role of U and 2.2.2 that L = ∅. Let L ∈ L and L ∈ Syl(L). As z ∈ T be conclude from 2.2.6.4 that T ≤ L. Hence for each L = K ∈ Comp(F), z ∈ T ≤ L ≤ CS (K), so by 2.2.3 and 2.2.2, C centralizes K. This completes the proof of (1). Next F0 is the product of O2 (F) with the components of F distinct from L. As z ∈ T ≤ CS (O2 (F)) and C and Comp(F) − {L} are in Comp+ (Fz ) by 2.2.3, it follows that C centralizes F0 , using 2.2.1.3 when C is solvable. Therefore C ≤ Y, establishing (2). By Theorem 6 in [Asc11], CF (F ∗ (F)) = Z(F ∗ (F)), so (3) follows. As L is quasisimple with Z(L) ≤ Z(Y), L+ is simple. By (3), L+ = F ∗ (Y + ), so (4) holds. If z ∈ Z(Y) then L is a component of Fz by 2.2.3. Then as T ≤ L and C ∈ / Comp(F). Comp+ (Fz ), it follows that C = L, contradicting the hypothesis that C ∈ Therefore z + = 1. Let E be the product of the Fz -conjugates of C. Then E ≤ CY (z) with E  Fz , so by 1.1.10 in [Asc19], E  CY (z) if C is quasisimple, and at least E is weakly normal if C is solvable. Then in the latter case condition (N1) holds by 2.2.1.3, so in any event C is subnormal in CY (z), and then C + is subnormal in CY (z)+ by 8.9.2 in [Asc08a]. Then as CY (z)+  CY + (z + ), we conclude that C + ∈ Comp+ (CY + (z + )), completing the proof of (5). 

2.3. Intrinsic SL2 [m]-components In this section we assume F is a saturated fusion system on a finite 2-group S. Often we also assume the following hypothesis: Hypothesis 2.3.1. For i = 1, 2, zi ∈ F f is an involution, such that the set Δi of components and solvable components of Fi = CF (zi ) on quaternion subgroups of S containing zi of order m is nonempty.

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35

Notation 2.3.2. When assuming Hypothesis 2.3.1, we adopt the following notation: e is an involution in S and φ ∈ A(e) with eφ = z2 . Pick Ci ∈ Δi , a system on Ti , set z = z1 , T = T1 , C = C1 , D = C2 , and D = T2 . Write Xi for the central product of the members of Δi , which is normal in Fi by Theorem 6 in [Asc11], 2.2.1, and 2.2.2. Lemma 2.3.3. Assume Hypothesis 2.3.1. Then CS (T ) centralizes C. Proof. Each R ∈ C f rc (distinct from T if m > 8) is isomorphic to Q8 , so the lemma follows from 2.3.5 in [Asc19].  Lemma 2.3.4. Assume ui is an involution in F for i = 1, 2 with u1 ∈ F f and u2 ∈ CF (u1 )f . Let α ∈ A(u2 ). Then u1 α ∈ CF (u2 α)f and α : CCF (u1 ) (u2 ) → CCF (u2 α) (u1 α) is an isomorphism. Proof. This is a special case of 2.2.2 in [Asc10].



Lemma 2.3.5. Assume Hypothesis 2.3.1 and CT (e) is quaternion. Then e centralizes C and zφ centralizes D. Proof. As z ∈ CT (e), e ∈ F1 , so replacing e by a member of eF1 ∩ F1f , we may assume e ∈ F1f . Set k = |CT (e)|. As Q = CT (e) is quaternion, either e centralizes T or m = 2k > 8, T = v, u with |v| = k, |u| = 4, v u = v −1 , and Q = w, u , where w = v 2 . Let R be the set of Q8 -subgroups of Q. For each R ∈ R, e acts on CR = O 2 (NC (R)) ∼ = SL2 [8]. By 2.3.3, e centralizes CR , so e centralizes Ce = CR : R ∈ R ∼ = SL2 [k]. Observe Ce ∈ Comp+ (CF1 (e)). Then using 2.3.4, B = Ce φ∗ ∈ Comp+ (CF2 (zφ)), so by 2.2.6.4, either B and D commute or B ≤ D, so that z = e. In either case zφ centralizes D, and in the latter e centralizes C. Finally in the former case, applying the isomorphism φ−∗ : CF2 (zφ) → CF1 (e) of 2.3.4, Ce commutes with D0 = Dφ−∗ ∈ Comp+ (CF1 (e)), so applying 2.2.6.4 to F1 , e, D0 , C in the roles of F, z, C, L, we conclude that C and D0 commute, so again e centralizes C.  Theorem 2.3.6. Assume Hypothesis 2.3.1 and adopt Notation 2.3.2. Then (1) If a ∈ CS (e) − e with [a, z] = 1 and aφ ∈ D, then a ∈ NS (T ). (2) If zφ ∈ NS (D) then e ∈ NS (T ). We establish Theorem 2.3.6 via a series of reductions. Thus in the remainder of this section we assume we are working in a counterexample to the theorem. In (1) as aφ ∈ D we have e ∈ a ≤ CS (z), while in (2) as zφ ∈ NS (D), again e ∈ CS (z). Thus in any event [e, z] = 1. Hence, replacing e by a member of eF1 ∩ F1f , we may assume e ∈ F1f . ¯ = aα. Let ϕ ∈ A(¯ a) For example let α ∈ AF1 (e), set e¯ = eα, and in (1) set a ¯ϕ = (aφ)ζ and aφ ∈ D, so with a ¯ϕ = z2 ; then ζ = φ−1 αϕ is an F2 -map. In (1), a ¯ an F2 -conjugate of D. Then if a as D ∈ Comp+ (F2 ), (aφ)ζ ∈ D ¯ ∈ NS (T1 ) for each ¯α−1 , so (1) holds. Similarly in (2) if F1 -conjugate T1 of T , the same holds for a = a ¯ and hence if e¯ ∈ NS (T1 ) for each T1 , then zφ ∈ NS (D) then zϕ = (zφ)ζ ∈ NS (D), the same holds for e. Set x = zφ. Lemma 2.3.7. x acts on D.

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Proof. If we are working in a counterexample to (1), then x centralizes aφ ∈ D, and as aφ = z2 , D is the unique member of Comp+ (F2 ) containing aφ, so x acts on D. On the other hand if we are in a counterexample to (2), then x ∈ NS (D) by hypothesis, so x acts on D.  Lemma 2.3.8. e acts on C. Proof. Assume otherwise. Suppose first that C is a component. Then by 10.11 in [Asc11]: (*) CCC e (e) = z × L, where L ∼ = C/z is a component of CF1 (e). On the other hand if C is solvable, then we apply 2.2.4 to F1 in the role of F and U = e , and, working in the group G of 2.2.4.1, CHH e (e) = z J where J∼ = A4 , so again (*) holds, except that this time L is a solvable component. In any case write L for a Sylow group of L. Now by (*) and 2.3.4, Lφ∗ ∈ Comp+ (CF2 (x)). Notice L is dihedral, so as D is quaternion, Lφ ≤ D. By 2.3.7, x acts on D, so we can appeal to 2.2.6.3 to conclude that each involution in Lφ centralizes some member of DF , and for some involution l ∈ L, lφ centralizes D. Suppose we are working in a counterexample to (1). Then e ∈ a2 . By the previous paragraph, aφ centralizes some involution lφ ∈ Lφ, so a centralizes l ∈ L ≤ CC e . Therefore a acts on {C, C e }, so e ∈ a2 acts on C, a contradiction. Therefore no such a exists, and we are working in a counterexample to (2). Suppose x centralizes d ∈ D − z2 . Let α ∈ A(x) with xα = z and β ∈ A(z2 α) with z2 αβ = z2 . Then ξ = αβ is an F2 -map, so ξ permutes Δ2 and hence dξ ∈ T2 Sylow in some member of Δ2 . Then as we’ve shown that e acts on C under the hypotheses of (1), and as zβ centralizes dξ ∈ T2 , we conclude that z2 α acts on each member of Δ1 . Hence as ζ = φα is an F1 -map and eζ = z2 α, we conclude that e acts on C, a contradiction. Thus we’ve shown that x centralizes no element of D − z2 . Then as x acts on D but centralizes no element of D − z2 , x is conjugate to xz2 in Dx = ND (x, z2 ). Then as lφ centralizes D, lφxz2 is conjugate to lφx under Dx , so (lze)F = (lz)F . Let Sz = CS (z) and S˜z = Sz /z . Let Q be a Q8 -subgroup of T , and L0 ˜ under the projection of L ˜ on T˜ and A = z, e L0 . the inverse image in L of Q 2 ∼ ∼ Then we may choose l ∈ L0 , A = E16 , O (AutL (L)) = Z3 lifts to Λ of order 3 in ˜ 0 ), so Λ acts on AutQ (A), and hence AutF (A), and E4 ∼ = AutQ (A) = CAutCCe (A) (L Σ = ΛAutQ (A) ≤ AutF (A) is isomorphic to A4 . Observe lze ∈ eΣ and lz ∈ lΣ . Therefore lF = (lz)F = (lze)F = eF = z2F , and indeed by 2.2.6.1, Lφ acts on D and hence also on Dx , so [Lφ, Dx ] ≤ z2 and hence Dx acts on Aφ. Thus l ∈ eθ , where θ = Σ, AutDA (A) ≤ AutF (A), and DA = AutDx (Aφ)φ−∗ . Hence there exists ψ ∈ A(l) with lψ = z2 . Suppose C0 ∈ Δ1 − {C, C e } is a system on T0 . As l ∈ CC e , l centralizes C0 . Hence by 2.2.6.4, zψ centralizes X2 . Now there exists ϕ ∈ A(zψ) with zψϕ = z, and by a second application of 2.2.6.4, z2 ϕ = lψϕ centralizes X1 . However ψϕ ∈ homF1 (l, CS (z)) and X1  F1 , so as lψϕ centralizes X1 , so does l, a contradiction. Therefore Δ1 = {C, C e }. Then as X1  F1 , Σ = O 2 (AutX1 (A))  AutF1 (A). Let u = lφ, μ ∈ A(u) with uμ = z2 , and η ∈ A(z2 μ) with z2 μη = z2 . As u centralizes D, the same argument says uμη = z2 η centralizes X2 , and then also u centralizes X2 . By symmetry, L0 φ centralizes X2 , and hence also Dx , so DA is

2.3.

INTRINSIC SL2 [m]-COMPONENTS

37

generated by a transvection on A with axis e, L0 and center e . Observe that Σ is indecomposible on A with chief series 1 < z < z L0 < A, so θ is irreducible on A. As A4 ∼ = Σ  Cθ (z), it follows from the list in [McL69] of irreducible subgroups of GL(A) containing a transvection that θ ∼ = O4− (2) or Sp4 (2) is the isometry group of a quadratic form or symplectic form f on A, in the respective case, and by an earlier observation, l ∈ eθ . In either case, z is a singular vector in A, e ∈ / z ⊥ = z, L0 , and l ∈ L0 = z, e ⊥ , with l nonsingular if f is quadratic. As l ∈ eθ , le is singular and hence in z θ if f is quadratic; then as l ∈ z, e ⊥ , we conclude that (z, l) ∈ (le, e)θ . Thus there exists an F-map ρ : (z, l) → ((le)φ, z2 ) and ν ∈ A((le)φ) with (le)φν = z. Further L0 , e φ centralizes X2 , so by 2.2.6.4, z2 ν = lρν centralizes X1 . However zρν = z, so as X1  F1 , l centralizes X1 , a contradiction. Thus the proof of 2.3.8 is at last complete.  Remark 2.3.9. Note that 2.3.8 establishes part (2) of Theorem 2.3.6, so it remains to prove part (1) of the theorem. Therefore in the remainder of the proof of Theorem 2.3.6, we work in a counter example to part (1) of the theorem. Thus we assume a ∈ CS (e, z ) is of order at least 4 with aφ ∈ D but C a = C. As aφ ∈ D and eφ = z2 is the involution in D, e is the involution in a . By 2.3.8, e acts on C, so we may choose a so that a2 acts on C. Recall we have chosen e ∈ F1f , so by 2.3.4, x ∈ F2f . Lemma 2.3.10. CT (e) and CD (x) are cyclic. Proof. Let (s, Y, Y) = (e, T, C) or (x, D, D). From 2.3.7 and 2.3.8, s acts on Y . Suppose CY (s) is noncyclic. Then by 2.3.5, s centralizes Y. Hence x centralizes D if s = x, while if s = e then Cφ∗ is a component of CF2 (x) by 2.3.4, so applying 2.2.6.4 to F2 , x, Cφ∗ , D in the role of F, z, C, L, we conclude that x centralizes D. Hence in any event, x centralizes D. Pick ϕ ∈ A(x) with xϕ = z. By 2.2.3, D ∈ Comp+ (CF2 (x)), so by 2.3.4, E = Dϕ∗ ∈ Comp+ (CF1 (z2 ϕ)). Hence by 2.2.6.4, either E and X1 commute or we may take Dϕ ≤ T . In either case aφϕ acts on T , so as φϕ ∈ homF1 (a, CS (z)) and  X1  F1 , a acts on C, contrary to the choice of a. Lemma 2.3.11. Let E = z, e . Then AutF (E) ∼ = S3 . Proof. By 2.3.7 and 2.3.8, E acts on T and Eφ acts on D. Then by 2.3.10,  [E, NT (E)] = z and [Eφ, ND (Eφ)] = z2 . Lemma 2.3.12. |CT (e)| > 2. Proof. Assume |CT (e)| = 2. Then T e is semidihedral. From 2.3.9, a2 acts on T and e ∈ a2 , so as CT (e) = z , a2 = e. Further [T, T a ] = 1 and T ∩ T a = z , with T a e semidihedral. Let v ∈ T be of order 4 inverted by e. Then u = [a, v] = v −a v ∈ T T a , so u is an involution centralizing e. Then as 2 e = a2 inverts v, ua = v −a v a = uz, so [u, a] = z. Set y = uφ and b = aφ. Then [y, b] = x ∈ [y, D] ≤ X2 , so x induces an inner automorphism on D. By 2.3.9, x ∈ F2f , so as x ∈ X2  F2 , we have x ∈ X2f . Let d be the projection of x on D; as X2 is a central product of the members of Δ2 , we have d ∈ Df . Let B = CD (x) and observe that B = CD (d). Claim B is cyclic of index 2 in D. If m = 8 this is clear, so take m > 8. Then D is a component of F2 so D is transitive on the members of D of order 4. Then as d ∈ Df , the claim follows.

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Next x = bb−y ∈ BB y with B y ≤ Dy and Dy Sylow in Dy . By 2.3.11 there is ρ ∈ AutF (Eφ) interchanging z2 and x. Indeed as z2 ∈ F f and x ∈ F2f , Eφ ∈ F2f , and then by 2.3.11, Eφ ∈ F f . Thus we can extend ρ to χ ∈ AutF (CS (Eφ)). Let ϕ ∈ A(x) with xϕ = z. Now B, y ≤ CS (Eφ), so B, y χϕ ≤ CS (Eφ)ϕ ≤ CS (x)ϕ ≤ S, with z2 χϕ = xϕ = z. Therefore as B ≤ D, Bχϕ ≤ C  ∈ Δ1 , so eφϕ ∈ Eφϕ ≤ (BB y )χϕ ≤ X1 . Then as X1  F1 , e ∈ X1 , contradicting  CT (e) = z . Lemma 2.3.13. e induces an inner automorphism on C and |T : CT (e)| = 2. Proof. Assume otherwise and let S1 be Sylow in X1 . By 2.3.9, e ∈ X1f . Then as X1  F1 , we have |CS1 (e)| ≥ |CS1 (f )| for each f ∈ eX1 . Hence as X1 is a central product of the members of Δ1 , we have |CT (s)| ≥ |CT (g)| for each g ∈ eC . Let V = v be an e-invariant cyclic subgroup of T of index 2, u ∈ T − V , and π ∈ V of order 4. Suppose first that CV (e) = z ; then by 2.3.12 we may take CT (e) = u . Set c = ue; then c centralizes Q = π, u ∼ = Q8 , so that lemma holds if m = 8, so we may assume m > 8. Then C ∼ = SL2 [m] is a component of Fz , so (as in the proof of 2.3.3) β ∈ AutC (Q) of order 3 with uβ = π extends to γ centralizing CS (Q). Hence eγ = c · uβ = cπ centralizes π. Therefore k = |V : CV (eγ)| ≤ 2. If k = 1 then |CT (eγ)| = m/2 > 4 = |CT (e)|, contrary to paragraph one. Thus k = 2, where CT (eγ) = v 2 , vu , for the same contradiction. Therefore |CV (e)| > 2, so j = |V : CV (e)| ≤ 2. If j = 1 then eπ centralizes T , so the lemma holds; thus we may assume j = 2 and hence m > 8 and v e = vz. But now e, z = Ω1 (V e ) is u-invariant, so eu = ez = ev , and hence e centralizes uv, contrary to 2.3.10.  By 2.3.13, e = ππ a y for some π of order 4 in T and y ∈ CS (T T a ). Let t ∈ T − CT (e), u = tta , w = uφ, and b = aφ. Then u is an involution centralizing e. Next z = t−1 tπ = t−1 te = t−1 ta t−a te = u−1 ua = [u, a], so x = [uφ, aφ] = [w, b] ∈ DDw . As z = e it follows that D = Dw . By 2.3.10 and 2.3.13, v = CT (e) is of order m/2. Then t1 = tv ∈ T −CT (e), so by symmetry between t and t1 , Dw1 = D, where w1 = (t1 ta )φ. Then as x ∈ DDw , / [v, a] , so if Dw = D w1 , so vφ acts on D. Then x centralizes [v, a]φ ∈ D and a ∈ [v, a] = e then x centralizes a Q8 -subgroup of D, contrary to 2.3.10. Hence [v, a] = e so |v| = 4 and hence m = 8 and v inverts a. Now x = [w, b] = b−w b and y = vφ inverts b. Let d ∈ D − b ; then [x, d] = z2 , y so d = dbz2 . Then dz2 = dx = (dbz2 )y = dbz2 b−1 z2 = d, for our final contradiction. This completes the proof of Theorem 2.3.6.

2.5.

BASIC RESULTS ON FUSION PACKETS

39

2.4. A sufficient condition for quaternion fusion packets In this section we assume F is a saturated fusion system on a finite 2-group S. We prove: Theorem 2.4.1. Assume z is an involution in S, z ∈ F f , and C is a component or solvable component of CF (z) on T such that z ∈ T and T is quaternion. Set Ω = T F . Then τ = (F, Ω) is a quaternion fusion packet. Proof. By construction, Ω is F-invariant, and condition (QFP1) is satisfied. For P ∈ Ω, let z(P ) be the involution in P , so that: (a) For P ∈ Ω, z(P ) is the unique involution in P . Set Fz = CF (z), E = T Fz , and Δ = Δz = {Cα∗ : α ∈ AutFz (E)}. Now C is a component or solvable component of Fz , so by Theorem 6 in [Asc11] and 2.2.2: (b) There exists a central product E of the members of Δ normal in Fz . Suppose D ∈ Ω − {T } with 1 = T ∩ D. Then by (a), z = z(T ) = z(D). By definition of Ω, there is α ∈ homF (T, S) with T α = D. Then zα = z(T )α = z(D) = z, so α ∈ homFz (T, CS (z)). Then by (b), T ∩ D = z and 1 = [T, D]. Hence (QPF2) is satisfied. We next establish (QFP3). Assume D ∈ Ω, d ∈ D − z(D) , and μ ∈ homF (d, S) with dμ ∈ CS (z). Replacing D by a suitable conjugate, we may assume z(D) ∈ F f . Set e = z(D)μ, a = dμ, and let φ ∈ A(e) with eφ = z(D). Thus Hypothesis 2.3.1 is satisfied with z(D) = z2 . As z(D)μφ = eφ = z(D) and d ∈ D ≤ D ∈ Δz2 , aφ = dμφ ∈ D ≤ D  ∈ Δz2 . Hence by Theorem 2.3.6, a ∈ NS (T ), so that (QFP3) is satisfied. Finally assume D ∈ Ω with z = z(D), v ∈ T , and α ∈ homFz (v, S). By (b), E  Fz , so by 3.3 in [Asc08a], α = ϕφ with ϕ ∈ AutFz (E), and φ ∈ homE (vϕ, E). By definition, Cϕ∗ ∈ Δ, and of course vϕ ∈ T ϕ. Further as φ is a E-map and vϕ ∈ T ϕ, also vα = vϕφ ∈ T ϕ. Finally by (b), either D = T ϕ or [D, T ϕ] = 1, so vα ∈ D or [vα, D] = 1. Thus (QFP4) is satisfied, completing the proof of the theorem.  2.5. Basic results on fusion packets In this section we assume that τ = (F, Ω) is a quaternion fusion packet, and we adopt the following notation: Notation 2.5.1. Set ZS = ZS (τ ) = {z(K) : K ∈ Ω} and  Z = Z(τ ) = zF . z∈ZS

For z ∈ ZS , set Ω(z) = {J ∈ Ω : z ∈ J} and Fz = CF (z). Lemma 2.5.2. Suppose E is a saturated subsystem of F on T ≤ S, Γ0 is a nonempty subset of Ω, and for some divisor k of m, Γ = {K ∩ T : K ∈ Γ0 } satisfies: (a) For each J ∈ Γ, J is nonabelian of order k. (b) ΓE = Γ. Then the pair (E, Γ) is a quaternion fusion packet.

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Proof. The pair satisfies condition (QFP1) by (a). The remaining conditions inherit from τ to the pair.  Lemma 2.5.3. Assume z ∈ ZS ∩ F f . Then (1) (Fz , Ω(z)) is a quaternion fusion packet. (2) If z centralizes Ω then (Fz , Ω) is a quaternion fusion packet. Proof. As z ∈ F f , Fz is a saturated fusion system by 1.2.1. Thus the hypotheses of 2.5.2 are satisfied by Fz in the role of E, m in the role of k, and Ω(z) in the role of Γ, so (1) follows. A similar argument establishes (2).  Definition 2.5.4. Define SL2 [8] to be the 2-fusion system of SL2 (3). Suppose m > 8 and let K ∈ Ω. Then K has two classes RiK , i = 1, 2, of subgroups Ri ∼ = Q8 . Define SL2 [m] to be the fusion system on K generated by O 2 (Aut(Ri )), i = 1, 2. Define SL12 [m/2] to be the fusion system on K generated by O 2 (Aut(R1 )). Lemma 2.5.5. Let K ∈ Ω and assume E is a saturated fusion system on K. Then: (1) Either E is isomorphic to FK (K) or SL2 [m], or m > 8 and E ∼ = SL12 [m/2]. ∼ ∼ (2) Assume K is Sylow in G = SL2 (q). Then FK (G) = SL2 [m], for m = (q 2 − 1)2 . (3) Assume K is Sylow in G such that F ∗ (G) ∼ = SL2 (q) and |G : F ∗ (G)| = 2. 1 2 ∼ Then FK (G) = SL2 [m/2], where m = 2(q − 1)2 . (4) FK (K), SL2 [m], and SL12 [m/2] are saturated. Proof. From 1.2.4, E is determined by its Alperin data : A set Q of representatives for the orbits of E on E f rc , and the systems AutE (Q) for Q ∈ Q. However for X ≤ K, either Aut(X) is a 2-group or X ∼ = S4 . We conclude = Q8 and Aut(X) ∼ that E has one of the following sets of Alperin data: (i) AutE (X) = AutK (X) for each X ≤ K and E f rc = {K}. (ii) m = 8, AutE (K) = O 2 (Aut(K)) ∼ = A4 , and E f rc = {K}. ∼ (iii) m > 8, AutE (Ri ) = Aut(Ri ) = S4 for i = 1, 2, and E f rc = {K, R1K , R2K }. (iv) For i = 1 or 2, AutE (Ri ) = Aut(Ri ) ∼ = S4 , AutE (R3−i ) = AutK (R3−i ) ∼ = f rc K = {K, Ri }. D8 , and E It follows that E is isomorphic to FK (K), SL2 [8], SL2 [m], SL12 [m/2] in cases (i)-(iv), respectively, since those 2-fusion systems have the respective Alperin data. This establishes (1). Then (1) implies (2) and (3), since the 2-fusion systems for the groups in (2) and (3) are saturated and have the Alperin data of the indicated fusion system. Finally (2) and (3) imply (4).  Definition 2.5.6. We define a fusion system OK on K ∈ Ω which is a subsystem of F. Set z = z(K). If m = 8, set OK = FK (K). If m > 8, define QK to be the set of Q8 subgroups of K and define OK = O 2 (AutF (Q)) : Q ∈ QK K , regarded as a fusion system on K. Lemma 2.5.7. Let K ∈ Ω. Then: (1) OK is isomorphic to FK (K), SL2 [m], or SL12 [m/2]. (2) OK is saturated.  (3) OK = O 2 (OK ). (4) AutF (K) ≤ Aut(OK ). Proof. From 2.5.6, OK is FK (K) or one of the three fusion systems defined in 2.5.4, so (1) follows. Then (2) follows from 2.5.5.4. If m = 8 then OK =

2.6.

THE CASE z ∈ Z(F ).

41

FK (K), so (3) holds in that case. If m > 8 then for R ≤ Q, O 2 (AutF (R)) ≤  O 2 (AutOK (R)), so again (3) holds. Part (4) follows as AutF (K) permutes QK ; cf. 1.2.4 for example.  Lemma 2.5.8. Let m > 8. Then: (1) L2 [m/2] is simple. (2) SL2 [m] is quasisimple. Proof. Suppose E is a fusion system on T isomorphic to L2 [m/2] or SL2 [m]. In the first case, E has one class of involutions and T is generated by involutions, so there is no proper nontrivial subgroup of T strongly closed in T with respect to E. Then as NF (T ) = T , E is simple by part (4) of Theorem 8 in [Asc11]. So assume E is SL2 [m]. Then Z = Z(E) is of order 2, and E/Z ∼ = L2 [m/2] is simple, so it remains to show E = O 2 (E). But O 2 (E)  E and Z is the unique subgroup of T of order 2, so Z ≤ O 2 (E), and hence E = O 2 (E) by 7.15 in [Asc11].  2.6. The case z ∈ Z(F). In this section we assume: Hypothesis 2.6.1. τ = (F, Ω) is a quaternion fusion packet and z ∈ ZS with z ∈ Z(F). Set O = O(z) = Ω(z) and F + = F/z . Let K ∈ Ω(z). Lemma 2.6.2. Suppose V ≤ K with |V | > 2 and let φ ∈ homF (V, S). Then (1) K is the unique member of Ω containing V . (2) V φ ≤ J for some J ∈ Ω(z). (3) J ∈ K F . (4) If K ∈ F f then V F ∩ F f ∩ K = ∅. Proof. Part (1) follows from (QFP2). Set X = V φ and let α ∈ A(X). Assume (2) fails. Suppose first that U = X ∩ J = z for some J ∈ Ω(z). As the lemma fails, U < X, so there exists v ∈ V with x = vφ ∈ / J. By (QFP4), x centralizes J. Hence x centralizes U , so U is cyclic and U ≤ x ≤ CS (J), contradicting U ≤ J. Therefore for each J ∈ Ω(z), J ∩ X = z , so X ≤ CS (O) by (QFP4). As ΩF = Ω, as z ∈ Z(F), and as O ≤ CS (X), Ω(z)α = Ω(z), so Oα = O and hence / Ω(z), so V = K and hence V < NK (V ). By Xα ≤ CS (O). In particular X ∈ 1.2.5.2 there is χ ∈ AutF (V ) such that χφα extends to ϕ ∈ homF (NK (V ), S). Then Y = NK (V )ϕ acts on V χφα = Xα, and by induction on the order of V , Y ≤ J ∈ Ω(z), contradicting X ≤ CS (O) ≤ CS (J). This completes the proof of (2). Assume (3) fails and choose a counter example with n = |K : V | minimal; then n > 1. By (2), Xα ≤ L ∈ Ω and then by (1) and (2), NJ (X)α ≤ L, so L ∈ J F by minimality of n. Thus replacing φ by φα, we may assume X ∈ F f . Hence by 1.2.5.2 there is χ ∈ AutF (V ) such that χφ extends to β ∈ A(V ). Then NS (V )β ≤ J by (1) and (2), so (3) follows from the minimality of n. Assume K ∈ F f and choose φ ∈ A(V ). By (3) there is γ ∈ A(J) with Jγ = K.  By (1), NS (X) ≤ NS (J), so Xγ ∈ F f . Thus (4) holds.

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Lemma 2.6.3. Assume K ∈ F f , z < P ≤ K, and ψ ∈ homF (P, K). Then ψ = ϕφ for some ϕ ∈ AutF (K) and φ ∈ homOK (P ϕ, K). Proof. Assume otherwise and pick a counter example P with n = |K : P | minimal. Then ψ ∈ / AutF (K), so n > 1. By 2.6.2.4 there is a member of P F ∩ F f contained in K. Suppose 2.6.3 holds whenever P ψ ∈ F f . Let β ∈ A(P ψ) with P ψβ ≤ K. Then ψβ = ϕ1 φ1 and β = ϕ2 φ2 , so −1 ψ = (ψβ)β −1 = ϕ1 φ1 φ−1 2 ϕ2 = ϕφ, −∗ where ϕ = ϕ1 ϕ−1 ∈ AutF (K) and φ = (φ1 φ−1 ∈ homOK (P ϕ, K), using 2 2 )ϕ2 2.5.7.4. Thus it suffices to assume P ψ ∈ F f . Let R = NK (P ); as n > 1, P < R. By 1.2.5.2, there exists χ ∈ AutF (P ) such that χψ extends to ρ ∈ homF (R, S). As P ψ ≤ K it follows from 2.6.2 that Rρ ≤ K. Thus as P < R it follows from minimality of n that ρ = ϕ3 φ3 . Next χψ = ψζ where ζ = χψ ∗ ∈ AutF (P ψ), so ψ = ϕ3 (φ3 ζ −1 ). If P is not isomorphic to Q8 then Aut(P ) is a 2-group, so ζ = cs for some s ∈ NS (P ψ) as P ψ ∈ F f . Thus as NS (P ψ) ≤ NS (K) by (QFP2), ζ = cs ∈ AutF (K), so ψ = ϕ4 φ4 with ϕ4 = ϕ3 ζ −1 ∈ AutF (K) and φ4 = φ3 ζ −∗ ∈ homOK (P ϕ4 , K), using 2.5.7.4. On the other hand if P ∼ = Q8 then ζ ∈ AutOK (P ψ), so ψ = ϕ3 φ4 with  φ4 = φ3 ζ −1 ∈ homOK (P ϕ3 , K). This completes the proof of the lemma. f , and α ∈ A(V ). Lemma 2.6.4. Assume K ∈ F f , z < V ≤ K with V ∈ OK ∗ Then NOK (V )α  NF (V α).

Proof. By 2.6.2, X = V α is contained in a unique member J of Ω, and by 2.6.2.4 we may take J = K. Set N = NF (X), T = NS (X), L = NK (X), and O = NOK (X). By 2.6.3, α|NK (V ) = ϕ1 φ1 for some ϕ1 ∈ AutF (K) and φ1 ∈ f f homOK (V ϕ1 , K). Therefore as V ∈ OK , X ∈ OK and NOK (V )α∗ = O is saturated by 1.2.1. Let Y ≤ L and φ ∈ homN (Y, T ). Then φ lifts to ϕ ∈ homN (XY, T ), and by 2.6.2, (XY )ϕ = X(Y ϕ) ≤ K and hence Y φ ≤ L. Therefore L is strongly closed in T with respect to N . By 2.5.2, ρ = (N , L) is a quaternion fusion packet. Define OL as in 2.5.6 with respect to ρ. By definition, OL ≤ O. Further if P ∈ Of and ψ ∈ AutO (P ), then either ψ ∈ AutL (P ) or P ∈ QL and ψ ∈ O 2 (AutO (P )) ≤ AutOL (P ), so indeed OL = O. As L is strongly closed in T with respect to N , OL is N -invariant by  construction of OL . By 2.5.7.3, OL = O 2 (OL ), so by Craven’s Theorem 1.2.6, O  N.  + Lemma 2.6.5. If K ∈ F f then OK is tightly embedded in F + . + Proof. As K ∈ F f , K + ∈ F +f , and as OK is saturated by 2.5.7.2, OK is saturated. Thus we must verify conditions (T1)-(T3) of Definition 3.1.2 in [Asc19] + for the embedding of OK in F + . Condition (T1) holds by 2.6.4, condition (T2) holds by 2.6.3, and condition (T3) holds by 2.5.7.4. 

Lemma 2.6.6.  (1) O is the central product of the members of Ω(z). (2) O + = J∈Ω(z) J + is a direct product. (3) Assume m > 8 and let Q be a Q8 -subgroup of K and R = CS (Q). Suppose α ∈ AutF (Q) is of order 3. Then α × 1 ∈ AutF (QR), where α × 1 is the automorphism of QR which acts as α on Q and 1 on R.

2.6.

THE CASE z ∈ Z(F ).

43

Proof. For distinct K, J ∈ Ω(z), K and J commute by (QFP4) applied to each v ∈ K with φ the identity map. Note that K ∩ J = K ∈ Ω = z as z = Z(K). Thus (1) holds and then (1) and (QFP2) imply (2). Thus it remains to prove (3), so we may assume the setup of (3). First suppose Q ∈ F f and let V = QR. Then α extends to β ∈ AutF (V ) by 1.2.5.1. Next as m > 8, α is inverted by cs for some s ∈ NK (Q). Then γ = [β, cs ] ∈ CAut(V ) (R) as s centralizes R. Further γ|Q = [α, cs ] = α, so γ = α × 1. In general, let β ∈ A(Q) and Qβ = P . Then αβ ∗ = δ is of order 3 in AutF (P ), so δ×1 ∈ AutF (P R ) for P ≤ J ∈ Ω(z) and R = CS (P ) by the previous paragraph.  Then α × 1 = (δ × 1)β −∗ ∈ AutF (V ), completing the proof. Definition 2.6.7. We define a fusion system O = O(z) on O. By construction, O is a subsystem of F. If m = 8, set O = FO (O). If m > 8, define notation as in 2.6.6 and set O = α × 1 ∈ AutF (QCO (J) : Q ∈ QJ , J ∈ Ω(z), α ∈ AutF (Q) of order 3 O , regarded as a fusion system on O. Here QJ is the set of Q8 -subgroups of J. Lemma 2.6.8. O is the central product of the subsystems OK , for K ∈ Ω(z), so O is saturated. Proof. If m = 8 then O = FO (O) is the central product of the systems OK = FK (K), as O is the central product of the members of Ω(z) by 2.6.6.1. Thus we may take m > 8, where the lemma follows from 2.3 in [Asc11].  Lemma 2.6.9. If m > 8 then O  F. + is tightly embedded in F + and as m > 8, Φ(K + ) = 1. Proof. By 2.6.5, OK By 2.6.2, the set T of Notation 3.3.2 in [Asc19] is empty. By 2.5.7.3, OK =  + O 2 (OK ). Thus F + , OK satisfies Hypothesis 3.6.1 of [Asc19]. Therefore O+  F + by Theorem 3.6.8 in [Asc19], so O  F by 8.10 in [Asc08a]. 

Lemma 2.6.10. If m = 8 then O  F. Proof. Assume otherwise and pick a minimal counter example τ . By minimality of τ and 2.5.2: (a) F = [O]F . Let Y = {P ≤ O : z < P } and Y f = Y ∩ F f . Let Z be the preimage in O of Z(F + ) ∩ O + and F ! = F/Z. Set N = NF (O). As O + is a weakly closed abelian subgroup of S + , it follows that (b) N + controls fusion in O + . (c) For each P ∈ Y, P F ∩ Y f = ∅. (d) For P ∈ Y f either P  F or O  NF (P ). As O + is abelian, τP = (NF (P ), Ω(z)) is a quaternion fusion packet by 2.5.2, so (d) follows from the minimality of τ . (e) For each P ∈ Y with P  F, P ≤ Z. By (a) and 8.10 in [Asc08a], F + = [O + ]F + . As O + is abelian, O + ≤ CS + (P + ), so F + centralizes P + , proving (e). (f) O ! is tightly embedded in F ! . We must verify conditions (T1)-(T3) of 3.1.2 in [Asc19] for the pair F ! , O ! . Condition (T1) follows from (c)-(e), (T2) follows from (b), and (T3) is trivial.

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2. SOME BASIC RESULTS

Define the set P ∗ as in Definition 3.1.9 of [Asc19] with respect to the pair F , O ! . Let B be the set of preimages in S of members of P ∗ . Then for B ∈ B, B F ∩ Y = ∅ and B is maximal subject to that constraint. As τ is a counter example to our lemma, it follows from 3.1.7 in [Asc19] that (g) There exists B ∈ B − {O}. From 3.1.12 in [Asc19]: (h) B ! is a TI-subgroup in S ! and B ∩ O = Z. Let A be the preimage in O of CO! (B ! ). As a consequence of (h): (i) [O ! , B ! ] ≤ A! , |A! | ≥ |B ! |, and for each 1 = b! ∈ B ! , A! = CO! (b! ). Pick B ! of maximal order. (k) We may choose B so that A is conjugate to B in AutF (AB). As B F ∩ Y = ∅ by (c), it follows that there is α ∈ A(B) with Bα ∈ Y. As A ≤ NS (B), Aα ≤ NS (Bα) and by 3.1.14 in [Asc19], the member B0 of B containing Aα is not O, so |A! | ≤ |B0! | ≤ |B ! |. Thus |A! | = |B0! | = |B ! | by (i), so B0 = Aα and (Bα)! = CO! (Aα). Further O acts on (AB)α and for x ∈ O −C0 (Aα), (Aα)x = Aα, so φ = cx α−∗ ∈ AutF (AB) with Aφ = A. Thus replacing B by Aφ, (k) holds. (l) K ∩ Z = z , so K ! ∼ = E4 . For if V = K ∩ Z = z then V  F, so K  F by 2.6.5, contrary to the choice of τ . (m) K ∩ A = z . If V = K ∩ A = z then by (h) and (k) there is φ ∈ homF (V, S) with V ≤ O, contrary to 2.6.2. (n) For b ∈ B − Z, K b = K. If (n) fails then 1 = CK ! (b! ) ≤ A! by (i), contrary to (m). By minimality of τ : (o) F is transitive on Ω(z). Let Ω(z) = {K1 , . . . , Kn }. By 2.6.6.2: (p) O + = K1+ × · · · × Kn+ . + + + By (p) for each 1 = x+ ∈ O + , x+ = x+ 1 · · · xn where xi is the projection of x + + + + + on Ki . Set I = {1, . . . , n}, I(x ) = {i ∈ I : xi = 1}, and r(x ) = |I(x )|. (q) For each 1 = x+ ∈ Z + , r(x+ ) = n. Thus |Z + | ≤ 4. For if i ∈ / I(x+ ) then x centralizes Ki , and then by (o), x centralizes O, a + contradiction. Thus r(x+ ) = n and for y + ∈ Z + − x+ , yi+ = x+ i , so |Z | ≤ + |K | = 4. We are now in a position to obtain a contradiction, and hence to establish 2.6.10. By (n), B ! acts semiregularly on Ω(z), so k = |B ! | divides n. On the other hand it follows from (i) that O/A is semiregular on b! A! for 1 = b! ∈ B ! , so j = |O : A| ≤ |A! | = k. Thus k2 ≥ kj = |O ! |. But |O + | = 4n , so |O ! | ≥ 4n−1 by (q). Therefore 4n−1 ≤ k2 ≤ n2 , while n ≥ 2 by (n), so we conclude n = k = j = 2. This is impossible as j = |O : A| ≥ |K ! | = 4 by (m). This completes the proof of 2.6.10.  Theorem 2.6.11. (1) O  F. (2) For each K ∈ Ω(z), OK is subnormal in F. (3) AutF (O) permutes Ω(z) and {OK : K ∈ Ω(z)}. !

Proof. Part (1) follows from 2.6.9 and 2.6.10. By 2.6.8, OK  O, so (1) implies (2). The first statement in (3) follows as Ω(z)F = Ω(z), and the second  statement in (3) follows from the first and the definition of OK in 2.6.7.

2.7.

F = SOτ

45

Lemma 2.6.12. CS (O) centralizes O. Proof. If m = 8 then O = O, so the lemma is trivial. If m > 8 then the lemma follows from 2.6.6.3 and the definition of O in 2.6.7.  Lemma 2.6.13. Either (1) OK = FK (K) and K ≤ O2 (F), or (2) OK ∼ = SL2 [m] or SL12 [m/2] and O 2 (OK ) is a component or solvable component of F. Proof. This is a consequence of 2.6.11.1, 2.5.7.1, and 2.5.8.



Lemma 2.6.14. (1) CF (O) = CF (O). (2) If |Ω(z)| ≤ 2 and m > 8, or if AutF (O) is a 2-group, then F = SOCF (O). (3) Assume m = 8 and Oz  F with O Sylow in Oz ∼ = SL2 [8] ∗ SL2 [8]. Then CF (O) = CF (Oz ) and F = SOz CF (Oz ). Proof. By 2.6.12, S0 = CS (O) is Sylow in Y = CF (O) and Y ≤ X = CF (O). As Y ≤ X and S0 is Sylow in both X and Y, it follows that X f c ⊆ Y f c . For U ∈ Y f c , set A(U ) = O 2 (AutX (U ))AutS0 (U ); thus A(U ) = AutX (U ). By 6.12 in [Asc11], AutY (U ) = A(U ), so as X f c ⊆ Y f c , it follows from Alperin’s Fusion Theorem that X = AutX (U ) : U ∈ X f c ≤ AutY (U ) : U ∈ Y f c = Y, and then as Y ≤ X , we have X = Y, establishing (1). If m > 8 then Aut(K) is a 2-group, and then as AutF (O) permutes Ω(z) by 2.6.11.3, if in addition |Ω(z)| ≤ 2 it follows that AutF (O) is a 2-group. Therefore we may assume AutF (O) is a 2-group. By 2.6.11.1, O  F. Set E = OCF (O) and let E = OS0 and G a model for NF (E). By 1.1.11.1 in [Asc19], E  F. As AutF (O) is a 2-group, O 2 (G) ≤ CG (O), so G = SCG (O) and hence NF (E) = SNCF (O) (S0 ). Finally by 1.3.2 in [Asc19], F = SE, NF (E) , so F = SE, establishing (2). Assume the hypothesis of (3). Then Oz = O1 ∗ O2 with Oi ∼ = SL2 [8] a solvable component of F. Hence CF (O) = CF (Oz ) by 2.2.1.3. Define E = Oz CF (Oz ), E = OS0 , and G as in the previous paragraph. Let H be the model for Oz normal in G. Then G = SHCG (O), so, as in the previous paragraph, F = SE, NF (E) , completing the proof of (3). 

2.7. F = SOτ Definition 2.7.1. Write T for the set of pairs τ = (F, Ω) such that F is a saturated  fusion system on a finite 2-group S, Ω is a set of subgroups of S, O 2 (F) = K∈Ω FK is a central product, where K is Sylow in FK , FK ∼ = SL2 [m], and S permutes {FK : K ∈ Ω}. For τ = (F, Ω) ∈ T and S Sylow in F, we say that τ is uniquely determined by S and Ω (or F is uniquely determined by S and Ω) if whenever τ0 = (F0 , Ω0 ) ∈ T with S Sylow in F0 and Ω = Ω0 , then we have F0 = F, so also τ0 = τ . Remark 2.7.2. Suppose τ = (F, Ω) ∈ T. Then τ is a quaternion fusion packet. Namely it is straightforward to see that 2.7.1 implies that the four defining conditions (QFP1)-(QFP4) in the definition of quaternion fusion packet are satisfied.

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The object of this section is to prove: Theorem 2.7.3. For each (F, Ω) ∈ T, F is uniquely determined by S and Ω.

Notation 2.7.4. Let Y = O 2 (F) and, using 2.7.2, set O = O(τ ), and Z = Z(O). Then Z = ZS = Z(Y), so Z  F. Set F˜ = F/Z. Lemma 2.7.5. F = SY. Proof. This is Theorem 5 in [Asc11].



Lemma 2.7.6. Theorem 2.7.3 holds if m = 8. Proof. Assume m = 8. Then O = O2 (Y), so F is constrained by 2.7.5, and  hence F has a model G. Set H = O 2 (G). By 2.7.1, H = K∈Ω HK is a central product of subgroups HK ∼ = SL2 (3) with K = O2 (HK ). Then O = O2 (H) and O ≤ O2 (G) ≤ R for each R ∈ F f rc . Set S0 = CS (O). Now [O2 (G), HK ] ≤ O2 (H) = O, so O2 (G) acts on HK = ˜ O 2 (HK O) and hence also on K = O2 (HK ). Then as HK is irreducible on K, + + O2 (G) ≤ KCS (K). Therefore O2 (G) = OS0 . Set G = G/S0 and F = F/S0 . Then G+ = AutG (O) = AutF (O) and F + = FS + (G+ ). Next [H, S0 ] ≤ CO (O) = Z ≤ Z(H), so H = O 2 (H) centralizes S0 . Let A be the subgroup of Aut(O) permuting Ω, A∗ the kernel of the action of A on Ω, and B = O 2 (A∗ ). Then AutF (O) = AutG (O) ≤ A and O 2 (AutF (O)) = B is determined by S and Ω. As S0  F, we have the bijection R → R+ of F f rc with (F + )f rc . Therefore as F + is determined by S and Ω, so is F f rc . By Alperin’s Fusion Theorem 1.2.2 , F = AutF (R) : R ∈ F f rc , so it remains to show AutF (R) is determined by S, Ω, and R, for each R ∈ F f rc . Let R ∈ F f rc , GR = NG (R), and HR = O 2 (GR ). Then AutF (R) = AutGR (R) = AutHR (R)AutS (R), + = O 2 (NB (R+ )) so it remains to show AutHR (R) is determined by R. Observe HR is determined by S, Ω, and R. Let Y ∈ Syl3 (HR ); thus as O ≤ R we have HR = Y OR , where OR = O2 (HR ) = [O, Y ] ≤ O is a product OR = K∈Γ K, where Γ = {K ∈ Ω : K = [K, Y ]}. Now R = OR CR (Y ) and AutHR (R) = AutOR (R)AutY (R) is determined by S, Ω, and AutY (R), with AutY (OR ) determined by Y + = AutY (O). Thus it remains to show that CR (Y ) is determined by S and Ω. Finally Y centralizes S0 , so CR (Y )+ = CR+ (Y + ), and hence CR (Y ) is the preimage in R of CR+ (Y + ). Therefore CR (Y ) is determined by CR+ (Y + ), which is in turn determined by S, Ω, and R. This completes the proof of 2.7.6. 

Because of 2.7.6, we may assume in the remainder of the section: Hypothesis 2.7.7. m > 8.

2.7.

F = SOτ

47

Notation 2.7.8. For K ∈ Ω let QK consist of the Q8 -subgroups of K. For ∅ = Γ ⊆ Ω let ΞΓ consist of the sets ξ that contain exactlyone member of QK for each K ∈ Γ} (so that |ξ| = |Γ|). For ξ ∈ ΞΓ set Qξ = X∈ξ X, Sξ = CS (Qξ ), and Rξ = Qξ Sξ . Write K(X) for the member of Ω containing X and set FX = O 2 (NFK(X) (X)). Pick ∅ = Γ ⊆ Ω and ξ ∈ ΞΓ . S f Lemma 2.7.9. (1) For K ∈ Ω and QK ∈ QK , QF K = QK , so QK ∈ F . F S f (2) Qξ = Qξ , so Qξ ∈ F . (3) Rξ ∈ F f rc .  (4) Let Gξ be a model for NF (Rξ ) and Hξ = O 2 (Gξ ). Then Hξ = X∈ξ HX is a central product where HX ∼ = SL2 (3) and X = O2 (HX ).  Proof. Set Q = Qξ , R = Rξ , etc. Set Γ = Ω − Γ and P = J∈Γ J. FK Assume the setup of (1). As Y is a central product of the FK , QY K = QK . K K Further as FK ∼ = SL2 [m], QF K = QK . Then (1) follows from 2.7.5. Next for φ ∈ homY (Q, S),   Xφ = X sX = Qs , Qφ = X∈Γ



X∈Γ

for some sX ∈ K(X) and s = X∈Γ sX by (1). Therefore QY = QO , so (2) follows. Let α ∈ A(R). By (2), Q ∈ F f , so Rα = QαCS (Qα). Define P = J ∈ Ω : J ≤ R and observe that ξ is the set of Q8 -subgroups of R contained in a member of Γ and centralizing P . Hence NS (Rα) acts on P α, and then also on ξα and hence on Qα. Let β ∈ A(Qα) with Qαβ = Q. Then NS (Rα) ≤ NS (Qα) and Rαβ = Qαβ · CS (Qα)β = R, so NS (Rα)β ≤ NS (R). Therefore R ∈ F f . As CS (R) ≤ CS (Q) ≤ R, R ∈ F c . Thus to complete the proof of (3), it remains to show R ∈ F r . Let G be a model for  NF (R). Then O 2 (G) is a model for O 2 (NF (R)) and there 2 is H  O (G) with H = X∈ξ HX , where HX is a model for O 2 (NFK(X) (X)). Set R0 = O2 (G). Then QP ≤ R ≤ R0 . For X ∈ ξ, since HX ≤ G, K(X) ∩ R0 = X, so Γ = {J ∈ Ω : J ≤ R0 } and ξ are G-invariant. As m > 8, Aut(J) is a 2-group for J ∈ Γ , so H = O 2 (G). Then [HX , R0 ] ≤ O2 (H) = Q, so R0 acts on HX = ˜ [R0 , X] ˜ = 1. O 2 (HX Q) and then also on X = O2 (HX ). As HX is irreducible on X, ˜ = 1, so R0 ≤ QCS (Q) = R. Therefore R = O2 (G), completing Therefore [R0 , Q] the proof of (3). Moreover in the process of proving (3), we also established (4).  Lemma 2.7.10. For each R ∈ F f rc , O2 (O 2 (NF (R))) = Qξ for some Γ ⊆ Ω and ξ ∈ ΞΓ . Hence NF (R) ≤ NF (Qξ ). Proof. Let N = NF (R), G a model for N , and H = O 2 (G). Then H = O (N ) ≤ O 2 (F) = Y, so H is model for H and Q = O2 (H) ≤ O. For K ∈ Ω, let ˜→K ˜ be the projection map. Set Γ = {K ∈ Ω : Qπ ˜ K = 1}. πK : O As H ≤ Y and Y is the central product of copies of SL2 [m], H is a {2, 3}group with an elementary abelian Sylow 3-subgroup Y . Further for y ∈ Y # , y = y1 × · · · × yr × 1 where yi ∈ AutFKi (QKi ) is of order 3 for some Ki ∈ Γ and ˜ K . Hence for K ∈ Γ, QK ∈ QK , QKi ∈ QKi . Thus QKi is the preimage in Ki of Qπ i ˜ where QK is the preimage of Qπ . K  Set Q0 = K∈Γ QK ; then R acts on Q0 , so [NQ0 (R), R] ≤ Q0 ∩ R = R0 . ˜ 0 = [R ˜0, Y ] ≤ Q ˜ ≤ Q ˜ 0 , so R0 = Q. Therefore NQ (R) centralizes R/Q, Now R 0 2

48

2. SOME BASIC RESULTS

˜ and Z, so NQ (R) ≤ R and hence Q0 ≤ R. Therefore Q0 = [Q0 , Y ] ≤ Q, so Q, 0 Q0 = Q. Further ξ = {QK : K ∈ Γ} ∈ ΞΓ , so Q = Qξ , completing the proof of the lemma.  We are now in a position to complete the proof of Theorem 2.7.3. By 2.7.6 we may assume m > 8, so Hypothesis 2.7.7 is satisfied. By Alperin’s Fusion Theorem, F = AutF (R) : R ∈ F f rc . Let R ∈ F f rc . By 2.7.10, NF (R) ≤ NF (Qξ ) = Fξ for some ∅ = Γ ⊆ Ω and ξ ∈ ΞΓ . Set τξ = (Fξ , ξ). By 2.5.2, τξ is a quaternion fusion packet, and by construction τξ ∈ T with mξ = |X| = 8 for X ∈ ξ. As NF (R) ≤ Fξ , AutF (R) = AutFξ (R) and R ∈ Fξf rc . As mξ = 8, AutFξ (R) is determined by NS (Qξ ) and ξ by 2.7.6. Therefore F = AutFξ (R) : R ∈ Fξf rc , ∅ = Γ ⊆ Ω, ξ ∈ ΞΓ is determined by S and Ω, proving Theorem 2.7.3.

2.8. Modules for groups with a strongly embedded subgroup In this section we assume the following hypothesis: Hypothesis 2.8.1. (1) G is a finite group with a strongly embedded subgroup. (2) a is an involution in G and H = aG . (3) V is a faithful F2 G-module such that V = [V, a]G . Lemma 2.8.2. Either  (1) H = O 2 (G) and H/O(H) is a Bender group, or (2) m2 (G) = 1 and H = a O(H) is the subgroup generated by the set aG of involutions of G. Proof. By the Bender-Suzuki classification of groups with a strongly embedded subgroup (cf. Theorem SE in [GLS99]) either   (i) O 2 (G)/O(O 2 (G)) is a Bender group, or (ii) m2 (G) = 1.  Suppose (i) holds and set L = O 2 (G). Then L/O(L) is simple so as H  L we conclude that (1) holds. Thus we may assume that (ii) holds. Hence by the  Z ∗ -Theorem , H = a O(H), so (2) holds. In the remainder of the section we assume: Hypothesis 2.8.3. Hypothesis 2.8.1 holds with Va = [V, a] of dimension 2. Lemma 2.8.4. (1) V = [V, O 2 (H)]. (2) If U is an O 2 (H)-submodule of V containing Va then U = V . Proof. By 2.8.2, O 2 (H) is transitive on aG , so the lemma follows from G.6.2 in [AS04]; note that lemma does not require the SQTK-condition. 

2.8.

MODULES FOR GROUPS WITH A STRONGLY EMBEDDED SUBGROUP

49

Lemma 2.8.5. Let U be an O 2 (H)-submodule of V . Then one of the following holds: (1) [U, H] = 0. (2) U = V . (3) a induces a transvection on U and V /U . Proof. See G.6.3 in [AS04].



Lemma 2.8.6. Assume a inverts x ∈ G of odd order p > 1. Then (1) p = 3 or 5. (2) If p = 5 then m([V, x]) = 4, Va = [V, x, a], and a centralizes CV (x). (3) If p = 3 then either m([V, x]) = 4, Va = [V, x, a] and a centralizes CV (x), or m([V, x]) = 2 and a induces a transvection on [V, x] and CV (x). Proof. See D.2.11 in [AS04].



Lemma 2.8.7. (1) If H/O(H) is Bender then H/O(H) ∼ = A5 or U3 (4). (2) If H ∼ = A5 and either = A5 or U3 (4) then H ∼ (a) m(V ) = 4 and V is the A5 -module for H, or (b) V /CV (H) is the 4-dimensional L2 (4)-module for H and m(CV (H)) ≤ 2. Proof. Let G∗ = G/O(H). Then H ∗ is Bender so H ∗ ∼ = L2 (q), Sz(q), or U3 (q) with q ≥ 4 even. It follows that either H ∗ is L2 (4) or U3 (4), or a∗ inverts some x∗ of odd order m > 5. But then a inverts a preimage x of x∗ of odd order at least |x∗ |, contrary to 2.8.6.1. This establishes (1). Assume H is A5 or U3 (4). In the latter case there is no nontrivial F2 H-module U with m([U, a]) = 2. Therefore H ∼ = A5 , so there are two nontrivial irreducible F2 H-modules U1 and U2 : U1 is the A5 -module and U2 the L2 (4)-module. In each case, m([Ui , a]) = 2, so we conclude from 2.8.4 and 2.8.5 that V = [V, H] with V¯ = V /CV (H) a 4-dimensional irreducible for H. If V¯ is the A5 -module then V¯ is projective, so CV (H) = 0 and (2a) holds. On the other hand if V¯ is the  L2 (4)-module then m(H 1 (H, V¯ )) = 2, so (2b) holds. Lemma 2.8.8. Op (H) ≤ Z(H) for each prime p ∈ / {3, 5}. Proof. Let P = Op (H). If a centralizes P then P ≤ Z(H) as H = aH . On the other hand if a does not centralize P then a inverts some x of order p in P , so p ∈ {3, 5} by 2.8.6.1.  Lemma 2.8.9. Assume O5 (H) ≤ Z(H). Then m(V ) = 4 and H ∼ = D10 . Proof. As in D.2.12 of [AS04], write X2 for the set of subgroups X of P = O5 (H) such that |X| = 5 and m([V, X]) = 4. As in the proof of 2.8.8, a inverts some X of order 5 in P , and by 2.8.6.2, we have X ∈ X2 , Va = [V, X, a] and a centralizes CV (X). In particular X2 = ∅. Let Q = X2 . By D.2.12 in [AS04], Q = X1 × · · · × Xs is the direct product of the members of X2 and [V, Q] = V1 ⊕ · · · ⊕ Vs where Vi = [V, Xi ]. As H permutes X2 it acts on Q and [V, Q]. We may take X = X1 ; then, as we just saw, a acts on V1 and centralizes CV (X) = V2 + · · · + Vs . Thus a acts on each Vi , so also H = aH acts on each Vi . Thus by 2.8.4.2, V = V1 is of dimension 4, so X2 = {X} and hence X  H. Now the lemma follows as NGL(V ) (X) is the multiplicative group of F = F16 extended by Aut(F ). 

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Lemma 2.8.10. Assume V is a decomposable H-module. Then m(V ) = 4, H = O(H)a , and a inverts O(H) ∼ = Z3 or E9 . Proof. By assumption, V = V1 ⊕V2 with V1 and V2 nontrivial H-submodules. Set V i = V /V3−i . As V = VaH also V i = [V i , a]H , so as V i is isomorphic to Vi , we have Vi = [Vi , a]H . In particular [Vi , a] = 0, so a induces a transvection on each Vi . Let H i be the image of H in GL(Vi ). Claim H i has a strongly embedded subgroup. If m2 (H) = 1 then ai is such a subgroup. On the other hand if H/O(H) is Bender then O(H) is contained in the strongly embedded subgroup M of H and as H/O(H) is simple, CH (Vi ) ≤ O(H), so M i is strongly embedded in H i , completing the proof of the claim. Now by 2.4.1 in [Asc19], m(Vi ) = 2 and  Hi ∼ = S3 , so the lemma follows using 2.8.2. Lemma 2.8.11. Assume V is an indecomposable H-module and H preserves a direct sum decomposition V = {Vi : 1 ≤ i ≤ n} in which n > 1 and dim(Vi ) > 1 for some i. Then m(V ) = 6, H = O(H)a with O(H) ∼ = 31+2 , and a centralizes Z(O(H)). Proof. As H preserves V and is indecomposable on V , it follows that H is transitive on V. Thus for each i, dim(Vi ) = m > 1. As H = aH is transitive on V, it follows that a moves some member of V, so we may take V1a = V2 . Then 2 ≤ m = m([V1 + V2 , a]) ≤ m(Va ) = 2, so we conclude that m = 2, Va = [V1 + V2 , a], and a centralizes Vi for i > 2. Then a induces a transposition on V, so as H = aH is transitive on V, it follows that H V = Sym(V). Then by 2.8.2, n = 2 or 3, so m(V ) = 4 or 6. In the first case H is decomposable on V , so the second holds. Then H is contained in the stabilizer Z3 wr S3 of V in GL(V ) = GL6 (2), so as  O(H) = [O(H), a] and m(Va ) = 2, the lemma follows. Definition 2.8.12. Let p be an odd prime and P a p-group. From Definition A.1.20 in [AS04], a supercritical subgroup of P is a characteristic subgroup X of P such that Φ(X) ≤ Z(X) ≥ [X, P ], X is of exponent p, and X contains all elements of order p in CP (X). Lemma 2.8.13. Let p be an odd prime and P a p-group. Then (1) P possesses a supercritical subgroup X. (2) X is of class at most 2. (3) Each p -automorphism of P is faithful on X. Proof. See A.1.21 in [AS04].



Lemma 2.8.14. Assume F ∗ (H) = O3 (H)Z(H). Then either (1) V is a decomposable H-module, or (2) H preserves a direct sum decomposition V = {Vi : 1 ≤ i ≤ n} with n > 1 and m(Vi ) > 1 for some i. Proof. Assume neither (1) nor (2) holds and set P = O3 (H). As F ∗ (H) = P Z(H), we have CH (P ) = Z(P )Z(H). By 2.8.13.1, P = O3 (H) contains a supercritical subgroup Q. By 2.8.13.3, each 3 -automorphism of P is nontrivial on Q, so CH (Q) = CP (Q)Z(G). In particular a is nontrivial on Q. Let X = Z(Q); as H is indecomposable on V , we have V = [V, X]. By definition of Q, Q is of exponent 3, so Φ(X) = 1. Thus if X is noncyclic then the set V of

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weight spaces of Q is a direct sum decomposition of V of size n > 1 whose members are noncyclic, so that (2) holds. Hence we may assume X ∼ = Z3 . If a inverts X then m(V ) = 2m(Va ) = 4, contradicting V an indecomposable H-module. Therefore a centralizes X, so X = Q. Hence Q is of class 2 and exponent 3 with Φ(Q) = Z(Q) ∼ = 31+2w . Further a inverts some Y ≤ Q = Z3 , so Q ∼ of order 3, and the set W of weight spaces for XY on V is W = {W1 , W2 , W3 } of order 3 with W1a = W2 and Q transitive on W. Thus 2 = m(Va ) = m(W1 ), so  m(V ) = 6 and Q ∼ = 31+2 . But now W satisfies (2), contrary to assumption. Theorem 2.8.15. Assume Hypothesis 2.8.3. Then one of the following holds: (1) m(V ) = 4 and H = O(H)a and O(H) ∼ = Z3 or E9 is inverted by a. (2) m(V ) = 4 and H ∼ = D10 . (3) m(V ) = 4 and V is the A5 -module for H ∼ = A5 . (4) H ∼ = A5 , V = [V, H] with V /CV (H) the L2 (4)-module for H, and m(CV (H)) ≤ 2. (5) m(V ) = 6, H = O(H)a with O(H) ∼ = 31+2 and a centralizes Z(O(H)). Proof. Suppose first that F ∗ (H) = F (H). Then from 2.8.2, H = E(H) with H/Z(H) Bender. Now by 2.8.7.1, H ∼ = A5 or U3 (4), and then (3) or (4) holds by 2.8.7.2. Therefore we may assume F ∗ (H) = F (H). Next by 2.8.8, F (H) = O3 (H)O5 (H)Z(H). Further if O5 (H) ≤ Z(H) then (2) holds by 2.8.9. Thus we may assume that F ∗ (H) = O3 (H)Z(H). Therefore by 2.8.14 either H is decomposable on V or H preserves a direct sum decomposition as in 2.8.14.2. In the first case, (1) holds by 2.8.10. Thus we may assume V is an indecomposable H-module and H preserve a direct sum decomposition, so (5) holds by 2.8.11. This completes the proof of the theorem. 

CHAPTER 3

Results on τ

In Chapter 3 we consider quaternion fusion packets τ = (F, Ω) and let S be Sylow in F. We develop and explore some of the most basic notions concerning such objects. For example in 3.1.5 we show that O(τ ) = Ω is a central product of the members of Ω. The set η(τ ) of collections η of cyclic subgroups of S, the subset ηS (τ ) of Sinvariant members of η(τ ), and the set W (τ ) of subgroups η for η ∈ ηS (τ ) are defined in Definition 3.1.9. By 3.1.10, ηS (τ ) is nonempty, and by 3.1.11, for each η ∈ η(τ ), η is abelian and its members are of order m/2. Then the Thompson group μ(τ ) is defined in Definition 3.1.26 as a certain subgroup of AutF (W ) for W ∈ W (τ ). From 3.1.25, μ(τ ) is a direct product of a group generated by 3-transpositions with copies of D12 ∼ = Weyl(G2 ). From 3.2.1, D12 -factors appear if and only if A(τ ) = ∅. The graph A(τ ) on Z(τ ) is discussed in section 3.2. The graphs D(τ ) and D∗ (τ ) are introduced in Definition 3.1.13 and discussed in section 3.1. Lemma 3.3.2 shows that if T is a normal subgroup of F such that Z(τ ) ∩ T = ∅ and F + = F/T then τ + = (F + , Ω+ ) is a quaternion fusion packet; this makes possible many inductive arguments. In 3.3.7 we find that there is a subsystem O(τ ) of F on O(τ ) which is the central product of the subsystems OK for K ∈ Ω, and O(τ )  NF (WS ), where WS = ZS (τ ) . Let W ∈ W (τ ) and M = [O(τ )]NF (W ) . From 3.3.11, M is constrained and hence has a model M (W ); we write M (τ ) for the collection of groups M (W ), W ∈ W (τ ). From 3.3.14, if F is transitive on Ω then either η(τ ), and hence also W (τ ) and M (τ ), has a unique member, or m = 8 and Δ(τ ) = Ω. Therefore, the exceptional case aside, W ∈ W (τ ) is weakly closed in S with respect to F and M (τ ) has a unique member M . Later we will see that μ(τ ) ∼ = M/W . Set Z = Z(F) and F + = F/Z; assume F = F ◦ . In 3.3.16 we find that if F + is tamely realized by a simple group L+ appearing as a conclusion in Theorem 1, then F = FS (L) for some covering group L of L+ . This makes it possible to lift statements about simple factor systems F + to F. Definition 3.4.2 introduces the subsystem E(τ ) of F. Section 3.4 explores E(τ ) and its relationship to F. In time we will see that often F = E(τ ). Lemma 3.4.5 is a step in that direction; for example it is used to prove 3.4.8, where it is shown that if Ω is of odd order and |Ω(z)| = 1 then generically we have F = E(τ ).

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3.1. Δ(τ ), η(τ ), and μ(τ ) In this section we assume τ = (F, Ω) is a quaternion fusion packet and adopt Notation 2.5.1. In addition we adopt the following notation: Notation 3.1.1. For z ∈ ZS define O(z) = Ω(z) and for z ∈ Z define Δ(z) = {(K ∩ CS (z)α)α−1 : K ∈ Ω(zα) and |K ∩ CS (z)α| > 2} for α ∈ A(z). In 3.1.3 we find that Δ(z) is independent of the choice of α ∈ A(z). If z ∈ ZS ∩ F f write O(z) for the normal subsystem of Fz on O(z) supplied by Definition 2.6.7 and Theorem 2.6.11. Similarly for K ∈ Ω(z), write OK for the normal subsystem of O(z) on K supplied by Definition 2.5.6 and 2.6.11. Set O(τ ) = Ω . We find in 3.1.5.1 that O(τ ) is a central product of the members of Ω. Lemma 3.1.2. Let z ∈ Z and α ∈ A(z). Then (1) If z ∈ ZS then Ω(z)α ⊆ Ω(zα), so Ω(z) ⊆ Δ(z). (2) zα ∈ ZS , so Z ∩ F f ⊆ ZS . (3) If z ∈ F f then α−1 ∈ A(zα) and Ω(z)α = Ω(zα). Proof. Suppose K ∈ Ω(z). Then Kα ≤ CS (z)α ≤ S, so Kα ∈ K F ⊆ Ω as τ is a quaternion fusion packet. Then as zα ∈ Kα, Kα ∈ Ω(zα), establishing (1). Let φ ∈ homF (z, S) with zφ ∈ ZS . Let β ∈ A(zφ) with zφβ = zα. By (1), Ω(zφ)β ⊆ Ω(zφβ) = Ω(zα), so (2) holds. If z ∈ F f then |CS (z)| ≥ |CS (zα)|, so α−1 ∈ A(zα), and then (3) follows from  (1) and symmetry between (z, α) and (zα, α−1 ). Lemma 3.1.3. Let z ∈ Z. Then (1) Δ(z) is independent of the choice of α ∈ A(z). (2) For D ∈ Δ(z), P ≤ D with |P | > 2, and φ ∈ homF (P, S), P φ ≤ E ∈ Δ(zφ). Proof. For i = 1, 2, let αi ∈ A(z) and set zi = zαi , Ti = CS (zi ), Si = CS (z)αi , and Δi = {(K ∩ Si )αi−1 : K ∈ Ω(zi )}. Then ψ = α1−1 α2 : S1 → S2 is an isomorphism. Let ϕ ∈ A(z2 ) with z2 ϕ = z1 . Then β = ψϕ ∈ homF (S1 , T1 ) with z1 β = z1 , so β is an Fz1 -map. By 3.1.2.3, Ω(z2 )ϕ = Ω(z1 ). Next for D ∈ Δ1 , D = (K1 ∩ S1 )α1−1 for some K1 ∈ Ω(z1 ), and as β is an Fz1 -map, (K1 ∩ S1 )β ≤ J1 ∈ Ω(z1 ) by 2.6.2. Thus (K1 ∩ S1 )ψ = (K1 ∩ S1 )βϕ−1 ≤ J1 ϕ−1 = K2 ∈ Ω(z2 ), so Dα2 = Dα1 ψ = (K1 ∩ S1 )ψ ≤ K2 . Further Dα2 = Dα1 ψ ≤ S1 ψ = S2 , so D ≤ E ∈ Δ2 . By symmetry, E ≤ D = (K1 ∩ S1 )α1−1 ∈ Δ1 , so D = E = D as D and E have order at least 4, so K1 = K1 by (QFP2). This shows Δ(z) is independent of α, establishing (1). Assume the hypothesis of (2). Let α ∈ A(zφ) and β ∈ A(z) with zβ = zφα = a. Then γ = β −1 φα is an Fa -map. As D ∈ Δ(z), Dβ ≤ K ∈ Ω(a), so by 2.6.2.2 and as |P | > 2, P φα = P βγ ≤ J ∈ Ω(a). Therefore P φ ≤ (J ∩ CS (zφ)α)α−1 ∈ Δ(zφ), proving (2).  Lemma 3.1.4. Suppose z ∈ Z and D ∈ Δ(z) is nonabelian and acts on T ≤ S with D ∩ T = 1. Then D centralizes T and DT = D × T .

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Proof. Let α ∈ A(z), a = zα, and W = DCT (z) . As D acts on T , so does W . Then Dα ≤ K ∈ Ω(a) is invariant under CS (z)α ∩ NS (K), so W α is the central product of the members of (Dα)CT (z)α by 2.6.6.1. Thus as D is nonabelian, z = Z(W ). Then as T ∩ W  W and z ∈ / T , T ∩ W = 1. Thus it remains to show that D centralizes T . Let R = CT (z). Then [R, W ] ≤ W ∩ T = 1, so we may assume R < T . Thus R < Y = NT (W R). Now [W, Y ] ≤ W R ∩ T = (W ∩ T )R = R ≤ CS (W ), so [Y, W, W ] = 1 = [W, Y, W ]. Therefore by the 3-Subgroups Lemma (cf. 8.7 in [Asc86]), Y centralizes [W, W ]. As z ∈ [W, W ], this contradicts R < Y .  Lemma 3.1.5. (1) O(τ ) is a central product of the members of Ω. (2) ZS = Z(O(τ )). (3) If z ∈ CZ (O(τ )) then z ∈ ZS . (4) If t ∈ ZS and α ∈ A(t), then Ωα = Ω and Ω(t)α = Ω(tα). (5) If t ∈ Z and some member of Δ(t) is nonabelian, then t ∈ ZS and Δ(t) = Ω(t). (6) AutF (O(τ )) is transitive on each orbit of F on Ω. Proof. We first prove (1). Let Σ be a subset of Ω maximal subject to T = Σ a central product of the members of Σ. Set E = z(J) : J ∈ Σ . We may assume K ∈ Ω with K ∈ / Σ. If {J ∈ Ω : J ≤ NS (T )} is contained in T then NS (NS (T )) = NS (T ), so T  S. Hence we may take K ≤ NS (T ). Let z = z(K). If K centralizes E, then by (QFP3), K acts on each J ∈ Σ. Then by 3.1.4, either K centralizes J or z = z(J), in which case K = J or K centralizes J by 2.6.6.1. However this contradicts K ∈ / Σ and the maximality of Σ. Therefore [K, E] = 1. Hence by 3.1.4, z ∈ E ≤ Z(T ). Then by (QFP3), each J ∈ Σ acts on K. Then by 3.1.4, either K centralizes J or z = z(J), for the same contradiction. This completes the proof of (1). Observe (1) implies (2) as z = Z(K) for K ∈ Ω(z). Assume the hypothesis of (3) and let α ∈ A(z). Then zα ∈ ZS by 3.1.2.2, and Ωα ⊆ CS (z)α ⊆ S, so Ωα = Ω as ΩF = Ω. Thus ZS α = ZS , so as zα ∈ ZS , (3) holds. Similarly (4) holds. Observe next that the proof of (1) (with Σ = Ω) shows that if t ∈ Z and D ∈ Δ(t) is nonabelian, then one of the following holds: (i) D ∈ Ω and hence t ∈ ZS . (ii) D centralizes Ω. (iii) t ∈ E = ZS . In case (i), t ∈ ZS , while in cases (ii) and (iii), t ∈ ZS by (3). Thus in any event, t ∈ ZS . Then for α ∈ A(t), Ωα = Ω by (4). Thus for V ∈ Δ(t), V α ≤ K ∈ Ω(tα), so as Ω(t)α = Ω(tα) by (4), we conclude that V = Kα−1 ∈ Ω(t) and (5) holds. Let K ∈ F f and J ∈ K F . Then there is α ∈ A(J) with Jα = K. By (4), O(τ )α = O(τ ), so (6) follows.  Note by 3.1.5.1 and the weak closure of O(τ ), the orbits of F on Ω are the same as those of AutF (O(τ )). In particular if F is transitive on Ω then so is AutF (O(τ )). Lemma 3.1.6. For K ∈ Ω, CZ (K) = CZ (O(z(K)). Proof. Let u = z(K), J ∈ Ω(u), t ∈ CZ (K), β ∈ A(t), and z = tβ. Suppose first that u ∈ F f . Then there exists α ∈ A(uβ) with uβα = u. Thus γ = βα is an Fu -map. As Kβ ≤ CS (t)β ≤ S, uβ ∈ ZS . Thus Ω(u) = Ω(uβ)α by 3.1.5.4. By 3.1.2.2, z ∈ ZS , so z centralizes Ω(uβ) by 3.1.5.2, and hence zα centralizes

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Ω(uβ)α = Ω(u). Thus tγ = zα ∈ CS (Ω(u)), so as O(u)  Fu by 2.6.11 and CS (O(u)) ≤ CFu (O(u)) by 2.6.12, we have t ∈ CS (Ω(u)). Thus the lemma holds when u ∈ F f . In general let φ ∈ A(u). By 3.1.5.4, Ω(u)φ = Ω(uφ), so as tφ centralizes Kφ, tφ centralizes Ω(uφ) by the previous paragraph. Then t centralizes Ω(uφ)φ−1 = Ω(u), completing the proof.  Lemma 3.1.7. Suppose t ∈ Z − ZS , V ≤ D ∈ Δ(t), K ∈ Ω, and assume V centralizes z = z(K) and |V | > 2. Let α ∈ A(t). Then (1) V acts on K. (2) V is cyclic. (3) CK (t) = CK (V ) contains each V -invariant cyclic subgroup of K. (4) V is a TI-subgroup of KV . (5) Either CK (t) is a cyclic subgroup of K of index 2, or t centralizes K. (6) If t centralizes K then t centralizes O(z) and zα ∈ ZS . (7) Assume [t, K] = 1, let U0 be the subgroup of CK (t) of order |V |, and X = U0 V . Then zα does not centralize O(tα) and AutF (X) induces GL(t, z ) on z, t . Proof. Part (1) follows from (QFP3). Part (2) follows from 3.1.5.5 and the hypothesis that t ∈ / ZS . Let U = CK (t). Now V acts on some cyclic subgroup B of K of index 2, while as |V | > 2, t ∈ Φ(V ), and hence CB (t) is of index at most 2 in B, and is of order at least 4. Now V α ≤ J ∈ Ω(tα) and U α ≤ NS (J) by (QFP3). Next [U α, V α] ≤ U α ∩ J = 1 as z = t, so U = CK (V ). Then as t is the unique involution in V , (4) holds. Further CB (t) = CB (V ), so as t ∈ Φ(V ), t centralizes B. Thus U = B or K, completing the proof of (3) and establishing (5). Suppose t centralizes K. Then by 3.1.6, t centralizes O(z). Now Ω(z)α ⊆ CS (t)α ≤ S, so as ΩF = Ω, Ω(z)α ⊆ Ω. Hence zα ∈ ZS , establishing (6). Assume the hypothesis of (7). Let γ ∈ A(z) and β ∈ A(zα) with zαβ = zγ. Set a = zγ, b = tγ, and φ = γ −1 αβ ∈ homF (CS (E)γ, S), where E = z, t . Then φ is an Fa -map and Eγ = a, b . Suppose [zα, J] = 1. Then by (6) applied to the pair zα, tα in the role of t, z, either zα ∈ ZS or tαβ ∈ ZS . In the first case, O(tα)β centralizes O(a) so again tαβ ∈ ZS . Thus in any event, bφ = tγφ = tαβ centralizes Ω(a), so as O(a)  Fa by 2.6.11 and CS (O(a)) ≤ CS (O(a)) by 2.6.12, also b centralizes Ω(a). By 3.1.5.4, Ω(z)γ = Ω(a), so t = bγ −1 centralizes Ω(z), a contradiction. Therefore zα does not centralize Ω(tα). Let k ∈ K − U and j ∈ CJ (zα). Then ck ∈ AutF (X) with [E, k] = z . Similarly cj ∈ AutF (Xα) with [Eα, j] = tα , so cj α−∗ ∈ AutF (X) with [E, cj α−∗ ] = t . Therefore AutF (X) induces GL(E) on E, completing the proof of (7).  Lemma 3.1.8. Let T ≤ S and Σ a T -invariant set of cyclic subgroups of T such that there exists a map D from Σ to the set of subgroups of S such that: (a) Each V ∈ Σ is of order at least 4 and contained in D(V ) ∈ Δ(z(V )), where z(V ) is the involution in V . (b) If U, V ∈ Σ are distinct then D(U ) = D(V ). Then Σ is abelian. Proof. Assume otherwise and choose a counter example with T minimal. Then T = Σ and there exists A, B ∈ Σ such that [A, B] = 1. As T is nonabelian,

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A < T , so X = AT < T , and hence X is abelian by minimality of T . Similarly Y = B T is abelian. Let a, b be the involutions in A, B, respectively, and let β ∈ A(b). As B ≤ D(B), Bβ ≤ J ∈ Ω(bβ) by 3.1.3.2. Suppose b ∈ NT (A). Then [b, A] ≤ Y ∩ A ≤ CA (B). If [b, A] = 1 then Aβ ≤ NS (J) for each J ∈ Ω(bβ) by (QFP3), so Aβ acts on Y β ∩ J. As Y β ∩ J is abelian, it is cyclic, so Aβ acts on Bβ. As D(A) = D(B), Aβ ≤ J. Therefore [Aβ, Bβ] = 1 by 3.1.5.1 and 3.1.7.3, contradicting [A, B] = 1. On the other hand if [b, A] = 1 then a ∈ [b, A] ≤ CA (B), so by symmetry we again have a contradiction. Therefore B acts regularly on AB . Let x, y be elements of order 4 in A, B, respectively, and set A0 = x . As Y is abelian, [x, y] ≤ Y ≤ CT (y), so x−1 xy = (x−1 xy )y = x−y xb , and hence ay = (xy )2 = xxb . Now b centralizes xxb , and hence −1 also ay . Then b centralizes (ay )y = a. Let α ∈ A(a). Then Aα ≤ K ∈ Ω(aα). As b centralizes a but does not act on A, and as [A, Ab ] = 1, [K, K bα ] = 1. Then as ay = xxb , (ay )α ∈ A0 α(A0 α)bα induces the inner automorphism cxα on K, so (ay )α does not centralize K. Next xb = x−1 ay , so Ab0 ≤ A0 Ay0 = Q. Then Qy = Ay0 Ab0 ≤ Q, so y acts on Q, and Q = Ay0 Ab0 . As A0 Ay0 = Ay0 Ab0 and (A0 α)bα centralize K but A0 α does not, it follows that (Ay0 )α does not centralize K. Therefore by 3.1.7.7, Λ = AutF (Q) induces GL(E) on E = a, ay . In particular setting c = aay , there is λ ∈ Λ with cλ = a, so as A0 , Ab0 ≤ Q and as C = xxy and C  = x−1 xy are the two cyclic subgroups of order 4 in Q containing c, Cλ = A0 or Ab0 . Hence C ≤ D ∈ Δ(c) by 3.1.3.2. Let γ ∈ A(c). As D ∈ Δ(c), Dγ ≤ J for some J ∈ Ω(cγ). However (xxy )y = xy xb = xy x−1 ay = x−y x−1 = (xxy )−1 , so y inverts C. But yγ ∈ D ∈ Δ(bγ), contrary to 3.1.7.3 applied to bγ, D , A0 γ, J in the role of t, D, V, K. This completes the proof of the lemma.  Definition 3.1.9. Set ZΔ = {z ∈ Z : Δ(z) = ∅} and  Δ = Δ(τ ) = Δ(z), z∈ZΔ

Define η(τ ) to consist of those sets Σ of cyclic subgroups of S such that: (i) Σ is Σ -invariant. (ii) Each V ∈ Σ is contained in some D(V ) ∈ Δ(τ ), and either V = D(V ), or D(V ) ∈ Ω and |D(V ) : V | = 2. (iii) The map V → D(V ) is a bijection of Σ with Δ(τ ). Write ηS (τ ) for the set of S-invariant members of η(τ ) and W (τ ) for {η : η ∈ ηS (τ )}. We find in 3.1.11.4 that |η(τ )| = 1 when m > 8. Lemma 3.1.10. ηS (τ ) = ∅. Proof. Let Δ1 = Δ − Ω. By 3.1.5.5, each member of Δ1 is cyclic. Pick a set {Ki : 1 ≤ i ≤ r} of representatives for the orbits of S on Ω, and for 1 ≤ i ≤ r, pick a NS (Ki )-invariant cyclic subgroup Vi of index 2 in Ki . Define  η = Δ1 ∪ ViS . 1≤i≤r

Then η ∈ ηS (τ ). Lemma 3.1.11. (1) η(τ ) = ∅. (2) For each η ∈ η(τ ), η is abelian.



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(3) (4) (5) (6) (7)

If η ∈ η(τ ) and φ ∈ homF (η, S) then ηφ ∈ η(τ ). If m > 8 then |η(τ )| = 1. For each η ∈ η(τ ) and V ∈ η, |V | = m/2. Each member of Δ(τ ) − Ω is cyclic of order m/2. For η ∈ η(τ ), Z ∩ η = ZΔ .

Proof. Part (1) follows from 3.1.10, and (2) follows from 3.1.8. Let η ∈ η(τ ), V ∈ η, z the involution in V , and φ ∈ homF (η, S). Then V ≤ D = D(V ) ∈ Δ(z) and V φ ≤ E = E(V ) ∈ Δ(zφ) by 3.1.3.2. If V  ∈ η − {V } and E(V ) = E(V  ) then either (a) V φ, V  φ is abelian, or (b) V φ, V  φ is nonabelian. In case (a), either V φ ≤ V  φ or V  φ ≤ V φ. But then V ≤ V  or V  ≤ V , contrary to the hypothesis in 3.1.9.iii that D : V → D(V ) is injective. Similarly in case (b), V, V  is nonabelian, so D(V ) = D(V  ) ∈ Ω, for the same contradiction. Therefore the map E : V → E(V ) is injective, while as D : η → Δ is a bijection, we have |η| = |Δ|. Thus E : η → Δ is a bijection, so condition (iii) of 3.1.9 is satisfied by ηφ. As φ : η → ηφ is an isomorphism, and as η satisfies condition (i) of 3.1.9, μ = ηφ also satisfies that condition. Thus it remains to show that μ satisfies condition (ii) of 3.1.9. As E : η → Δ is a bijection, we must show that either E(V ) ∈ Δ − Ω or E(V ) ∈ Ω and |E(V ) : V φ| = 2. For ν ∈ {η, μ}, set ν  = {X ∈ ν : |X| = m/2} and ν¯ = ν − ν  . Then |ν  | = |Ω| + t, where t = |Δ | and Δ = {X ∈ Δ − Ω : |X| = m/2}. As E : η → Δ is a bijection with |V | = m/2 for D(V ) ∈ Ω, it follows that E(η  ) = Ω ∪ Δ and φ : η  → μ is a bijection. Thus as |η| = |μ|, also φ : η¯ → μ ¯ is a bijection, and E(¯ η) = φ(¯ η) = μ ¯ = Δ − (Ω ∪ Δ ), completing the proof of (3). Suppose m > 8. If D(V ) ∈ Ω then |D(V ) : V | = 2, so |V | = m/2 and V is the unique cyclic subgroup of D(V ) of order m/2. On the other hand if D(V ) ∈ /Ω then D(V ) is cyclic. In either case V is the unique subgroup of D(V ) of its order, so η is the unique member of η(S). Thus (4) is established. Further if V ∈ η with |V | < m/2, then V = D(V ) ∈ Δ − Ω. Let z be the involution in V and α ∈ A(z). Then by (2), ηα ⊆ CS (z)α ≤ S, and hence by (3), ηα ∈ η(τ ). Therefore by (4), ηα = η. In particular V α ∈ ηα = η. However by 3.1.2.2, zα ∈ ZS , so Δ(zα) ⊆ Ω, and hence |D(V α) : V α| = 2, contradicting |V | < m/2. Therefore (5) holds in this case. On the other hand if m = 8 then as |V | ≥ 4 = m/2, we have |V | = m/2, completing the proof of (5). Observe (5) implies (6). Let z ∈ Z ∩ η and α ∈ A(z). By (3), ηα ∈ η(τ ) and by 3.1.2.2, zα ∈ ZS , so  zα ∈ V ∈ ηα. Then z ∈ V α−1 ∈ η, so (7) holds. Lemma 3.1.12. Let η ∈ η(τ ) and K ∈ Ω. Then (1) K acts on η and η acts on K. (2) If t ∈ ZΔ centralizes K then so does each member of Δ(t). Proof. As η ∈ η(τ ), the member U of η contained in K is of index 2 in K. In particular U  K. Let z = z(K), V ∈ η − {U }, and t the involution in V . By 3.1.11.2, W = η is abelian, so V centralizes U and V acts on K by (QFP3). If t centralizes K then so does V by 3.1.7.3, so (2) holds and in particular K acts on V . Thus we may assume [t, K] = 1. Therefore D(V ) ∈ / Ω by 3.1.5.1, so

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V = D(V ) ∈ Δ − Ω. By 3.1.3.2, Δ − Ω is S-invariant, so V K ⊆ Δ − Ω ⊆ η. Thus  in each case, V K ⊆ η, completing the proof of (1) and the lemma. Definition 3.1.13. Let D, D∗ be the set of pairs (a, b) ∈ Z ×Z such that a = b and for some φ ∈ homF (a, b , S), (aφ, bφ) ∈ ZS × ZS , ZΔ × ZΔ , respectively. Visibly D and D∗ are symmetric relations on Z, so we can regard D and D∗ as undirected graphs on Z. Also both D and D∗ are F-invariant: that is if (a, b) is an edge in the graph and φ ∈ homF (a, b , S), then also (aφ, bφ) is an edge in the graph. Lemma 3.1.14. Let (z1 , z2 ) ∈ Z × Z and E = z1 , z2 . Then the following are equivalent: (1) (z1 , z2 ) ∈ D. (2) For each ϕ ∈ homF (E, S) with Eϕ ∈ F f , (z1 ϕ, z2 ϕ) ∈ ZS × ZS . (3) For some i ∈ {1, 2} and φ ∈ homF (E, S), there exists K ∈ Ω(zi φ) centralized by z3−i φ. (4) For each i ∈ {1, 2} and φ ∈ homF (E, S) with zi φ ∈ ZS , z3−i φ centralizes O(zi φ). Proof. We first prove (2) implies (4), so assume the setup of (2) and let i ∈ {1, 2} and φ ∈ homF (E, S) with zi φ ∈ ZS . Let α ∈ A(zi φ) and ψ = φα. By 3.1.5.4, Ω(zi φ)α = Ω(zi ψ). If Eψ centralizes Ω(zi ψ) then Eφ = Eψα−1 centralizes Ω(zi ψ)α−1 = Ω(zi φ), so that (4) holds. Hence, replacing φ by ψ, we may assume a = zi φ ∈ F f . Next δ = φ−1 ϕ ∈ homF (Eφ, S) and as a ∈ F f there exists β ∈ A(zi ϕ) with aδβ = a. Then γ = δβ is an Fa -map and as (z1 , z2 )ϕ ∈ ZS × ZS , we conclude that z3−i φγ = z3−i ϕβ centralizes Ω(a) by 3.1.5.4. But O(a)  Fa by 2.6.11 and CS (O(a)) ≤ CS (O(a)) by 2.6.12. Therefore also z3−i φ ∈ CS (Ω(a)), so (4) holds. That is (2) implies (4). Trivially (4) implies (3). Assume (3) and let α ∈ A(z3−i φ). By 3.1.2.2, z3−i φα ∈ ZS , and by 3.1.7.6, zi φα ∈ ZS , so (z1 , z2 )φα ∈ ZS × ZS , and hence (1) holds. That is (3) implies (1). Finally assume (1) holds, so that there is φ ∈ homF (E, S) with (z1 , z2 )φ ∈ ZS × ZS . Choose ϕ as in (2), and let α ∈ A(Eφ) with Eφα = Eϕ. Then Ωα ⊆ CS (Eφ)α ≤ CS (Eφα), so Ωα = Ω centralizes Eφα = Eϕ. Therefore (z1 , z2 )ϕ ∈ ZS × ZS , proving that (1) implies (2).  Lemma 3.1.15. Let (z1 , z2 ) ∈ Z × Z and E = z1 , z2 . Then the following are equivalent: (1) (z1 , z2 ) ∈ D ∗ . (2) For each ϕ ∈ homF (E, S) with Eϕ ∈ F f , (z1 ϕ, z2 ϕ) ∈ ZΔ × ZΔ . (3) For some i ∈ {1, 2} and φ ∈ homF (E, S), there exists K ∈ Ω(zi φ) such that |CK (z3−i φ)| > 2. (4) For each i ∈ {1, 2} and φ ∈ homF (E, S) such that zi φ ∈ ZS , z3−i φ induces an inner automorphism on each K ∈ Ω(zi φ). Proof. The proof is similar to that of 3.1.14. Again, trivially (4) implies (3). We begin the proof that (2) implies (4) as in 3.1.14; in particular, as in that proof, we may assume a = zi φ ∈ F f and define δ, β, and γ as in the proof of 3.1.14. Then as (z1 , z2 )ϕ ∈ ZΔ × ZΔ , Eϕ ≤ η for some η ∈ η(τ ), so Eφγ = Eϕβ ≤ ηβ with ηβ ∈ η(τ ) by 3.1.11.3. Let d = z3−i φ, b = dγ, and Ω(a) = {K1 , . . . , Kn }.

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As ηβ ∈ η(τ ), b = ux with u ∈ CS (O(a)) and x ∈ O(a). By 2.6.12, u centralizes O(a), so b induces the inner automorphism cx on O(a). Hence as dγ = b and γ ∈ homFa (Eα, Eαγ), with O(a)  Fa by 2.6.11, it follows that cd is also an inner automorphism of O(a). But then d induces an inner automorphism on each Ki , proving that (2) implies (4). We also begin the proof that (1) implies (2) as in 3.1.14. Then Eφ ≤ η for some η ∈ η(τ ), so Eϕ = Eφα ≤ ηα and ηα ∈ η(τ ) by 3.1.11. Therefore (1) implies (2) by 3.1.11.7. Finally assume (3) and let β ∈ A(z3−i φ). Set U = CK (z3−i φ)β, a = zi φβ, and b = z3−i φβ. By 3.1.3.2, U ≤ A ∈ Δ(a), so (a, b) ∈ ZΔ × ZΔ and hence (1) holds.  Definition 3.1.16. For i = 1, 2, let Vi ≤ Di ∈ Δ(τ ) with Vi cyclic of order m/2. Let zi be the involution in Vi and assume V1 , V2 = V1 × V2 and (z1 , z2 ) ∈ / D. Let αi ∈ homF (V1 V2 , S) with Vi αi ≤ Ki ∈ Ω. Define A(V1 , V2 , α1 , α2 ) = cki αi−∗ : i = 1, 2 ≤ AutF (V1 V2 ), for ki ∈ Ki − Vi αi , where cki ∈ Aut((V1 V2 )αi ) is conjugation by ki . In the next lemma we see that cki αi−∗ is independent of the choice of αi , so also A(V1 , V2 , α1 , α2 ) is independent of the choice of α1 and α2 . Thus we write A(V1 , V2 ) for A(V1 , V2 , α1 , α2 ) and ci for cki αi−∗ . Lemma 3.1.17. Assume the hypothesis of 3.1.16. Then (1) A(V1 , V2 , α1 , α2 ) is independent of the choice of α1 and α2 . (2) ci = cki αi−∗ is independent of αi . (3) AutA(V1 ,V2 ) (z1 , z2 ) = GL(z1 , z2 ). Proof. Let E = z1 , z2 and X = V1 , V2 . By hypothesis, X = V1 × V2 , and Vi αi ≤ Ki ∈ Ω. Then as V3−i centralizes Vi , V3−i αi acts on Ki by (QFP3). As / D, the equivalence of parts (1) and (3) of 3.1.14 says that z3−i αi does (z1 , z2 ) ∈ not centralize Ki , so cki induces the transvection on Eαi with center zi αi . Hence (3) holds. Let i ∈ {1, 2}, β1 ∈ A(zi ), a = zi β1 , β2 ∈ AFa (Xβ1 ), and set β = β1 β2 ; thus Xβ ∈ Faf . Let Vi β ≤ Ji ∈ Ω. Then γ = αi−1 β|Xαi ∈ homF (Xαi , S). Observe that CAut(X) (zi ) is a 2-group as X is abelian of rank 2, so B = AutFa (Xβ) is a 2-group (cf. 24.3 in [Asc86]). Further by choice of β, AutS (Xβ) is Sylow in B, so B = AutS (Xβ). Thus cki ∈ Nγ (cf. the definition of saturation in section 2), and hence γ extends to γˆ ∈ homF (Ki Xαi , S). Then as Vi αi γ = Vi β ≤ Ji , Ki αi γˆ = Ji , so cki γ ∗ = cj for j ∈ Ji − Vi β. Therefore cki αi−∗ = cki γ ∗ β −∗ = cj β −∗ . As this  holds for each choice of αi , (1) and (2) follow. Lemma 3.1.18. Assume the hypothesis of 3.1.16, and let A = A(V1 , V2 ), X = V1 V2 , and z = z2 α2 . Then either (1) A ∼ = S3 and given a generator v1 for V1 , we can pick a generator v2 for V2 such that [v3−i , ci ] = vi for i = 1, 2, or (2) A ∼ = D12 and there exists δ ∈ A of order 3 such that V1 δα2 ≤ J2 ∈ Ω(z) − {K2 }, α1−1 δα2 ∈ isoF (Xα1 , Xα2 ) extends to ρ ∈ homF (K1 (V2 α1 ), S) with K1 ρ = J2 , z1 α2 ∈ Φ(V1 α) ≤ K2 J2 , and Ω(z) = {K2 , J2 }. Proof. Let E = z1 , z2 and B = CA (E). As A = c1 , c2 is generated by a pair of involutions, A ∼ = D2r is dihedral, where r = |γ| and γ = c1 c2 . By 3.1.17.3,

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∼ AutA (E) = GL(E). As X is abelian and E = Ω1 (X), B is a 2-group, (cf. A/B = 24.3 in [Asc86]) we have r = 3 · 2a . Pick a generator v1 for V1 . As A/B acts faithfully as GL(E) on E, z1c2 = z1 z2 , so [v1 α2 , k2 ] ∈ V2 α2 is inverted by c2 and of order m/2, and hence (a) [v1 , c2 ] = v2 generates V2 , and (b) Vi = I(ci ), where I(ci ) = {x ∈ X : xci = x−1 }. Suppose first that a = 0, so that A ∼ = S3 . Then cA 2 = {c1 , c2 , c2 γ} is the set of involutions in A, so it follows from (b) that (c) V1A = I(c1 )A = {I(c) : c ∈ cA 1 } = {V1 , V2 , V3 } is of order 3, where V3 = I(c2 γ) = I(c2 c1 c2 ). But by (a), v1c1 = v1 v2 , so as V3 = I(cc12 ) = I(c1 )c2 = V1c2 , we conclude that (d) V3 = v1 v2 . Similarly [v2 , c1 ] = v1b for some odd integer b. Thus V3 = v2 v1b , and then b = 1 by (d). Therefore (1) holds in this case, so we may assume a > 0. As a > 0, A2 = c2 , cδ1 is a Sylow 2-subgroup of A, where δ ∈ A is of order 3 with z1 δ = z2 . Then z2 ∈ V1 δ = I(c1 )δ = I(cδ1 ). By 3.1.3.2, V1 δ is of index at most 2 in D(V1 δ) ∈ Δ(z2 ). If V1 δ = V2 , then I(cδ1 ) = I(c2 ), so ζ = cδ1 c2 centralizes V2 , and then as ζ centralizes δ, ζ centralizes X = V2 (V2 δ), contradicting |A2 | > 2 so ζ = 1. Therefore V2 = V1 δ, so as D(V1 δ) ∈ Δ(z2 ), V2 α2 and V1 α2 (δα2∗ ) are of index 2 in K2 and J2 ∈ Ω(z), respectively. As V2 and V2 δ commute and are distinct, K2 = J2 . As in the proof of 3.1.17, we may choose α2 so that Xα2 ∈ Fzf , and hence AutS (Xα2 ) = AutFz (Xα2 ). Then as ξ = α1−1 δα2 ∈ isoF (Xα1 , Xα2 ) maps z1 α1 to z, we conclude that ck1 ∈ Nξ , so ξ extends to ρ ∈ homF (K1 (V2 α1 ), CS (z)). Then as V1 α1 ξ = V1 δα2 ≤ J2 , K1 ρ = J2 . Hence cδ1 α2∗ = ck1 α1−∗ δ ∗ α2∗ = c1 ρ∗ = cj for j ∈ J2 − V1 δα2 . Further as K2 = J2 , [K2 , J2 ] = 1, so cδ1 α2∗ = cj commutes with ck2 = c2 α2∗ , and hence [cδ1 , c2 ] = 1. Therefore the Sylow 2-subgroup cδ1 , c2 of A is isomorphic to E4 , so a = 1 and A ∼ = D12 . Next V1 δα2 ∩ V2 α2 ≤ K2 ∩ J2 = z , so V1 δV2 is of index 2 in X. Therefore (v1 α2 )2 ∈ (V1 δV2 )α2 ≤ K2 J2 , so z1 α2 ∈ K2 J2 . Therefore z1 α2 centralizes each member of Ω(z) − {K2 , J2 }, so Ω(z) = {K2 , J2 } by 3.1.6. Therefore (2) holds in this case.  Definition 3.1.19. Let η ∈ η(τ ) and set W = η . For V ∈ η and α ∈ homF (W, S) with V α ≤ K ∈ Ω, define dV,η,α = ck α−∗ ∈ AutF (W ), for k ∈ K −V α, where ck ∈ Aut(W α) is conjugation by k. Observe that by 3.1.11.3, ηα ∈ η(τ ), so K acts on ηα by 3.1.12.1. Thus dV,η,α acts on η, and is indeed a member of AutF (W ). In the next lemma we find that dV,η,α is independent of the choice of α, so we write dV,η for this map. When the choice of η is clear, we also write dV for dV,η . Set Dη = {dV : V ∈ η} and μη = Dη . Thus μη ≤ AutF (W ). Lemma 3.1.20. Assume the hypothesis of 3.1.19, and for U ∈ η, write z(U ) for the involution in U . Then (1) dV = dV,η,α is independent of the choice of α. (2) dV inverts V and centralizes V  ∈ η − {V } whenever z(V  ) centralizes dV . (3) [W, dV ] ≤ V and [ZΔ , dV ] ≤ z(V ) . (4) For ϕ ∈ homF (W, S), dV,η ϕ∗ = dV ϕ,ηϕ . (5) μη  AutF (W ).

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(6) The map V → dV is an equivalence of the representations of AutF (W ) on η and Dη via conjugation. Proof. Assume the setup and notation of 3.1.19, and let E = ZΔ and z = z(V ). Let β1 ∈ A(z), a = zβ1 , β2 ∈ AFa (W β1 ), and set β = β1 β2 ; thus W β ∈ Faf , so AutSa (W β) is Sylow in AutFa (W β), for Sa Sylow in Fa . Let γ = α−1 β, regarded as an F-isomorphism from W α to W β mapping V α to V β. From 3.1.12.1, [Eα, ck ] ≤ zα , so (a) [Eβ, ck γ ∗ ] ≤ a . As W is abelian, CAut(W ) (E) is a 2-group (cf. 24.3 in [Asc86]), and from (a), (ck γ ∗ )|Eβ is in O2 (AutFa (Eβ)), so ck γ ∗ ∈ O2 (AutFa (W β)). Therefore γ extends to γˆ ∈ homF (W αK, S). Then K γˆ = J ∈ Ω and ck γ ∗ = cj , where j = kˆ γ ∈ J − V β. Thus cj β −∗ = ck α−∗ for each choice of α. This proves (1). Next V α ≤ K ∈ Ω(zα) with (dV α∗ )|W α conjugation by k ∈ K − V α, so as k inverts V α, dV inverts V . If z  = z(V  ) centralizes dV then z  α centralizes K = V α, k , so V  α centralizes K by 3.1.12.2. Thus dV centralizes V  , completing the proof of (2). Similarly by 3.1.12.1, [W α, K] ≤ W α ∩ K = V α, so [dV , W ] ≤ V , proving (3). Let ϕ ∈ homF (W, S). Then γ = ϕ−1 α ∈ homF (W ϕ, W α) and dV ϕ,ηϕ,γ = ck γ −∗ . Therefore dV,η ϕ∗ = ck α−∗ ϕ∗ = ck γ −∗ = dV ϕ,ηϕ , establishing (4). Then (4) implies (6), and by (6), AutF (W ) acts on Dη , so (5) follows.  Lemma 3.1.21. Let η ∈ η(τ ), and for i = 1, 2, let Vi ∈ η, zi = z(Vi ), and di = dVi . Then exactly one of the following holds: (1) z1 = z2 and [d1 , d2 ] = 1. (2) (z1 , z2 ) ∈ D and d1 , d2 ∼ = E4 centralizes z1 , z2 . (3) (z1 , z2 ) ∈ D ∗ − D and d1 , d2 ∼ = A(V1 , V2 ) ∼ = S3 or D12 . Proof. Set W = η , E = z1 , z2 , X = V1 V2 , di = dVi , and B = d1 , d2 ≤ AutF (W ). Suppose first that z1 = z2 and let α ∈ A(z1 ). Then Vi α ≤ Ki ∈ Ω(z1 ) and di = cki α−∗ , where ki ∈ Ki − Vi α. Thus either V1 = V2 and d1 = d2 , or V1 = V2 , so K1 = K2 , and hence [k1 , k2 ] ≤ [K1 , K2 ] = 1, so that also [d1 , d2 ] = 1. Thus (1) holds in this case. Next assume (z1 , z2 ) ∈ D, and let ϕ ∈ homF (W, S) with Eϕ ∈ F f . Then by 3.1.14, zi ϕ ∈ ZS for i = 1, 2, so Vi ϕ ≤ Ki ∈ Ω(zi ϕ) with [K1 , K2 ] = 1. Hence, as in the previous paragraph, (2) holds in this case. / D. As Vi ∈ η, zi ∈ ZΔ , so (z1 , z2 ) ∈ D ∗ . Finally suppose z1 = z2 and (z1 , z2 ) ∈ / D, the hypothesis of 3.1.16 are satisfied for any αi ∈ homF (W, S) Also as (z1 , z2 ) ∈ with Vi αi ≤ Ki ∈ Ω. Then A = A(V1 , V2 ) = ¯ cki βi−∗ : i = 1, 2 , where ki ∈ Ki − Vi αi , c¯ki ∈ Aut(Xαi ) is conjugation by ki , and βi = αi|X . On the other hand c¯ki = (cki )|Xαi , so c¯ki βi−∗ = di|X , and hence A = AutB (X) = Bρ, where ρ : B → A is the restriction map ρ : b → b|X . Now B is dihedral of order 2k, where k is the number of involutions in B conjugate to d1 or d2 . By 3.1.20.6, k is also the number of conjugates of V1 or V2 under B, and as all these conjugates are in X, this is also the number of conjugates of V1 or V2 under A, which, from the proof of 3.1.18, is the number of involutions

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conjugate to d1 ρ or d2 ρ. That is |B| = 2k = |A|, so ρ : B → A is an isomorphism. Hence (3) holds by 3.1.18, completing the proof.  Lemma 3.1.22. Let η ∈ η(τ ) and for i = 1, 2 let Vi ∈ η, zi = z(Vi ), and di = dVi . Assume z1 = z2 . Then the following are equivalent: (1) [z1 , d2 ] = 1. (2) [d1 , d2 ] = 1. (3) [d1 , z2 ] = 1. (4) [V1 , d2 ] = 1. (5) [d1 , V2 ] = 1. Proof. By 3.1.20.2, (1) implies (4) and (3) implies (5), while trivially (4) implies (1) and (5) implies (3). By 3.1.21, (2) is equivalent to (1) and to (3), completing the proof.  Lemma 3.1.23. Let η ∈ η(τ ), μ = μη , W = η , and for i = 1, 2 let Vi ∈ η, zi = z(Vi ), and di = dVi . Suppose ν = d1 , d2 ∼ = D12 and set ην = V1ν ∪ V2ν . Then: μ ν (1) Vi = Vi is of order 3 for i = 1, 2, and ην is of order 6. (2) |Ω(zi αi )| = 2 for i = 1, 2 and αi ∈ A(zi ). (3) μ = ν × ν  , where ν  = dV : V ∈ η − ην . (4) ν  centralizes V1 V2 , ν centralizes W  = η − ην , and W = V1 V2 × W  . Proof. Part (2) follows from 3.1.21 and 3.1.18. Next Viμ = Viν from (3) and (4), and hence |ηnu | = 6 by 3.1.20.6. Therefore (1) follows from (3) and (4), so it remains to prove those assertions. Let η  = η − ην and V ∈ η  . By 3.1.20.3, dV centralizes a hyperplane of F = ZΔ , so dV centralizes some member of E # , where E = z1 , z2 , which we may take to be z2 . Then by 3.1.22, dV centralizes V2 and the member V1 δ of V1ν with z2 ∈ V1 δ. But by 3.1.18, z1 ∈ V1 δV2 , so dV centralizes E and hence dV centralizes X = V1 V2 and ν by 3.1.22. Therefore ν  centralizes ν and X. Similarly as ν centralizes ν  , ν centralizes W  by 3.1.22, so ν is faithful on X, and then μ = ν × ν  . For δ of order 3 in ν, CX (δ) = 1, so W = X × W  , completing the proof of (3) and (4) and the lemma.  ˆ = {K ∈ Ω : Δ − {K} ⊆ CS (K)}, kˆ = |Ω|, ˆ and for Definition 3.1.24. Set Ω ˆ ˆ ˆ ˆη = Dη . η ∈ η(τ ), set ηˆ = {η ∩ K : K ∈ Ω}, Dη = {dV : V ∈ ηˆ}, and μ ˆ =D ˆη. Lemma 3.1.25. Let η ∈ η(τ ), μ = μη , W = η , D = Dη , μ ˆ=μ ˆη , and D Then there exist subsets ηi of η and Di of D such that: (1) We have   ˆ η = ηˆ  ηi and D = D Di , 0≤i≤r

0≤i≤r

such that μ = μ ˆ × μ0 × μ1 × · · · × μr , where μi = Di , μ ˆ∼ = E2kˆ , D0 is a set of for 1 ≤ i ≤ r. D 3-transpositions of μ0 , and μi ∼ = 12 ˆW ¯ with W ˆ ∩W ¯ ≤ Ω1 (CW (μ0 )), where W ¯ = W0 × W1 × · · · × Wr (2) W = W 0 ˆ ¯ , and μi centralizes W ˆ and W = ˆ η , and for 0 ≤ i ≤ r, Wi = ηi , μ ˆ centralizes W and Wj for j = i. (3) For each Σ ∈ η(τ ),  ˆ Σ=Σ ηi , and μΣ ∼ = μη .

0≤i≤r

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¯ = {dV : V ∈ η¯}, etc. By definition of Ω, ˆ each K ∈ Ω ˆ Proof. Let η¯ = η − ηˆ, D centralizes WK = η − {VK } , where VK = W ∩ K. Thus dVK centralizes WK , and then by 3.1.22, dVK ∈ Z(μ). Then as dVK inverts VK ,  μ ˆ= dVK ∼ = E2kˆ , ˆ K∈Ω

ˆW ¯ with W ˆ ∩W ¯ ≤ C ˆ (ˆ ˆ and W = W ¯ centralizes W μ) ≤ Ω1 (W ). Moreover by 3.1.22, μ ˆ W , and then μ = μ ˆ×μ ¯. Next by 3.1.23,   ¯ = η¯ = ηi , D Di , 0≤i≤r

0≤i≤r

¯ = W0 × W1 × · · · × Wr , where Di = {dV : V ∈ ηi }, Wi = ηi , μi = Di , and W and for i > 0, μi ∼ = D12 with Wi = [W, μi ] and μi centralizes Wj for j = i. Finally by 3.1.18, D0 is a set of 3-transpositions of μ0 . Thus (1) and (2) hold. η ∪ Δ ), Let Σ ∈ η(τ ). Then Δ = Δ − Ω ⊆ Σ ∩ η. Further for V ∈ η − (ˆ  ¯ V = CK (U ) for some K ∈ Ω and U ∈ Δ , so by symmetry, V ∈ Σ. Thus η¯ = Σ. ˆ ¯ ˆ ¯ ¯ ¯ As W = W W , as Σ = Σ Σ , and as W = Σ , it follows that ¯ =μ ¯Σ , μ ¯ = dV,η : V ∈ η¯ ∼ = dV,Σ : V ∈ Σ ˆΣ , also μΣ ∼ so as μ ˆ∼ =μ = μ, completing the proof of (3) and the lemma. = E2kˆ ∼



Definition 3.1.26. By 3.1.25, μη is independent of the choice of η ∈ η(τ ). Thus we can define the Thompson group of τ to be μ(τ ) ∼ = μη for any η ∈ η(τ ). Lemma 3.1.27. (1) Δ(τ ◦ ) = Δ(τ ). (2) η(τ ◦ ) = η(τ ). (3) μ(τ ◦ ) = μ(τ ). Proof. Recall τ ◦ = (F ◦ , Ω), where F ◦ = [Ω]F . Let T be Sylow in F ◦ ; then T is strongly closed in S with respect to F. Let V ∈ Δ(τ ); then there exists K ∈ Ω and φ ∈ homF (V, K). As T is strongly closed, V ≤ T . Indeed either V ∈ Ω or using 3.1.21, |K : V φ| = 2 and we may choose K and φ so that V φ = CK (V ), φ acts on X = V φV , and S3 ∼ = Σ = ck , φ ≤ AutF (X) for k ∈ K −V φ. As F ◦  F, AutF ◦ (X)  AutF (X), so ◦ as ck ∈ AutF ◦ (X), we have Σ = cΣ k ≤ AutF ◦ (X). Thus V ∈ Δ(F ), establishing (1). Next for η ∈ η(F), η = Δ(τ ) − Ω ∪ η0 , where η0 is some choice of cyclic subgroups of index 2 from members of Ω. Hence (1) implies (2). Finally set W = η and μ = μ(τ ); by (2), W ∈ W (τ ◦ ). Now μ = D ≤ AutF (W ), where D is the set of AutF (W )-conjugates of the ck for k ∈ K − W and K ∈ Ω. Then as AutF ◦ (W )  AutF (W ), D ⊆ AutF ◦ (W ), so μ ∈ μ(τ ◦ ), proving (3).  3.2. The graph A In this section assume τ = (F, Ω) is a quaternion fusion packet, and adopt the notation from earlier sections. Given a graph G and a vertex v of G, write G(v) for the set of vertices adjacent to v in G.

3.2. THE GRAPH A

65

Definition 3.2.1. Given (z1 , z2 ) ∈ Z × Z, define Φ(z1 , z2 ) to be the set of φ ∈ homF (z1 , z2 , S) such that (z1 , z2 )φ ∈ ZΔ × ZΔ , and for some η ∈ η(τ ), A(V1 , V2 ) ∼ = D12 , where zi φ ∈ Vi ∈ η. Note that by 3.1.21, Φ(z1 , z2 ) = ∅ when (z1 , z2 ) ∈ D. Define A = {(z1 , z2 ) ∈ Z × Z : Φ(z1 , z2 ) = ∅}. Visibly A is a symmetric F-invariant relation on Z, so we can regard A as a graph on Z. Set W(z) = {t ∈ Z : D(z) = D(t)}. Lemma 3.2.2. Let z ∈ ZS and t ∈ A(z). Then ˜ is of order 2. (1) Ω(z) = {K, K} ˜ (2) t ∈ K K. (3) t ∈ D∗ (z) − D(z). Proof. Let z1 = z and z2 = t. As t ∈ A(z) there exists φ ∈ Φ(z, t), and η ∈ η(τ ) such that zi φ ∈ Vi ∈ η with A(V1 , V2 ) ∼ = D12 . By definition of η(τ ) in Definition 3.1.9, and by 3.1.11.5, Vi ≤ Di ∈ Δ with Vi cyclic of order m/2. Thus there is αi ∈ A(zi φ) with Vi αi ≤ Ki ∈ Ω(si ), for i = 1, 2, where si = zi φαi ; ˜ 1 } and that is the hypothesis of 3.1.16 is satisfied. By 3.1.18, Ω(s1 ) = {K1 , K ˜ 1 . Let α ∈ A(z) with zα = s1 . By 3.1.2.1, Ω(z)α = Ω(s1 ), so (1) s = z2 φα1 ∈ K1 K holds. Also ζ = α1−1 φ−1 α ∈ homFs1 (s, CS (s1 )), so as O(s1 )  Fs1 by 2.6.11, and ˜ 1 , it follows that tα = sζ ∈ K1 K ˜ 1 . Hence (2) holds. By (2), t induces as s ∈ K1 K a nontrivial inner automorphism on K, so (3) follows from 3.1.14 and 3.1.15.  Lemma 3.2.3. Let (z1 , z2 ) ∈ Z × Z with E = z1 , z2 ∼ = E4 . Then the following are equivalent: (1) z2 ∈ A(z1 ). (2) For each i ∈ {1, 2}, α ∈ A(zi ), and γ ∈ AFzi α (Eα), we have αγ ∈ Φ(z1 , z2 ). (3) There exists φ ∈ homF (E, S) with (z1 , z2 )φ ∈ ZΔ × ZΔ , and for each such φ we have φ ∈ Φ(z1 , z2 ). Proof. Assume (1) and let α ∈ A(zi ) and γ ∈ AFa (Eα), where a = zi α. Set ρ = αγ. Then there exists ϕ ∈ Φ(z1 , z2 ), we have δ = ρ−1 ϕ ∈ homF (Eρ, S), and there exists β ∈ homF (CS (zi ϕ), S) with aδβ = a. As ϕ ∈ Φ(z1 , z2 ), there is η ∈ η(τ ) and for j = 1, 2, zj ϕ ∈ Vj ∈ η such that A(V1 , V2 ) ∼ = D12 . Now by 3.1.11.3, ηβ ∈ η(τ ), and zj ϕβ ∈ Vj β ∈ ηβ. Moreover D12 ∼ = A(V1 , V2 )β ∗ = A(V1 β, V2 β). Finally by 3.1.17.3, AutF (Eϕ) = GL(Eϕ), so by 2.5 and as Eρ ∈ Faf , there is χ ∈ AFa (Eϕβ) with z2 ϕβχ = z2 ρ. Then D12 ∼ = A(V1 β, V2 β)χ∗ = A(V1 βχ, V2 βχ), so ρ ∈ Φ(z1 , z2 ). That is (1) implies (2). Trivially (3) implies (1), so it remains to assume (2) and show that (3) holds. Assume the setup of (2), and let α ∈ A(z1 ) and γ ∈ AFa (Eα), where a = z1 α and ρ = αγ. By (2), ρ ∈ Φ(z1 , z2 ). Next choose φ as in (3); then there exists β ∈ A(z1 φ) with z1 φβ = a. As (z1 , z2 )φ ∈ ZΔ × ZΔ , zi φ ∈ Vi ∈ η ∈ η(τ ). By 3.1.11.3, ηβ ∈ η(τ ), and as AutF (Eρ) = GL(Eρ) by 3.1.17.3, it follows from 1.2.5 that there is χ ∈ AFa (Eφβ) with z2 φβχ = z2 ρ. Then A(V1 , V2 )(βχ)∗ ∼ = D12 , = A(V1 βχ, V2 βχ) ∼ so indeed (2) implies (3).  Lemma 3.2.4. Assume z ∈ ZS with A(z) = ∅. Then (1) |Ω(z)| = 2. (2) There exists a unique 4-subgroup E of S containing z such that A(z)∩ZΔ = E − z . (3) E ≤ O(z).

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(4) (5) (6) (7) (8)

A(z) = D∗ (z) − D(z). A(z) ⊆ O(z). A(z) = Z ∩ O(z) − {z}, W(z) ∩ D∗ (z) = A(z). If z ∈ F f then Fz is transitive on A(z).

Proof. Let t ∈ A(z), E = z, t , and η ∈ η(τ ). Replacing (z, t) by (z, t)αγ for α ∈ A(z) and γ ∈ AFzα (Eα), and appealing to 3.2.3, we may assume z ∈ F f and t ∈ ZΔ with z ∈ V ∈ η, t ∈ U ∈ η, and A(V, U ) ∼ = D12 . By parts (1) and (2) of 3.2.2, (1) and (3) hold with E = z, t in (3). For K ∈ Ω(z), [E, K] = z , so as t ∈ A(z), also tz ∈ A(z). By 3.1.25.1, μ = μ(τ ) = μ1 × μ2 , where μ1 = dV , dU ∼ = A(V, U ) and D = D1 ∪D2 , where Di = D ∩μi . In particular for x ∈ ZΔ −E, and x ∈ X ∈ η, dX centralizes μ1 , so x ∈ D(z) by 3.1.21. Therefore (2) holds, and (∗)

ZΔ − D(z) = E #

By 3.2.2.3, A(z) ⊆ D∗ (z) − D(z). Let y ∈ D∗ (z) − D(z), Y = z, y , and ϕ ∈ AFz (Y ). By 3.1.21, AutF (Y ) = GL(Y ), so by 2.5, Y ϕ ∈ F f , and then by 3.1.25, yϕ ∈ ZΔ . Then yϕ ∈ M ∈ η, so by (*), yϕ ∈ {t, tz}. Therefore as {t, tz} ⊆ A(z), also y ∈ A(z), completing the proof of (4). Let s ∈ A(z). From paragraph one there is γ ∈ AFz (s) with sγ ∈ V  ∈ Δ, so sγ ∈ E by (2), proving (8). Then as O(z)  Fz by 2.6.11, s ∈ O(z) by (3), proving (5). Let B = Z ∩ O(z) − {z}; by (5), A(z) ⊆ B. On the other hand if b ∈ B then b induces a nontrivial inner automorphism on K, so by 3.1.15, b ∈ D∗ (z) − D(z) and hence b ∈ A(z) by (4). This proves (6). Suppose w ∈ D(z). Then w centralizes Ω(z) by 3.1.14, so for β ∈ A(w), Ω(z)β centralizes Ω(wβ) by 3.1.5.1. Then by (5), A(z)β centralizes Ω(wβ), so A(z) ⊆ D(w). That is D(z) ⊆ D(s) for s ∈ A(z). But as AutF (z, s ) is transitive on z, s # , |D(z)| = |D(s)|, so D(z) = D(s); that is A(z) ⊆ W(z). On the other hand let w ∈ W(z). Then w ∈ / D(z) as w ∈ / D(w). Hence if  w ∈ D∗ (z) then w ∈ A(z) by (4), completing the proof of (7).

3.3. More basic lemmas In this section we assume τ = (F, Ω) is a quaternion fusion packet and adopt Notation 2.5.1 and 3.1.1. Lemma 3.3.1. Let z ∈ ZS ∩ F f . Then τz = (Fz , Ω) is a quaternion fusion packet. Proof. By 3.1.5.1, z centralizes Ω, so the lemma follows from 2.5.3.2.



Lemma 3.3.2. Let T be a subgroup of S strongly closed in S with respect to F such that T ∩ Z = ∅. Set F + = F/T , let Θ : F → F + be the natural surjection, and for P ≤ S and φ ∈ homF (P, S) set P + = P Θ and φ+ = φΘ. Then (1) T ≤ CS (Ω). (2) τ + = (F + , Ω+ ) is a quaternion fusion packet. (3) The map K → K + is a bijection of Ω with Ω+ . (4) Assume T  F and F = [Ω]F . Then T ≤ Z(F) and F + = [Ω+ ]F + .

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Proof. As T ∩ Z = ∅, (1) follows from 3.1.4. We next prove (2). By 1.2.7.1, F + is saturated, and as Ω is F-invariant, Ω+ is F + -invariant. Thus it remains to verify that τ + = (F + , Ω+ ) satisfies (QPF1)(QPF4). Let z ∈ ZS and K ∈ Ω(z). As Z ∩ T = ∅, K ∼ = K + , so τ + satisfies (QPF1). + Let t ∈ Z; then t = z + iff t ∈ zT . Suppose t+ = z + . Then t ∈ zT so by (1) and 3.1.5.1, t ∈ CS (Ω), and hence t ∈ ZS by 3.1.5.3. Thus there is J ∈ Ω(t), and if J = K then by (1), K ∩ JT ≤ Z(K) = z . Hence (3) holds and K + ∩ J + ≤ z + , so τ + satisfies (QPF2). Assume V ≤ K with |V | > 2 and φ+ ∈ homF + (V + , S + ). Then zφ ∈ zT , say z+

zφ = t. By 3.1.3.2, V φ ≤ D ∈ Δ(t), and by the previous paragraph, t ∈ ZS , so D ∈ Ω(t) and hence D+ ∈ Ω+ (z + ). Therefore τ + satisfies (QPF4). Assume L+ ∈ Ω+ , u+ ∈ L+ − Z(L+ ), and ψ + ∈ homF + (u+ , CS + (z + )). We may assume i = z(L) ∈ F f , and choose α ∈ A(iψ) with iψα = i. Set w = uψ. Then ψα is an Fi -map, so wα = uψα ∈ L ∈ Ω(i) by 2.6.2. Then w ∈ D ∈ Δ(iψ), and D acts on K by 3.1.12. Therefore w+ acts on K + , so τ + satisfies (QPF3), completing the proof of (2). Assume the hypothesis of (4). As T  F, CF (T )  F by Theorem 4 in [Asc11]. Then by (1) and as F = [Ω]F , T ≤ Z(F). Let E + = [Ω+ ]F + . As T ≤ Z(S), the preimage E of E + in F under Θ is normal in F by 1.2.7. As Ω+ ⊆ E + , Ω ⊆ E, so  E = F as F = [Ω]F . Therefore E + = F + , completing the proof of (4). Lemma 3.3.3. Let T be a subgroup of S strongly closed in S with respect to F, let K ∈ Ω, and Ω0 = (K ∩ T )F . (1) If K ∩ T is nonabelian then |K : K ∩ T | ≤ 2 and (F, Ω0 ) is a quaternion fusion packet. (2) Assume |K ∩ T | > 2 and OK = FK (K). Then K ∩ T is nonabelian and if K ∩ T = K then OK ∼ = SL12 [m/2]. Proof. Set J = K ∩ T ; then J is strongly closed in K with respect to OK , and in particular J  K. Suppose J is nonabelian. Then as J  K, |K : J| ≤ 2, and in any event J is quaternion. Now it is straightforward to verify that (F, Ω0 ) is a quaternion fusion packet, proving (1). Assume the hypotheses of (2). As OK is not FK (K), it follows from 2.5.5.1 that OK ∼ = SL2 [m] or SL12 [m/2]. Then as J is strongly closed in K with respect to OK , it follows that either K = J or OK ∼ = SL12 [m/2] and |K : J| = 2 with J nonabelian. This proves (2).  Recall the graphs D and D∗ on Z defined in Definition 3.1.13. For z ∈ Z, set z ⊥ = {z} ∪ D(z). Lemma 3.3.4. Let z ∈ ZS . Then z ⊥ = CZ (O(z)). Proof. First if t ∈ CZ (O(z)) then condition (3) of 3.1.14 is satisfied for (z, t) with φ = 1, so t ∈ z ⊥ by 3.1.14. Therefore CZ (O(z)) ⊆ z ⊥ . Next let t ∈ D(z). Then t ∈ CZ (O(z)) by the equivalence of parts (1) and (4) of 3.1.14 with φ = 1. 

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Lemma 3.3.5. Let z ∈ ZS and z = t ∈ Z. Then the following are equivalent: (1) t ∈ D∗ (z). (2) |CK (t)| > 2 for some K ∈ Ω(z). (3) t induces an inner automorphism on each member of Ω(z). Proof. Observe (2) holds iff condition (3) of 3.1.15 is satisfied for (z, t) with φ = 1. Thus as conditions (1), (3), and (4) of 3.1.15 are equivalent, the lemma follows.  Lemma 3.3.6. Let K ∈ Ω, z = z(K), and assume t ∈ Z − {z}. Set E = z, t . Then exactly one of the following holds: (1) E ∼ = D8 and K = K t . (2) E ∼ = E4 and K = K t ∈ Ω(z). (3) E ∼ = E4 , [t, K] = 1 and t ∈ D(z). (4) E ∼ = E4 , t induces a nontrivial inner automorphism on K, and t ∈ D∗ (z) − D(z). (5) E ∼ = E4 and t induces an outer automorphism on K with CK (t) = z . In particular Kt is semidihedral. Proof. If K = K t then [K, K t ] = 1 by 3.1.5.1, so either z t = z and (1) holds, or z = z and (2) holds. Thus we may assume t acts on K. If [K, t] = 1 then (3) holds by 3.1.14, so assume [K, t] = 1. If t induces an inner automorphism on K then (4) holds by 3.3.5. So assume t induces an outer automorphism on K. Then as parts (2) and (3) of 3.3.5 are equivalent, (5) holds.  t

Recall from the introduction to the chapter that WS = ZS . Lemma 3.3.7. Set N = NF (WS ), Γ = {OK : K ∈ Ω}, and O(τ ) = [O(τ )]N . Then (1) O(τ ) is a central product of the members of Γ. (2) O(τ )  N . (3) AutF (O(τ )) permutes Γ. Proof. As WS  S, WS ∈ F f , so N is saturated. By 2.5.2, (N , Ω) is a quaternion fusion packet, so, replacing F by N , we may assume WS  F. Set O = O(τ ) and O = O(τ ). Part (2) holds by definition of O(τ ) and (3) follows as Ω is F-invariant and AutF (K) ≤ Aut(OK ) by 2.5.7.4. Let K ∈ Ω, z = z(K), and Q ∈ QK . If α ∈ AutF (Q) is of order 3 then by 2.6.6.3, α × 1 ∈ AutF (QCS (Q)), so by 2.3 in [Asc11], the central product E of  the members of Γ is a subsystem of F. By 2.5.7.3, E = O 2 (E), so E ≤ O. Also O ≤ CF (WS ) = D, and D  F by Theorem 4 in [Asc11]. By 2.5.2, ρ = (D, Ω) is a quaternion fusion packet. By 2.6.11, O(z)  D. Then by Theorem 3 in [Asc11], E  D, so E = [O(τ )]D . Now as Ω is F-invariant, E  F by 7.4 in [Asc11], so E = O, completing the proof of (1) and the lemma.  Lemma 3.3.8. Set S0 = O2 (F) and assume K ∈ Ω with K0 = K ∩ S0 nonabelian. Set Ω0 = K0F , Ω1 = K F , and let O1 be the product of OJ , J ∈ Ω1 in O(τ ). Then (1) τi = (F, Ωi ) is a quaternion fusion packet for i = 0, 1. (2) z F = {z(J) : J ∈ Ω1 } and E = z F  F. (3) O1  F is the central product of the OJ for J ∈ Ω1 .

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(4) |K : K0 | ≤ 2 and in case of equality, m = 16 and OK ∼ = SL12 [8]. (5) O1 = [Ω1 ]F centralizes [Ω − Ω1 ]F . Proof. Suppose (3) holds and let Ω2 = Ω − Ω1 . By (3), O1 = [Ω1 ]F . By 3.3.7.1, Ω2 ⊆ CF (O1 ), so as CF (O1 )  F by (3) and Theorem 4 in [Asc11], we conclude that [Ω2 ]F ≤ CF (O1 ), completing the proof of (5). Thus (3) implies (5), so it remains to prove (1)-(4). First τ1 is a quaternion fusion packet by 2.5.2. Thus as (1)-(4) are statements about τ1 , replacing Ω by Ω1 , we may assume F is transitive on Ω. By 3.3.3.1, τ0 is a quaternion fusion packet, completing the proof of (1); further |K : K0 | ≤ 2 by 3.3.3.1. Let z = z(K); as Ω = K F , Z = z F . But for φ ∈ homF (z, S), φ extends to ϕ ∈ AutF (S0 ), so zφ = zϕ generates Z(K0 ϕ), and hence Z = ZS . Therefore E = ZS = Z  F, establishing (2). Then (3) follows from 3.3.7. Finally suppose |K : K0 | = 2. Then by (3), OK = FK (K), so OK ∼ = SL12 [8] by  3.3.3.2 and the fact that K0 ≤ O2 (OK ), completing the proof of (4). Lemma 3.3.9. Let η ∈ η(τ ) and assume W = η  F. Then (1) For K ∈ Ω, OK = FK (K) and AutF (K) is a 2-group. (2) M = [O(τ )]F is constrained. (3) Let M be a model for M. Then W  M = O(τ )M and M/CM (W ) ∼ = μ(τ ). Proof. As V = K ∩W is cyclic of index 2 in K and normal in OK , (1) follows. Replacing F by M, we may assume F = M. Let z = z(K); we may assume z ∈ F f . As z ∈ W ≤ O2 (F), D = E(F) = E(Fz ) by 10.3 in [Asc11]. By (1) and 2.6.11, K ≤ O2 (Fz ) ≤ CFz (E(Fz )) = CF (D), so F = [O(τ )]F ≤ CF (D) and hence D = 1. Then (2) follows from 14.2 in [Asc11]. ∗ As F = [O(τ )]F , M = O(τ )M . Set M ∗ = M/CM (W ); then M ∗ = O(τ )∗M , and from Definition 3.1.19, K ∗ = dV , so M ∗ = dU : U ∈ η = μ(τ ), completing the proof of (3).  Lemma 3.3.10. Let η ∈ η(τ ). Then (1) η = (Δ(τ ) − Ω) ∪ {VK : K ∈ Ω} for some choice of cyclic subgroups VK of index 2 in K. (2) η is weakly closed in CS (η) with respect to F. Proof. From conditions (ii) and (iii) of 3.1.9, η = {VD : D ∈ Δ}, where VD = D if D ∈ / Ω and |D : VD | = 2 if D ∈ Ω. Therefore (1) holds. Suppose φ ∈ homF (η, CS (η)) with ξ = ηφ = η. By 3.1.11.3, ξ ∈ η(τ ), so by (1) applied to ξ we have ξ = (Δ − Ω) ∪ {UK : K ∈ Ω} with |K : UK | = 2. Therefore for some K ∈ Ω, VK = UK . This is a contradiction as UK ≤ CS (η) ≤ CS (VK ).  Notation 3.3.11. Recall from Definition 3.1.9 that W (τ ) = {η : η ∈ ηS (τ )}. For W ∈ W (τ ) set τW = (NF (W ), Ω). As W = η for some η ∈ ηS (τ ), we have W  S, so W ∈ F f , and hence NF (W ) is saturated by 1.2.1. Hence τW is a quaternion fusion packet by 2.5.2. By 3.3.9.2 applied to τW in the role of τ , [O(τ )]NF (W ) is constrained, and hence has a model M (W ). By 3.3.9.3, M (W )/CM (W ) (W ) ∼ = μ(τ ). Write M (τ ) for the set of groups M (W ), W ∈ W (τ ). As W is normal in S, so is CS (W ), so NF (CS (W )) is saturated and hence has a model GW . By 3.3.10.2, W  NF (CS (W )), so [O(τ )]NF (W )  NF (CS (W )), so we may take M (W )  GW by 1.2.8.4.

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Lemma 3.3.12. Set S0 = O2 (F) and assume K ∈ Ω such that K0 = K ∩ S0 is cyclic of order at least 4. Set Ω1 = K F , η1 = K0F , and E = z F . Then (1) τ1 = (F, Ω1 ) is a quaternion fusion packet. (2) OK = FK (K) and AutF (K) is a 2-group. (3) z F ⊆ ZΔ but z F is not contained in ZS . (4) |K : K0 | = 2, {η1 } = η(τ1 ), and η1 is contained in each member of η(τ ). ˜ F where Ω(z) = {K, K}; ˜ (5) If A(z) = ∅, set Ω2 = Ω1 ∪ K otherwise set ˜ ˜ Ω2 = Ω1 . Then [Ω2 ]F centralizes [Ω − Ω2 ]F , and K ∩ S0 is cyclic of index 2 in K. (6) η1  F. Proof. Part (1) follows from 2.5.2. As K0 is cyclic of order at least 4 and invariant under AutF (K), (2) follows. By construction, η1 is an F-invariant subset of S0 , so W1 = η1  F, proving (6). Suppose (4) holds; we prove (5) holds, so adopt the notation of (5), let J ∈ Ω − Ω2 with t = z(J) ∈ F f , and set Ω3 = J F . As (4) holds, η1 is contained in each η ∈ η(τ ), and we can apply 3.1.25 to η. In particular we obtain the direct product ˜ it follows decomposition of μ in that lemma, and as J is not fused to K or K, that dK0 and dJ0 are in different factors of μ, and hence commute. Therefore J centralizes K by 3.1.22. As this holds for each K ∈ Ω1 , it follows that J centralizes W1 . Then E = [Ω3 ]F centralizes W1 , and hence also centralizes O(z) as J centralizes O(z). Therefore [Ω2 ] centralizes E; that is (4) implies (5). Further (3) is a statement about τ1 , so replacing Ω by Ω1 and appealing to (1), we may assume F is transitive on Ω, keeping in mind that to prove (4) in this case we must show η1 is the unique member of η(τ ). Define a map ξ : η1 → Δ, where for U ∈ η1 , ξ(U ) is the member of Δ containing U . If U and V are distinct members of η1 and ξ(U ) = ξ(V ), then as |U | = |V | > 2, U, V is a nonabelian subgroup of ξ(U ) ∩ S0 . But ξ(U ) ∩ S0 is conjugate to the cyclic group K0 , a contradiction. Thus ξ is an injection. As Ω = K F , Z = z F . Then for each t ∈ Z, t ∈ K0 φ for some φ ∈ AutF (S0 ), so Z = ZΔ . Hence ξ is a bijection. Suppose ZΔ = ZS . Then WS = ZS  F, so by 3.3.7, K ≤ S0 , contrary to hypothesis. This completes the proof of (3). / ZS . Let Set z = z(K). By (3) there is ψ ∈ AutF (S0 ) such that t = zψ ∈ / ZS , D ∈ η for each η ∈ η(F, Ω). By D0 = K0 ψ and D = ξ(D0 ) ∈ Δ(t). As t ∈ 3.1.5.3 and the transitivity of F on Ω, we may choose D0 so that [K, t] = 1. Hence by parts (3) and (5) of 3.1.7, V = CK (t) is cyclic of index 2 in K and the unique such D-invariant subgroup of K. Therefore as W = η is abelian, V = K ∩ W . Further K0 = K ∩ S0 is KD-invariant, so CK (K0 ) is cyclic of index 2 in K, and hence V = CK (K0 ). It follows that η = {Cξ(X) (X) : X ∈ η1 }, so in particular η(τ ) = {η} and W1 ≤ W . Thus W ≤ CF (W1 ) ≤ Fz , so W ∩ O2 (Fz ) ≤ O2 (CF (W1 )) ≤ S0 . By (2), V ≤ O2 (Fz ), so V ≤ K ∩ S0 = K0 and hence V = K0 . Thus η1 = η, completing the proof of (4) and the lemma.  Lemma 3.3.13. Assume K ∈ Ω such that z = z(K) ∈ O2 (F) and OK = FK (K). Then O 2 (OK ) is a component or solvable component of F. Proof. As z ∈ O2 (F), it follows from 10.3 in [Asc11] that E(F) = E(Fz ). As OK = FK (K), C = O 2 (OK ) is a component or solvable component of Fz by 2.6.13.2. In the former case as E(F) = E(Fz ), C is a component of F, so we may

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assume C ∼ = SL2 [8]. Then OK centralizes E(Fz ) = E(F), so OK ≤ CF (E(F)) = Y. Therefore replacing F by Y, we may assume E(F) = 1. Therefore F is constrained by 14.2 in [Asc11], so F has a model G. Set Q = O2 (G) and L = NG (WS ). Then Q ≤ S ≤ L. By 3.3.7, a model H for O(τ ) is normal in L and H = H1 · · · Hn is a central product of the models Hi for OKi , where {K1 , . . . , Kn } = Ω. Take K = K1 ; then H1 ∼ = SL2 (3)(1) by u 3.3.3.2. Now if u ∈ Q − NQ (K) then H1 centralizes H1 , so [H1 , u] is not a 2-group, a contradiction. Therefore Q ≤ NG (K) ≤ CG (z). Then as L1 = O 2 (H1 ) is a solvable component of CG (z), it follows from 2.2.4.4 that Q1 = [Q1 Q, L1 ], where Q1 = O2 (L1 ). Hence as CG (Q) ≤ Q, we have Q1 ≤ Q. Therefore by 3.3.8.3, H1G is the central product of the conjugates of H1 , so H1 is a solvable component of G, completing the proof of the lemma.  Lemma 3.3.14. Assume F is transitive on Ω. Then either (1) η(τ ) = {η} is of order 1, or (2) m = 8, Δ(τ ) = Ω, and for each η ∈ η(τ ), NF (η) ≤ NF (O(τ )). Proof. By 3.1.11.4, either (1) holds or m = 8. Similarly for η, Σ ∈ η(τ ), it ˆ = ∅, where Ω ˆ = {J ∈ Ω : Δ − {J} ⊆ follows from 3.3.10.1 that either Σ = η or Ω ˆ ˆ and then CS (J)}. But if Ω = ∅, then as F is transitive on Ω we have Ω = Ω, Δ = Ω. Set W = η . By 3.3.9 and as Δ = Ω, O(τ )  NF (W ), so (2) holds, completing the proof of the lemma.  Lemma 3.3.15. Assume F is transitive on Ω, K ∈ Ω, z = z(K) ∈ F f , and Ω(z) = {K}. (1) Assume s, t ∈ Z − D∗ (z) with E = z, t, s ∼ = E8 . Then AutF (E) = CGL(E) (e) for some e ∈ E. (2) Assume D ∗ (z) = ∅ and z = t ∈ Z. Then Z = {z} ∪ tK . Proof. Assume the setup of (1). As Ω(z) = {K}, E acts on K and then Kt and Ks are semidihedral of order 2m by 3.3.6. Let F = z, t ; then NK (F ) = v ∼ = Z4 with [v, t] = z. As F is transitive on Ω and z ∈ F f there is α ∈ A(t) with tα = z. Then v = NK (E) and [E, v] = [t, v] with [t, v] = z, so v induces a transvection cv = γ on E with center z . Similarly NK (Eα) = v and cv α−∗ = β ∈ AutF (E) is a transvection with center t and [z, β] = t. Thus γ, β induces GL(F ) on F . By symmetry there is a transvection δ ∈ AutF (E) with [z, δ] = s. Now A = γ, β, δ ≤ GL(E) is generated by transvections, so A = CGL(E) (e) for some e ∈ E, proving (1). Next assume (2) fails. As D∗ (z) = ∅ and Ω(z) = {K}, also Ω = {K}, so K  S. As (2) fails there is s ∈ Z − Kt and D = s, t is dihedral. We may choose s so that s and t induce the same automorphism on K, so x = st ∈ CS (K). Set / D E = z, t, s . If |D| = 4 then as s ∈ / tK , x = z so E ∼ = E8 . If |D| > 4 but z ∈ replacing D by CD (t) we may assume |D| = 4. Finally if z ∈ D then we may choose x of order 4. Then vx is an involution commuting with t and vxt = vs ∈ sK ⊆ Z, so replacing v by vs, again E ∼ = E8 . But now by (1), A = AutF (E) = CGL(V ) (e)  for some e ∈ E, so v centralizes a member of z A − {z}, a contradiction. Lemma 3.3.16. Assume τ = (F, Ω) is a quaternion fusion packet with F = F ◦ transitive on Ω. Let Z = Z(F) and F + = F/Z and assume that F + = FS + (L+ ) for some simple group L+ appearing in Theorem 1. Then (1) We may choose L+ to tamely realize F + .

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(2) F = FS (L) for some quasisimple group L such that Z(L) = Z and L/Z =

Proof. As L+ appears in Theorem 1, L+ is a known simple group, so by Theorems 3.5 and 3.6 in [AO16], we may choose L+ to tamely realize F + . Hence (1) holds. Then by 2.22 in [AO16], F = E(F)Z and F = FS (L) for some group L = E(L)Z. Let K ∈ Ω with z = z(K) ∈ F f . As L+ appears in Theorem 1, K + ≤ 2 O (CL+ (z + )). Then by 3.3.2.3, K ≤ O 2 (CL (z)). Hence as F = F ◦ , also F = O 2 (F), so L = O 2 (L). Then as L = E(L)Z, we conclude that L is quasisimple and (2) holds. 

3.4. Generating F In this section we assume the following hypothesis: Hypothesis 3.4.1. τ = (F, Ω) is a quaternion fusion packet, K ∈ Ω, z = z(K) ∈ F f , and F is transitive on Ω.  Definition 3.4.2. For ∅ = θ ⊆ ZS set O(θ) = t∈θ O(t). Define E = E(τ ) = SO(τ ), NF (O(θ)), NF (η) : η ∈ η(τ ) and ∅ = θ ⊆ ZS with O(θ) ∈ F f . (Recall the definition of T F0 from 8.19 in [Asc11].) Write F e for the set of Fessential subgroups of S. Let R ∈ F e , G a model for NF (R), and G∗ = G/R. Then NG (NS (R)) ≤ M ≤ G with M∗ strongly embedded in G∗ . Lemma 3.4.3. (1) Either (i) η(τ ) = {η} is of order 1, or (ii) m = 8, Δ = Ω, and for each η ∈ η(τ ), NF (η) ≤ NF (O(τ )). (2) Let 1 = X ≤ ZS = WS and α ∈ A(X). Then Xα ≤ WS , α ∈ NF (O(τ )), and α ∈ homE(τ ) (NS (X), S). Hence if NF (Xα) ≤ E(τ ) then NF (X) ≤ E(τ ). (3) For each ∅ = θ ⊆ ZS , NF (O(θ)) ≤ E(τ ). (4) For each ∅ = θ ⊆ ZS , NF (θ ) ≤ E(τ ). Proof. Part (1) is 3.3.14. Set O = O(τ ). Assume the setup of (2). Then O ≤ CS (WS ), so Oα ≤ S and hence Oα = O as O is weakly closed in S. Further WS  NF (O), so Xα ≤ WS . As NF (O) ≤ E by 3.4.2, we have α ∈ homE (NS (X), S). Finally suppose NF (Xα) ≤ E; then NF (X)α∗ ≤ NF (Xα) ≤ E, so as α is an E-map, we have (NF (X)α∗ )α−∗ ≤ E, completing the proof of (2). Now assume ∅ = θ ⊆ ZS and let P = O(θ) and α ∈ A(P ). Again P  O, so Oα = O and α is an E-map. As O(θα) = P α ∈ F f , NF (P α) ≤ E by 3.4.2, so NF (P ) ≤ E as in the previous paragraph, establishing (3). Set X = θ , β ∈ A(X), and Θ = ZS ∩ Xβ. Let Γ be the union of the Ω(t) for t ∈ Θ, N = NF (Xβ), and ρ = (N , Γ). Then ρ is a quaternion fusion packet by 2.5.2. Hence O(Θ)  N by 3.3.7.2 applied to ρ, so by 1.3.2 in [Asc19], N = NS (Xβ)O(Θ), NN (O(Θ)) . By (3), NF (O(Θ)) ≤ E and NS (Xβ)O(Θ) ≤ SO(τ ) ≤ E, so N ≤ E. Now (4) follows from (2). 

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Lemma 3.4.4. Let Q ≤ R with Q  G and set H = O 2 (CG (Q)). (1) If H ≤ R then CS (Q) ≤ R. (2) If H ≤ R then CS∩G (O 2 (H ∗ )) = R and AutF (R) = AutH (R)AutNM (S∩H) (R). Proof. Suppose H ≤ R and let X = RCS (Q). As G∗ ≤ M∗ , Z(G∗ ) is of odd order. However [G, NX (R)] ≤ [G, RH] = [G, R] ≤ R, so NX (R)∗ ≤ Z(G∗ ) and hence NX (R) ≤ R, proving (1). Suppose H ≤ R. Then H ∗ is normal in G∗ of even order, so by a Frattini argument, G = HNG (S ∩ H) and NG (S ∩ H) = NM (S ∩ H) as M∗ is strongly embedded in G∗ . This establishes the second statement in (2). Also G∗ = H ∗ M∗ , so as M∗ is proper in G∗ , H ∗ ≤ M∗ . Thus O 2 (H ∗ ) ≤ M∗ , so as M∗ is strongly  embedded in G∗ , CS ∗ (O 2 (H ∗ )) = 1. Hence (2) holds. Lemma 3.4.5. Assume P ∈ Δ with |P ∩ R| > 2. Then AutF (R) = AutE(τ ) (R). Proof. Assume otherwise and choose R of maximal order subject to J = R∩P of order at least 4 for some P ∈ Δ and AutF (R) = AutE (R). Observe: (a) If R < T ≤ S then homF (T, S) = homE (T, S). For let α ∈ homF (T, S). By the Alperin-Goldschmidt Fusion Theorem, α = α1 · · · αn where αi ∈ AutF (Ri ), Ri = S or Ri ∈ F e , T0 = T ≤ R1 , and Ti = T α1 · · · αi ≤ Ri+1 for i < n. Moreover if Ri ∈ F e then AutF (Ri ) = AutE (Ri ) by maximality of |R| and as R < T , while NF (S) ≤ NF (O(τ )) ≤ E. This proves (a). Let ΔR = {P ∈ Δ : |P ∩ R| > 2} and J = {P ∩ R : P ∈ ΔR }. For J ∈ J there is a unique DJ ∈ Δ with J ≤ DJ . Set Q = J and Q1 = CR (Q). Suppose for some J ∈ J that J is nonabelian and let j = z(J), θ = j G , and X = θ . As J is nonabelian, DL ∈ Ω for each L ∈ J G , so θ ⊆ ZS . Hence by 3.4.3.4, NF (R) ≤ NF (X) ≤ E, contrary to the choice of R. Therefore (b) Each member of J is cyclic. By (b) and 3.1.8: (c) Q is abelian. (d) CS (Q1 ) ≤ Q1 . Let S1 = CS (Q1 ) and S2 = NS1 (R). Then [S2 , R] ≤ CR (Q) ≤ Q1 ≤ CR (S2 ), so S2 ≤ R. Therefore S2 = S1 so S1 ≤ Q1 , establishing (d). (e) We may assume Q1 ∈ F f . Let α ∈ A(Q1 ); as Q1  G, Rα ∈ F e . By (a), α is an E-map, so if AutF (Rα) = AutE (Rα) then also AutF (R) = AutE (R), contrary to the choice of R. Hence, replacing R by Rα if necessary, (e) holds. By (d) and (e), Q1 ∈ F cr , so there is a model G1 for NF (Q1 ). As Q1  G, we may take G = NG1 (R). Set  NG1 (J). G2 = J∈J

Set J1 = {J ∈ J : DJ ∈ / Ω}, J3 = {J ∈ J : DJ ∈ Ω and [L, DJ ] = 1 for all L ∈ J1 }, and J2 = J − (J1 ∪ J3 ). As G = NG1 (R) and R is radical: (f) O2 (G1 ) ≤ R. By 3.1.11: (g) DJ ∈ η for all J ∈ J1 . By 3.1.5.1: (h) For distinct J, L ∈ J − J1 , we have [DJ , DL ] = 1.

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(i) J3 = ∅. Suppose J ∈ J3 . By (h), D = DJ centralizes each member of J − {J}, so X = ND (Q1 ) ≤ G2 . But X  NG1 (J), so X  G2 , and hence X ≤ O2 (G2 ) ≤ O2 (G1 ) ≤ R by (f). Therefore X = D ∩ R = J, so D = J ≤ R, contrary to (b). Hence (i) holds. (j) J ≤ W for each J ∈ J . If J ∈ J1 this is a consequence of (g), so by (i) we may take J ∈ J2 . Set D = DJ ; as J ∈ J2 there is L ∈ J1 with V = CD (L) < D. By 3.1.7.3, V = CD (DL ) and by 3.1.7.5, V is cyclic of index 2 in D. Thus as DL ∈ η, also V ∈ η, so J ≤ V ≤ W , establishing (j). By (j), Q ≤ W , so Q centralizes VJ = DJ ∩ W for each J ∈ J . Set X = NVJ (Q1 ); then X  NG1 (J), so X  G2 , and hence X ≤ R by (f). Therefore X = J, so X = VJ , proving: (k) J ⊆ η.  (l) Let H = O 2 (CG (Q)); then H ≤ R. For if H ≤ R then CS (Q) ≤ R by 3.4.4.1. But by (k), W ≤ CS (Q), while by (i), Δ = Ω, so W is weakly close in S by 3.4.3.1, so NF (R) ≤ NF (W ) ≤ E by 3.4.2, contrary to the choice of R. Set Ω1 = {K ∈ Ω ∩ NS (Q1 ) : K ∩ R ∈ J } and J4 = {K ∩ R : K ∈ Ω1 }. (m) Ω1 = ∅. Let J ∈ J and α ∈ A(J); then Jα is of index 2 in some K ∈ Ω by (k). Now [Q1 α, K] ≤ Jα ≤ Q1 α, so K ≤ NS (Q1 α). By (e) there is β ∈ A(Q1 α) with Q1 ζ = Q1 , where ζ = αβ. Then K0 = Kβ ≤ G1 is in Ω and K0 = DJζ , with Jζ = K0 ∩ Q1 = K0 ∩ R ∈J . That is K0 ∈ Ω1 . Set O1 = Ω1 , G4 = J∈J4 NG1 (J), and G5 = NG1 (J4 ). Therefore G4  G5 as G5 permutes J4 . Also O1 ≤ G4 and for J ∈ J4 , we have DJ  NG1 (J), so that DJ  G4 and hence DJ ≤ O2 (G4 ) ≤ O2 (G5 ). Hence (n) O1  G5 . Next R permutes Ω1 and hence R acts on O1 . Similarly H ≤ G4 ≤ G5 , so (o) O1  O1 RH ≤ G5 . Set Θ = {z(J) : J ∈ J4 }. Then O1 = O(Θ), so by 3.4.3.3, NF (O1 ) ≤ E. Then by (o): (p) AutH (R) ≤ AutE (R). By (a), (l), and the proof of (e), we have AutNM (S∩H) (R) ≤ AutE (R), so by (p) and 3.4.4.2, AutF (R) = AutE (R), contrary to the choice of R. This final contradiction completes the proof of the lemma.  Lemma 3.4.6. Assume Ω(z) = {K} and z ∈ Z(R). Then either AutF (R) = AutE (R) or the following hold: (1) Ω = {K}. (2) Z = {z} ∪ tK with Kt semidihedral of order 2m. (3) G = H × CG (H) with H = NK (R)G ∼ = S4 , O2 (H) = z G , R = CS (O2 (H)), and CG (H) is 2-closed. Proof. Assume R is a counter example. Then by 3.4.5: (a) |P ∩ R| ≤ 2 for each P ∈ Δ. Set Z = z G . As z ∈ Z(R), Z ≤ Z(R). Let v ∈ K be of order 4 with v  CS (z). As z ∈ Z(R) and Ω(z) = {K}, R acts on K, so [R, v] ≤ z ≤ R, and

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hence v ∈ NS (R). By (a), v ∈ / R, so [R, v] = z . Set H = v G . We conclude from 2.4.3 in [Asc19] that the following two results hold: (b) H ∼ = S4 and G = H × CG (H) with Z = O2 (H). (c) R = CS (Z) and CG (H) is 2-closed. Let t ∈ Z − z . (d) t induces an outer automorphism on K and Kt ∼ = SD2m . By (a) and (c), CK (t) = z , so the result follows from 3.3.6. (e) D∗ (z) = ∅. Assume D∗ (z) = ∅. Then D(z) = ∅, so (1) holds, and (2) follows from (d) and 3.3.15.2. Finally (3) follows from (b) and (c), contrary to the choice of R as a counter example. Notice that (e) shows that Lemma 3.4.6 holds when D∗ (z) = ∅. In the proof of Lemma 7.1.7, we appeal to this special case of Lemma 3.4.6. At the end of the proof of Lemma 3.4.6 we appeal to Lemmas 7.1.12 and 7.1.13, which are proved under the hypothesis that μ(τ ) ∼ = S3 . These lemmas do not depend on Lemma 3.4.6. (f) Suppose s ∈ tCS (K) ∩ R is not in Z, and set E = z, t, s . Then st or zst is in CS (H). For from (b) and (c), H acts on E, so AutH (E) centralizes e equal to st or zst, so e ∈ CS (H). (g) CZS (H) = ∅. Suppose X ∈ Ω with x = z(X) ∈ CS (H). Then z centralizes O(x) by 3.1.5.1, so Z = [Z, H] centralizes O(x) as O(x)  Fx . Therefore X ≤ CS (Z) = R, contrary to (a). (h) ZS ∩ R = {z}. Assume X ∈ Ω − {K} with x = z(X) ∈ R, and set θ = xR . Then R acts on O(θ). Let U be a CS (x)-invariant subgroup of X of order 4 and Y = U R . By (a), t inverts U , so t inverts Y . Thus Y0 = Ω1 (Y ) = CY (Z) = R ∩ Y . Then from (b) and (c), Y0 is of index 2 in Y1 = NY (R). Then z = [Y1 , t] ≤ B = θ . As t inverts Y , tx ∈ tY ⊆ tCS (K) , so by (f), x or xz is in CS (H); therefore xz ∈ CS (H) by (g). Therefore for all a, b ∈ θ, ab = (az)(bz) ∈ CS (H). Also n = |θ| is even, or else R acts on U and hence [t, U ] ≤ Z ∩ U = 1, a contradiction. Therefore z ∈ CB (R) ≤ ab : a, b ∈ θ ≤ CS (H), a contradiction. (i) Set R0 = CR (H) ∩ CR (K). Then R0  G and we may take R0 ∈ F f . The first statement in (i) follows from (b). Let α ∈ A(R0 ). As K ≤ NS (R0 ), 3.4.5 and the proof of statement (a) in the proof of 3.4.5 implies that α is an E-map. Then the proof of statement (e) in the proof of 3.4.5 shows that we may replace R by Rα to obtain R0 ∈ F f . (j) Ω = {K}. Assume otherwise. By (h), t is fixed point free on Γ = Ω−{K}. Let K1 , . . . , Kn be a set of representatives for the orbits of t on Γ, zi = z(Ki ), ui = zi zit , Ji = n CKi Kit (t), and J = i=1 Ji . Then Ji ∼ = K and J is the central product of the Ji , 1 ≤ i ≤ n. Observe that J ≤ CS (Z) ≤ R, so |J : J0 | ≤ 2, where J0 = CJ (H). In particular ui ∈ Φ(Ji ) ≤ CS (H). Also J centralizes K, so J0 ≤ R0 and in particular R0 = 1. As |J : J0 | ≤ 2, K is the unique member of Ω in CS (J0 ), and hence also in CS (R0 ). Let F0 = NF (R0 ); by (i), R0 ∈ F f and then by 2.5.2, τ0 = (F0 , {K}) is a quaternion fusion packet. Set F1 = [K]F0 . As H ≤ CG (J0 ), we conclude

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from (d) and Theorem 2 that z ∈ / O2 (F1 ), so, applying Theorem 7.1.29 in an inductive setting, we conclude that F1 ∼ = L2 [2m](1) , L− 3 [m], G2 [m], or M12 . Let t E = zi , zi : 1 ≤ i ≤ n . Observe E ∩ R0 ≤ CE (t) < E so there is e ∈ NE (R0 ) − R0 and 1 = [e, t] = f ∈ u1 , . . . , un ≤ J0 . Now e ∈ CS (K) acts on R0 , so e acts on the Sylow group S0 of F1 . Then f ∈ CS0 (K) impossible as z is the unique involution in CS0 (K) while f ∈ J0 centralizes F1 . This completes the proof of (j). We are now in a position to complete the proof of the lemma. By (j), (d), and (e), Hypothesis 7.1.8 is satisfied. Thus we can adopt notation 7.1.10. By 7.1.12, t inverts W , so E ≤ CS (Z) = R. Let A = ZE; by (b), H acts on A, so as E  S, HR acts on A. By 7.1.13.2, AutF (A) = CGL(A) (e) for some e ∈ E − z . Then as e ∈ z NF (W ) and NF (W ) ≤ E ≥ Fz , it follows that AutH (R) ≤ AutE (R). Similarly AutCG (Z) (R) ≤ AutFz (R) ≤ AutE (R), so AutF (R) = AutE (R) by (b), completing the proof of the lemma.  Lemma 3.4.7. z ∈ Z(S) iff |Ω|/|Ω(z)| is odd. Proof. As F is transitive on Ω, A = AutF (O(τ )) is transitive on Ω by 3.1.5.6.  Hence |z A | = n = |Ω|/|Ω(z)| and z ∈ Z(S) iff n is odd. Lemma 3.4.8. Assume Ω(z) = {K} and |Ω| is odd. Then either F = E(τ ) or conclusions (1)-(3) of Lemma 3.4.6 hold. Proof. By 3.4.7, z ∈ Z(S), so for each R ∈ F e , z ∈ CS (R) ≤ Z(R). Then the lemma follows from 3.4.6 and the Alperin-Goldschmidt Fusion Theorem.  Lemma 3.4.9. Assume H  G with H ∗ of even order and Q  R. (1) If [CG (Q), H] ≤ R then CS (Q) ≤ R. (2) If Q is H-invariant and H ∗ −CH (Q)∗ contains an involution then CS (Q) ≤ R. Proof. As H  G and H ∗ is of even order, G∗ = H ∗ M∗ , so H ∗ ≤ M∗ . Suppose S1 = CS (Q) ≤ R. As Q  R, R acts on S1 and S2 = NS1 (R) ≤ R, so S2∗ = 1. Then NG∗ (S2∗ ) ≤ M∗ . Now S2 ≤ CG (Q), so if [CG (Q), H] ≤ R then H ∗ ≤ CG∗ (S2∗ ) ≤ M∗ , a contradiction. Thus (1) holds. It remains to prove (2), so assume Q is H-invariant, S1 ≤ R, and t∗ is an involution in H ∗ −CH (Q)∗ . As S2∗ is a nontrivial 2-group, S2∗ contains an involution s∗ . Let L = HS2 . Then CL (Q)∗  L∗ as Q is H-invariant. We showed G∗ = H ∗ M∗ and H ∗ ≤ M∗ . Thus (M ∩ L)∗ is strongly embedded in L∗ , so L∗ is transitive on ∗  its involutions, and hence t∗ ∈ s∗L ⊆ CL (Q)∗ , a contradiction.  Lemma 3.4.10. Assume |Ω(z)| ≤ 2, NK (R) is nonabelian, and AutF (R) = AutE (R). Set Q = z G , J = NK (R), H = J G , and ΩH = J H ∩ S. Then (1) |J| = 8, ΩH = {J, J h } for some h ∈ H − NH (J), τH = (FS∩H (H), ΩH ) is a quaternion fusion packet, H/Z(H) ∼ = AE5 , and Z(H) = zz h . (H) with C (Q) ≤ R. (2) R = QCS S (3) If z = z h then Q ∼ = Q8 D8 and we may take h ∈ z H . Proof. By 2.5.2, τH is a quaternion fusion packet. As F ∗ (G) = R = O2 (G), z ∈ R by 3.3.2.4. Then J ∩ R = z by 3.4.5. Hence H is described in Theorem 2, so as H ∗ has a strongly embedded subgroup we conclude that H/Z(H) ∼ = AE5 . In ˜ = G/Z(H). Then particular (1) holds. Then (3) follows from (1) and 5.1.7. Set G

3.4.

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˜ is a projective H ∗ -module, so R ˜=Q ˜ ⊕ C ˜ (H). Hence R = QCR (H) using (3). Q R  Finally CS (Q) ≤ R by 3.4.9.2, so (2) follows. Lemma 3.4.11. (1) Let θ ⊆ ZS with Q = O(θ) ∈ F f . Then NF (Q) ≤ NF (O(τ )), NF (O(ξ)) for each ∅ = ξ ⊆ θ such that NS (Q) is transitive on ξ. (2) E(τ ) = SO(τ ), NF (O(ξ)), NF (W ) : W ∈ W (τ ), ξ ∈ Ξ , where Ξ consists of those ∅ = ξ ⊆ ZS such that NS (ξ) is transitive on ξ and O(ξ) ∈ F f . Proof. It follows from the definition of E(τ ) in 3.4.2 and from 3.4.3.4 that (1) implies (2). So choose θ and ξ as in (1) and set N = NF (Q). Observe that P = O(ξ) ≤ Q. Set O = O(τ ) and X = NF (O), NF (P ) ; we must show that N ≤ X. Set T = NS (Q) and Y = CF (Q)T . By 1.3.2 in [Asc19] and as O is weakly closed in QCS (Q), we have N = Y, NN (O) , so it remains to show that Y ≤ X . But as P ≤ Q, CF (Q) ≤ NF (P ) ≤ X . As ξ is T -invariant, P  T . Thus indeed Y ≤ X , completing the proof.  Lemma 3.4.12. Assume Δ(τ ) = Ω. Then (1) E(τ ) = SO(τ ), NF (O(ξ)) : ξ ∈ Ξ , where Ξ is defined in 3.4.11. (2) If m = 8 then E(τ ) = NF (O(ξ)) : ξ ∈ Ξ . Proof. If m = 8 then O(τ ) = O(τ ), so (1) implies (2). Thus it remains to prove (1), so by 3.4.11.2, it suffices to show that NF (W ) ≤ NF (O), for W ∈ W (τ ) and O = O(τ ). But as Δ = Ω, this follows from the proof of 3.3.14.  Lemma 3.4.13. Let x be an involution in F f . Assume τx◦ = τ1 ∗ τ2 where τi = (Fi , Ωi ) with Fi  Fx and Ωi = ∅ for i = 1, 2. Then Fx ≤ E(τ ). Proof. Let Oi = O(τi ) and Ti Sylow in Fi . As Fi  Fx for i = 1, 2, it follows from 1.3.2 in [Asc19] that Fx = CS (x)F1 , NFx (T1 ) . Further O2  CS (x)F1 and  O1  NFx (T1 ), and as Ωi = ∅, NF (Oi ) ≤ E by 3.4.3.3, so the lemma holds.

CHAPTER 4

W (τ ) and M (τ )

In Chapter 4, τ = (F, Ω) is a quaternion fusion packet and S is Sylow in F. Recall from 3.1.26 that μ = μ(τ ) = D for a certain set D of involutions in AutF (W ), for W ∈ W (τ ). Let W  M ∈ M (τ ). From 3.3.11, M is a model for [O(τ )]NF (W ) and μ = AutM (W ). Further τW = (NF (W ), Ω) is a quaternion fusion ◦ packet, so τW satisfies Hypothesis 4.2.1, and from 4.2.4, μ = μ(τW ). Also from 3.1.25, generically D is a set of 3-transpositions in μ. Therefore in section 4.1 we are led to consider 3-transposition groups. Our reference for 3-transposition groups is [Asc97]. In 4.2.14 we find that when D is a set of 3transpositions of μ that D satisfies Hypothesis 4.1.12 in μ, while 4.1.25 determines the groups μ generated by a conjugacy class of 3-transpositions satisfying 4.1.12. Indeed by that result, μ/Z(μ) is isomorphic to Weyl(Φ)/Z(Weyl(Φ)) for some Φ Coxeter diagram Φ of type A, D, or E. ¯ = ω In 4.3.1 we define a universal group G ¯ (Φ, m) and show in 4.3.8 that ¯ ¯ Moreover there is a surjective homomorphism π : G → W with ker(π) ≤ Z(G). ∼ W = CM (W ), so μ = M/W and μ = Weyl(Φ). Hence M is determined up to isomorphism by Φ, m, and ker(π). Further the structure of M can be retrieved ¯ from the structure of G. ¯ The group G = ω ¯ (Φ, m) is isomorphic to M (τ ) for τ the Lie packet of a suitable universal group of Lie type.

4.1. 3-transposition groups In this section we assume the following hypothesis: Hypothesis 4.1.1. G is a finite group and D is a G-invariant set of involutions in G such that G = D . In addition we adopt the following notation: Notation 4.1.2. Let d ∈ D and set d⊥ = CD (d), Dd = d⊥ − {d}, and Ad = D − d⊥ . Let D(D) be the commuting graph on D; that is a, b ∈ D are adjacent in D(D) iff b ∈ Da . Define Vd = {a ∈ D : a⊥ = d⊥ } and Wd = {a ∈ D : Da = Dd }. 79

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Recall that D is a set of 3-transpositions if Hypothesis 4.1.1 is satisfied and |ab| ≤ 3 for all a, b ∈ D. In the remainder of the section we assume: Hypothesis 4.1.3. D is a set of 3-transpositions of G. Our reference for 3-transposition groups is [Asc97]. Lemma 4.1.4. If D = D1



D2 is a G-invariant partition of D then [D1 , D2 ] =

1. Proof. See 8.2.1 in [Asc97].



Lemma 4.1.5. If G = a, b, d with b ∈ Dd and a ∈ Ad ∩ Ab then G ∼ = S4 and D is the set of ordinary transpositions in G. Proof. This follows as G is an image of the Coxeter group S4 of type A3 .  Lemma 4.1.6. D(D) is disconnected iff G is transitive on D and G has a factor group isomorphic to S3 . Proof. Assume G is transitive on D and H  G with G∗ = G/H ∼ = S3 . As G = D , D is not contained in H, so as G is transitive on D, H ∩ D = ∅. We must show that D(D) is disconnected; it suffices to show Dd ⊆ dH, so assume b ∈ Dd − dH. Then 2 = |bd| = |b∗ d∗ |, contradicting G∗ ∼ = S3 . Conversely assume D(D) is disconnected and let Γ be the set of connected components of D(D), and for d ∈ D let Σd be the member of Γ containing d. By 4.1.4, G is transitive on D. Let K be the kernel of the action of G on Γ and set G+ = G/K. If Σd = Σa then d does not act on Σa , or else d ∈ Da by 4.1.4, a contradiction. Thus d is semiregular on Γ − {Σd }, so in particular d+ = 1. Further for b ∈ Dd and a ∈ D−Σd , a ∈ Ad ∩Ab , so X = a, b, d ∼ = S4 by 4.1.5. In particular abd ∈ Da , so bd ∈ K. Therefore Dd+ = ∅, so by 8.6 in [Asc97], G+ = O3 (G+ )d+ . Now let M + be a maximal subgroup of O3 (G+ ) normal in G+ and M the preimage  of M + in G. Then G/M ∼ = S3 , completing the proof. Lemma 4.1.7. Assume G is transitive on D. Then (1) Vd = dO2 (G) and (2) Wd = dO3 (d) . Proof. See 9.2 in [Asc97].



Lemma 4.1.8. Assume that G/O2 (G) ∼ = S3 and D(D) is disconnected. Then 2 = |Vd | for some positive integer n, and G is generated by n elements of d⊥ together with an element of Ad . n−1

Proof. Let G be a minimal counter example, Q = O2 (G), a ∈ Ad , and π : G → G/Q the natural map. By 4.1.6, G is transitive on D, so for b ∈ π −1 (d), b = dg for some g ∈ G. Then gπ ∈ CGπ (dπ) = dπ , so b ∈ dQ . Hence by 4.1.7.1, π −1 (d) = dQ = Vd , so indeed |Vd | = |Q : CQ (d)| = 2n−1 for some positive integer n. If Q = 1 then S3 ∼ = G = d, a , so the lemma holds with n = 1 in this case. Therefore as G is a counter example, Q = 1. Let M be a minimal normal subgroup

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of G contained in Q and set G∗ = G/M . By minimality of G, G∗ = d∗1 , . . . , d∗m , a∗ where 2m−1 = |Vd∗ |, d = d1 , and d∗i ∈ d∗⊥ . Set H = d1 , . . . , dm , a , so that H ∗ = G∗ and hence G = HM . As D is a set of 3-transpositions of G, di ∈ d⊥ . Further n ≥ m. If d centralizes M then as G = D is transitive on D, M ≤ Z(G). Then as G = M H, H  G, so as G = D is transitive on D, we have G = H. Then as n ≥ m, we have a contradiction to the choice of G as a counter example. Therefore [M, d] = 1. Next G is irreducible on M , so M ≤ Z(Q). Then as [M, d] = 1, we have Q = CG (M ), so AutG (M ) ∼ = S3 and hence as G is irreducible on M we have M ∼ = E4 , so that M is transitive on the involutions in dM . Therefore CG∗ (d∗ ) = CG (d)∗ , so n = m + 1 and G = H, dx for x ∈ M − CM (d), again contradicting the choice of G.  Remark 4.1.9. Denote by Weyl(D4 ) the Coxeter group of type D4 ; recall Weyl(D4 ) is the split extension W of Q28 by S3 in which F ∗ (W ) = O2 (W ) and the two Q8 -subgroups of O2 (W ) are conjugate in W . Lemma 4.1.10. Assume G/O2 (G) ∼ = S3 with D(D) disconnected and |Vd | = 4. Then G is isomorphic to Weyl(D4 ) or Weyl(D4 )/Z(Weyl(D4 )). Proof. By 4.1.8, G = d1 , . . . , d3 , a with d = d1 , di ∈ d⊥ , and a ∈ Ad . Then a ∈ Adi for each i, so G is an image of the Coxeter group W of type D4 , with D the image of the reflections in W . Then the lemma follows from Remark 4.1.9.  A D-subgroup of G is a subgroup X of G generated by D ∩ X. The width of G is |D ∩ S| for S ∈ Syl2 (G). At various times we assume one of the following hypotheses: Hypothesis 4.1.11. (1) D is a set of 3-transpositions of G. (2) If X is a D-subgroup of G of width 1 then X ∼ = Z2 or S3 . Hypothesis 4.1.12. Hypothesis 4.1.11 holds and (*) if X = E ≤ G with E an X-invariant subset of D such that D(E) is disconnected then |eO2 (X) | ≤ 4 for e ∈ E. Hypothesis 4.1.13. Hypothesis 4.1.11 holds and (**) if X = E ≤ G with E an X-invariant subset of D such that D(E) is disconnected then X ∼ = S3 or S4 .

∅.

Lemma 4.1.14. The following are equivalent: (1) Hypothesis 4.1.11 holds. (2) For each D-subgroup X of G isomorphic to S3 and for each d ∈ D, d⊥ ∩X =

Proof. Suppose X and d are as in (2) with d⊥ ∩ X = ∅. Then by 5.5.4 in [Asc97], X, d is of width 1, so (1) fails. Thus (1) implies (2). Conversely assume (2) holds and X is a D-subgroup of G of width 1 not isomorphic to Z2 or S3 . Then there exist distinct a, b ∈ D ∩ X, and as X is of width 1, a ∈ Ab , so Y = a, b ∼ = S3 . Then there exists c ∈ (D ∩ X) − Y , and by (2), c  commutes with a, b, or ab , contradicting X of width 1. Lemma 4.1.15. Each of Hypotheses 4.1.11, 4.1.12, and 4.1.13 inherit to Dsubgroups of G.

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Proof. This is immediate from the definitions.



Lemma 4.1.16. Hypothesis 4.1.11 inherits to homomorphic images of G. ¯ = G/H. Suppose Hypothesis 4.1.11 holds and Proof. Let H  G and G ¯ = ¯ ¯ ¯ ¯ a, ¯b with b ∈ D and let X be a D-subgroup of G isomorphic to S3 . Then X ⊥ ∼ ¯ = ∅. a ∈ Ab , so X = a, b = S3 . Let d ∈ D; then by 4.1.14, d ∩ X = ∅, so d¯⊥ ∩ X Then another application of 4.1.14 completes the proof.  Lemma 4.1.17. Assume Hypothesis 4.1.11 with G transitive on D, and suppose Y  G such that O3 (G/Y ) ≤ Z(G/Y ). Then Y = O2 (G) and G/Y ∼ = S3 . ¯ in G, a ∈ D, ¯ = G/Y and let L be the preimage in G of O3 (G) Proof. Set G ¯ L and a ¯ = θ. Then ν = θ is of width 1, so by 4.1.16 and 4.1.11, ν ∼ = S3 . But by ¯ − θ, D¯b ∩ θ = ∅, so D ¯ − θ centralizes 4.1.7, Wa¯ = θ, while by 4.1.14, for each ¯b ∈ D ¯ = ν¯ ∼ ν¯. Hence as G is transitive on D, it follows that G = S3 . ⊥ We next claim that aY ∩D ⊆ a . If not there is c ∈ aY ∩D with a, c ∼ = S3 . Let ¯, |¯ c¯b| = 3, so |cb| = 3. As this also holds for ca , b⊥ ∩ a, c = ∅, b ∈ aL ∩ Aa ; as c¯ = a so 4.1.14 supplies a contradiction, completing the proof of the claim. ¯ ∼ Set Q = O2 (G) and G∗ = G/Q; as G = S3 , we have Q ≤ Y . By the claim ⊥ ∼ ¯ and as G = S3 , aY ∩ D = a = Va , so by 4.1.7.1, aY ∩ D = aQ ⊆ aQ. Thus ¯ ∼ ¯ = 3 and hence G∗ ∼ {a∗ } = a∗ Y ∗ ∩ D∗ , so as G = S3 , |D∗ | = |D| = S3 , so that ∗  Y = 1 and hence Y = Q. This completes the proof. Lemma 4.1.18. Assume Hypothesis 4.1.11. (1) If D(D) is disconnected then G/O2 (G) ∼ = S3 . (2) If G is transitive on D and O3 (G) ≤ Z(G) then G ∼ = S3 . Proof. Part (1) follows from 4.1.6 and 4.1.17. Assume the hypothesis of (2).  Applying 4.1.17 with Y = 1, we conclude that G ∼ = S3 , proving (2). Lemma 4.1.19. Hypotheses 4.1.12 and 4.1.13 are inherited by homomorphic images of G. ¯ = G/H. Assume Hypothesis 4.1.12 or 4.1.13 holds, Proof. Let H  G and G ¯ ¯ ¯ ¯ and some X-invariant ¯ ¯ of D ¯ such that and X = E for some X ≤ G subset E ¯ ¯ D(E) is disconnected. By 4.1.15 and 4.1.16, X satisfies Hypothesis 4.1.11, so by ¯ ∼ ¯ is transitive on E, ¯ so X ¯ = d¯X¯ for d ∈ D ¯ 2 (X) 4.1.18.1, X/O = S3 . By 4.1.6, X ¯ Let Y be the preimage of X ¯ in G, E = dY , and X = E . By 4.1.6, with d¯ ∈ E. D(E) is disconnected. ¯ ∼ Suppose Hypothesis 4.1.13 holds. Then X ∼ = S3 or = S3 or S4 and hence X ¯ S4 , so G satisfies Hypothesis 4.1.13. ¯ = Finally suppose Hypothesis 4.1.12 holds. By 4.1.18, X/O2 (X) ∼ = S3 , so O2 (X) ¯ O2 (X) O2 (X) ¯ O2 (X), and hence |d | ≤ |d | ≤ 4. This completes the proof.  Lemma 4.1.20. Assume G satisfies Hypotheses 4.1.11 and G is a subgroup of a group L transitive on D via conjugation. Assume that G is of width 2. Then G∼ = S3 × S3 , E4 , S4 or S5 . Proof. Let d ∈ D, let  Di , 1 ≤ i ≤ n, be the orbits of G on D and set Gi = Di . By 4.1.4, G = i Gi is a central product. As L is transitive on D, L permutes the factors Gi transitively, so Gi ∼ = Gj for all i, j. Let w be the width of Gi . Then 2 = nw so either n = 1 or n = 2 and w = 1. In the latter case by

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Hypothesis 4.1.11, Gi ∼ = Z2 or S3 , so that the lemma holds in this case. Therefore we may assume that G is transitive on D. If D(D) is disconnected then by 4.1.18.1, G/O2 (G) ∼ = S3 . Then d⊥ = Vd = O2 (G) d by 4.1.7, so as G is of width 2, |Vd | = 2 and hence G ∼ = S4 . Therefore we may assume D(D) is connected. Similarly if Vd = {d} then as G is of width 2, d⊥ = Vd , so D(D) is disconnected, contrary to assumption. Therefore Vd = {d}, so O2 (G) ≤ Z(G) by 4.1.7. Also as D(D) is connected, O3 (G) ≤ Z(G) by 4.1.18.2. We’ve shown that O2 (G) ≤ Z(G) ≥ O3 (G) and D(G) is connected. It follows from 9.4.4 and 9.5.4 in [Asc97] that H = Dd is transitive on Dd and |Dd | > 1. But as G is of width 2, H is of width 1, so H ∼ = S3 by 4.1.11. Therefore H = d3 , d4 with d4 ∈ Ad3 . Set d1 = d. As G is transitive on D and D(D) is connected, there exists d2 ∈ Ad with d3 ∈ Dd2 . If d4 ∈ Dd2 , then d2 ∈ Wd , contradicting 4.1.7 and O3 (G) ≤ Z(G). Therefore setting Δ = {d1 , . . . , d4 } and G0 = Δ , (G0 , Δ) is a Coxeter system of type A4 , so G0 ∼ = S5 . Finally by 9.5.5 in [Asc97], G = G0 , completing the proof.  Lemma 4.1.21. If Hypotheses 4.1.13 holds with G transitive on D, then G ∼ = Sn for some n ≥ 2. Proof. Assume otherwise and let G be a minimal counter example. If |D| = 1 then G ∼ = S3 or S4 by 4.1.13, = S2 , so |D| > 1. If D(D) is disconnected then G ∼ contrary to the choice of G. Therefore D(D) is connected. Observe that O3 (G) ≤ Z(G) by 4.1.18.2. Claim O2 (G) ≤ Z(G). Assume ¯ = G/O2 (G). By 4.1.19 and minimality of G, G ¯∼ otherwise and set G = Sn for some n. We will show: ¯ has no D-subgroups ¯ (!) G isomorphic to S4 . It follows from (!) that n = 3, so D(D) is disconnected by 4.1.6, contrary ¯ is a D¯ to an earlier reduction. Thus it remains to establish (!). So suppose X ¯a ¯ = d, ¯, ¯b with b ∈ Dd and a ∈ Ad ∩ Ab . Let subgroup isomorphic to S4 . Then X X = Vd , a, b and E = dX . Then D(E) is disconnected by 4.1.6, so X ∼ = S4 by 4.1.13, whereas by 4.1.7, Vd ∪ {b} ⊆ dO2 (X) , so |dO2 (X) | > 2, a contradiction. This completes the proof of the claim. We’ve shown that D(D) is connected and O2 (G) ≤ Z(G) ≥ O3 (G). At this point we could appeal to Theorem Z in section 15 of [Asc97] to finish off the proof fairly quickly. Instead we will give most of the details in this special case, as most are fairly easy, and it is only a little more difficult to include these details than to exclude some of the conclusion appearing in Theorem Z. By 9.4.4 and 9.5.4 in [Asc97], H = Dd is transitive on Dd . By minimality of G, H ∼ = Sn for some n. If G is of width 2 then G ∼ = S5 by 4.1.20, so n ≥ 4. Let a ∈ Ad and X = a, d . As H ∼ = Sn with n ≥ 4, it follows that for b ∈ Dd , CH (b) is transitive on Ab ∩ Dd , so: (a) G is transitive on triples (u, v, w) with w ∈ Av and u ∈ Dv ∩ Dw . Thus CG (X) is transitive on E = Da ∩ Dd . Set Y = E ; we will show that: (b) Y is transitive on E. Then by (b), Y ∼ = Sm for some integer m by minimality of G. We will also show: (c) m = n − 1. By (c), Y = Δ where Δ = {d1 , . . . , dn−2 } has Coxeter diagram An−2 , and there exists d0 ∈ Dd ∩ Aa such that Δ0 = {d0 , d1 , . . . , dn−1 } has Coxeter diagram

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An−1 and H = Δ0 . Then Γ = {d, a, d0 , . . . , dn−2 } has Coxeter diagram An , so G0 = Γ ∼ = Sn+1 . Further G0 = H, d, a , so G = G0 by 9.5.5 in [Asc97], contrary to the choice of G. Therefore it remains to prove (b) and (c). Let b ∈ Dd , c ∈ Dd ∩ Ab , and W = b, c . Then CD (W d ) = d × K where K = CDd (W ) ∼ = Sn−3 . Suppose Y is transitive on E. Now for e ∈ E, CY (e) = e × L where L ∼ = Sm−2 , so by (a), either m = n − 1 or (n, m) = (4, 2). However in the latter case, the rank 3 parameters for the graph D(D) (cf. section 6 in [Asc97]) are k = 6 and λ = μ = 1, so δ = (λ − μ)2 + 4(k − μ) = 20 is not a square, contrary to 6.3.3 in [Asc97]. Thus if Y is transitive on E then m = n − 1; that is (b) implies (c), so it remains to prove (b). (b) fails. Then by 4.1.4 and as CG (X) is transitive on E, E = So assume  E1 · · · Er , for some r > 1, and Y = Y1 × · · · × Yr , where Yi = Ei ∼ = St for some t ≥ 2. Then for e ∈ E1 , Ye = CY (e) = e × X1 × Y2 × · · · × Yr , where X1 ∼ = St−2 . However by (a), Ye ∼ = Z2 × Sn−3 , so we conclude that r = 2, t = n − 3, and X1 = 1. As X1 = 1, t ≤ 3, so as 2 ≤ t = n − 3, n = 5 or 6. Suppose n = 5. Then k = 10, λ = 3, and μ = 2, so δ = 33 is not a square, again a contradiction. Therefore n = 6. Here k = 15 and λ = μ = 6, so δ = 62 is a square. Indeed a second group exists in this case: by 15.6 in [Asc97], G is either S8 or O6− (2). However in the latter case, the centralizer of a 2-central involution in G contains a D-subgroup isomorphic to Weyl(D4 ), contrary to 4.1.13. This completes the proof of the lemma.  Lemma 4.1.22. Assume Hypotheses 4.1.12 with G transitive on D and |Vd | > 1 for d ∈ D. Then (1) G ∼ = Weyl(Dn ) or Weyl(Dn )/Z(W eyl(Dn )) for some n ≥ 3. (2) Either |Vd | = 2 or n = 4 and |Vd | = 4. ¯ = G/Q. By 4.1.7, Vd = dQ , so for Proof. Let d ∈ D, Q = O2 (G), and G ¯ v ∈ D, d = v¯ iff v ∈ Vd . Claim: ¯ D ¯ satisfy Hypothesis 4.1.13. (a) G, ¯ is a subgroup of G ¯ generated by an X-invariant ¯ ¯ of For suppose X subset E ¯ ¯ ¯ D such that D(E) is disconnected. Let X0 be the preimage of X in G, let d ∈ D ¯ and set E = dX0 and X = E . Arguing as in the proof of 4.1.19, with d¯ ∈ E, ¯ ¯ ¯ X/O2 (X) ∼ = S3 and D(E) is disconnected. Let q = |d¯O2 (X) |. Then |dO2 (X) | ≥ q|dQ | = q|Vd | ≥ 2q, ∼ S3 ¯ = so by 4.1.12 either |Vd | = 4 and q = 1 or |Vd | = 2 and q ≤ 2. It follows that X or S4 , establishing (a). It also follows that ¯∼ (b) Either |Vd | = 2, or |Vd | = 4 and X = S3 . ¯∼ By (a) and 4.1.21, G = Sn for some n, and as |Vd | > 1 and G is transitive on ¯ = 1, n = 4. If n = 3 then D(D) is disconnected by 4.1.6. D, n ≥ 3. As O2 (G) Then by (b), |Vd | = 2m for m = 1 or 2. In the first case, G ∼ = Weyl(D3 ), and = S4 ∼ in the second the lemma follows from 4.1.10. Therefore we may assume n ≥ 5, so |Vd | = 2 by (b), establishing (2). As ¯ ∼ ¯ Δ) ¯ with Δ ¯ = {d¯1 , . . . , d¯n−1 } of type An−1 G = Sn there is a Coxeter system (G, ¯ Let Γ = {d1 , . . . , dn } where di ∈ D is a preimage of d¯i for 1 ≤ i < n and and d¯i ∈ D. {dn−1 , dn } = Vdn−1 . As D is a set of 3-transpositions of D, (G0 , Γ) is a Coxeter ¯ = G ¯ 0 is transitive on D ¯ and system of type Dn , where G0 = Γ . Further G

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0 S4 ∼ = dn−2 , dn−1 , dn = G1 ≤ G0 , so dn ∈ dG n−1 . Hence G0 is transitive on D, so  that G = G0 . This completes the proof of the lemma.

Lemma 4.1.23. Assume Hypotheses 4.1.12 with D(D) disconnected. Then G ∼ = S3 , S4 , Weyl(D4 ) or Weyl(D4 )/Z(Weyl(D4 )). Proof. By 4.1.18.1, G/O2 (G) ∼ = S3 , so either G ∼ = S3 or O2 (G) ≤ Z(G). In  the latter case |Vd | > 1 by 4.1.7, so the lemma holds by 4.1.22. Lemma 4.1.24. Assume Hypotheses 4.1.12 with G transitive on D and O2 (G) ≤ Z(G) ≥ O3 (G). Then G/Z(G) is one of the following: (1) Sn for some n ≥ 5. (2) O6− (2). (3) Sp6 (2). (4) O8+ (2). Proof. Passing to G/Z(G), we may assume Z(G) = 1. Thus by 9.4.1 in [Asc97], F ∗ (G) = L is simple. Then by 4.1.18.1, D(D) is connected, so by 9.4.4 in [Asc97], |G : L| ≤ 2. Now by Fischer’s Theorem in the Introduction to [Asc97], G is in one of the six classes of groups listed there. We will examine each class. Let M be the set of proper D-subgroups M of G such that M is transitive on M ∩ D and O2 (M ) ≤ Z(M ) ≥ O3 (M ). By 4.1.15 and induction on the order of G, each M ∈ M appears on the list of this lemma. Suppose G ∼ = O4− (2) ∼ = S5 , = On (2). As G is almost simple, either n ≥ 6 or G ∼ + − and we may assume the former. Now O6 (2) ∼ = S8 and O6 (2) and O8+ (2) are on our list. Further if n > 8 then there is M ∈ M isomorphic to O8− (2). Thus we may assume G is O8− (2). Hence there is a maximal parabolic X of G which is the extension of O2 (X) by O6− (2) with O2 (X) the natural module for X/O2 (X). Then there is a D-subgroup Y of X such that Y /O2 (Y ) ∼ = S3 and |dO2 (Y ) | = 8 for  d ∈ Y ∩ D, contrary to 4.1.12. Therefore G is not On (2). Suppose G ∼ = S6 and Sp6 (2) are on our list, we may = Spn (2). As Sp4 (2) ∼ assume n ≥ 8. But now there is a member of M isomorphic to O8− (2), a contradiction. Suppose G ∼ = Un (2). As G is almost simple, n ≥ 4, and if n > 4 there is a member of M isomorphic to U4 (2), so we may take n = 4. Hence there is a D-subgroup X = O3 (X)d with O3 (X) ∼ = 31+2 , contrary to 4.1.11. −,π μ,π ∼ Suppose G = P On (3). As G is almost simple either G ∼ = P O4 (3) ∼ = S6 or n ≥ 5, and we may assume the latter. If n = 5 then G is P O5π,π (3) ∼ = O6− (2) or P O5−π,π (3) ∼ = U4 (2), so the lemma holds in this case as we’ve already eliminated U4 (2). Therefore n > 5, so there is a member of M isomorphic to P O5−π,π (3), for our usual contradiction. This leaves the case in Fischer’s Theorem where G is one of the three Fischer groups. But then there is M ∈ M with M/Z(M ) ∼ = U6 (2), for our final contradiction.  Lemma 4.1.25. Assume Hypotheses 4.1.12 with G transitive on D. Then one of the following holds: (1) G ∼ = S3 . (2) |Vd | > 1 for d ∈ D and G ∼ = Weyl(Dn ) or Weyl(Dn )/Z(Weyl(Dn ) for some n ≥ 3.

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(3) O2 (G) ≤ Z(G) ≥ O3 (G) and G/Z(G) is isomorphic to Sn for some n ≥ 5, O6− (2), Sp6 (2), or O8+ (2). Proof. If O3 (G) ≤ Z(G) then (1) holds by 4.1.18.2. If O2 (G) ≤ Z(G) then d ∈ D does not centralize O2 (G), so |Vd | > 1 by 4.1.7.1, and hence (2) holds by 4.1.22. This leaves the case where O2 (G) ≤ Z(G) ≥ O3 (G), where (3) holds by 4.1.24. 

4.2. The groups in M (τ ) In this section we assume the following hypothesis: Hypothesis 4.2.1. (1) τ = (F, Ω) is a quaternion fusion packet. (2) η ∈ η(τ ) and W = η  F. (3) F = F ◦ . Lemma 4.2.2. (1) For K ∈ Ω, OK = FK (K) and AutF (K) is a 2-group. (2) F is constrained. (3) Z = ZΔ . Proof. Parts (1) and (2) follow from 3.3.9. By definition of Z, for each t ∈ Z there exists φ ∈ homF (t, S) and K ∈ Ω with tφ = z(K). Hence as W  F by 4.2.1.2, t ∈ W . Now (3) follows from 3.1.11.7.  Notation 4.2.3. Let μ = μ(τ ). By 4.2.2.2 there exists a model G for F, and as W  F also W  G. Set G∗ = G/CG (W ) and define D = Dη as in Definition 3.1.19; thus D = {dV : V ∈ η} ⊆ G∗ . Set Λ = {K g : K ∈ Ω, g ∈ G}. Lemma 4.2.4. (1) G = O(τ )G = Λ . (2) For each V ∈ η there exists a unique K(V ) ∈ Λ containing V . Moreover the map L → L ∩ W is a G-equivariant bijection of Λ with η with inverse V → K(V ). (3) For V ∈ η, dV = K(V )∗ . Proof. Part (1) follows from 3.3.9.3. Let J ∈ Ω. By definition of η, U = J ∩ W ∈ η, and by (QFP2), J is the unique member of Ω containing U . Set z = z(J). By 4.2.2.1 and 2.6.11, O(z)  CG (z), so J  NG (U ) and hence J is the unique member of Λ containing U . Then (2) follows. By Definition 3.1.19, dU = j ∗ for j ∈ J − U , so (3) follows from (2).  Lemma 4.2.5. Let K ∈ Ω, V = K ∩ W , and U ∈ η with dU , dV ∼ = S3 . Set M = K, K(U ) . Then (1) Choose a generator v for V . Then we can choose a generator u for U with [v, dU ] = u and [u, dV ] = v. (2) CM (U V ) = U V = U × V = M ∩ W and M/U V ∼ = M ∗ = dU , dV ∼ = S3 . M   (3) K = {K, K(U ), K(U )}, where U = uv . (4) M splits over U V and is determined up to isomorphism. (5) For each k ∈ K − V and j ∈ K(U ) − U , |kj| = 3.

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Proof. Let t, z be the involutions in U, V , respectively. First, the hypothesis of 3.1.16 is satisfied with V = V1 , U = V2 , and A(U, V ) ∼ = S3 . Therefore (1) follows from 3.1.18. By (1), v dU = vu and udV = uv, so U V  M . As U does not centralize K, z = t, so U V = U × V . Then M = k, j, U V with k ∈ K − V , j ∈ K(U ) − U , and k2 , j 2 ∈ U V . Next v dU dV = (vu)dV = v −1 uv = u, so (3) follows. Set M + = M/U V . By (3), M K = K(U )M is of order 3, so k+M = j +M is of order 3, and hence M + ∼ = S3 . Hence (2) follows. Let Y ∈ Syl3 (M ). Now X = Ω1 (U V ) = z, t and CX (Y ) = 1, so CUV (Y ) = 1. Therefore H = NM (Y ) is a complement to U V in M . Let a, b be the involutions in H centralizing z, t, respectively. Then a∗ = dV and b∗ = dU , so [u, a] = v and [v, b] = u by (1). This gives a presentation for M , so M is determined up to isomorphism and (4) holds. Indeed H = a, b ∼ = S3 with CUV (H) = 1, so (m/2)2 = |U ||V | = |U V | = UV |(a, b) |. But for c ∈ {a, b}, c U V ∼ = Zm/2 wr Z2 , so U V is transitive on the set I(c) of involutions in cU V with |I(c)| = m/2. Thus P = I(a) × I(b) is of order (m/2)2 = |(a, b)UV |, so P = (a, b)UV . Hence for each (a1 , b1 ) ∈ P , |a1 b1 | = 3. Next I(a) = aV as V is the subgroup of U V inverted by a, and similarly I(b) = bU . As k∗ = dV = a∗ , k = a1 u1 for some a1 ∈ I(a) and u1 ∈ U . Then as z = k2 , we have u1 = t. Similarly j = b1 z for some (a1 , b1 ) ∈ P . Hence kj = a1 tb1 z = ys where 2 y = a1 b1 is of order 3 and s = tz. Therefore (kj)3 = (ys)3 = sy sy s = tzs = 1, proving (5).  Lemma 4.2.6. Let K ∈ Ω, V = K ∩ W , z = z(K), and assume t ∈ A(z). Then ˜ for some K. ˜ (1) Ω(z) = {K, K} ˜ and K ˜ ∈ ˜ G and U = J˜ ∩ W ; (2) t ∈ z G is contained in K K / K G . Let t ∈ J˜ ∈ K D . then dU , dV ∼ = 12 ˜ Then CM (U V ) = M ∩ W = U V = U × V = U × (K ˜ ∩W) (3) Set M = K, J . ∗ ∼ ∼ and M/U V = M = dU , dV = D12 . (4) Let Y ∈ Syl3 (M ). Then NM (Y ) is a complement to U V in M and the involution in Z(NM (Y )) inverts U V . (5) M is determined up to isomorphism. ˜ by 3.2.2.2. By definition of Proof. Part (1) follows from 3.2.2.1 and t ∈ K K A(z) in Definition 3.2.1, there is U ∈ η with t ∈ U such that Hypothesis 3.1.16 is satisfied with V1 = V , V2 = U , and A(V, U ) ∼ = D12 . Then by 3.1.17.3, t ∈ z G , so there are two members J, J˜ of Λ containing t, and as A(V, U ) ∼ = D12 we can choose ˜ ˜ ˜ ¯ notation so that U = J ∩W is conjugate to V = K ∩W and U = J ∩W is conjugate to V under A(V, U ). In particular (2) holds. ¯ = K, J is described in 4.2.5. In particular M ¯ ∩W = Next dV , dU¯ ∼ = S3 , so M ∼ ¯ ¯ ¯ ¯ U × V , NM¯ (Y ) = S3 is a complement to U V in M , and M is determined up to isomorphism. By 3.1.12, [K, W ] ≤ K ∩ W = V and then also [J˜, W ] ≤ U , so M acts on ¯ ∈ V M¯ ⊆ V M , U ¯ × V ≤ U V , so U ¯ × V = U V = U × V . Set U V . Then as U + M = M/U V and B = k, j , where k ∈ K − V and j ∈ J˜ − U . By 3.1.23.1, ˜ M is of order 6, so B + ∼ KM ∪ K = D12 and hence U V = CM (U V ), so (3) holds. Then as CUV (Y ) = CU¯ V (Y ) = 1, NM (Y ) is a complement to U V in M . Let c be ˜ = K, ˜ J . ˜ Then c = ad where a ∈ NM¯ (Y ) and the involution in Z(NM (Y )) and M ˜ d ∈ N ˜ (Y ) are involutions centralizing z. Then a ∈ KW inverts V and d ∈ KW M

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¯ ∈ V NM¯ (Y ) , c inverts U ¯ V = U V , completing centralizes V , so c inverts V . Then as U ¯ is determined up to isomorphism and c centralizes the proof of (4). Moreover as M  NM¯ (Y ) and inverts U V , (5) follows. Definition 4.2.7. Define D(D) to be the commuting graph on D; that is the edges in D(D) are the pairs (dU , dV ) such that dU , dV ∼ = E4 . A bimorphism φ : G → H of graphs is a map of vertices such that for each pair of vertices u, v of G with uφ = vφ, we have u adjacent to v if and only if uφ is adjacent to vφ. Lemma 4.2.8. (1) The map δ : V → dV is a G-equivariant bijection of η and D. (2) For V ∈ η let z(V ) be the involution in V . Then the map ζ : dV → z(V ) is a G-equivariant surjection of D onto Z. (3) The fiber ζ −1 (z) for z ∈ Z is {dV : z = z(V )} of order |Ω(z g )| for g ∈ G with z g ∈ ZS . In particular ζ −1 (z) is elementary abelian. (4) ζ : D(D) → D(τ ) is a bimorphism of graphs. (5) D(D) is connected iff D(τ ) is connected. (6) |Ω| = |D ∩ AutS (W )| is the width of μ. (7) If μ is transitive on D then F ◦ is transitive on Ω. Proof. Part (1) follows from 3.1.20.6. Let ρ : η → Z be the map ρ : V → z(V ). Then ρ is a G-equivariant surjection, so as ζ = δ −1 ρ, (1) implies (2). Part (3) is trivial. Let K ∈ Ω, z = z(K), V = K ∩ W , V = U ∈ η, and t = z(U ). Then [dU , dV ] = 1 iff [dV , U ] = 1 by 3.1.22, so as K ∗ = dV , it follows that [dU , dV ] = 1 iff K centralizes U iff t ∈ z ⊥ by 3.3.4. Thus (4) holds. Then (5) follows from (3) and (4) and (1) implies (6). It remains to prove (7), where by 3.1.27 we may assume F = F ◦ . Assume μ is transitive on D and let K, J ∈ Ω, V = K ∩ W , and U = J ∩ W . As μ is transitive on D, dU ∈ dμV , so by (1) there is g ∈ G with dU = dV g . Then by 4.2.4.2, K g = J, establishing (7).  Definition 4.2.9. For V ∈ η, we define a subset η(V ) of η as follows: (1) If A(z(V )) = ∅ set η(V ) = V G ∪ V˜ G , where {V, V˜ } = {U ∈ η : z(U ) = z(V )}. (2) If A(z(V )) = ∅, set η(V ) = V G . In either case set G(V ) = K(U ) : U ∈ η(V ) and D(V ) = {dU : U ∈ η(V )}. Lemma 4.2.10. (1) {η(V ) : V ∈ η} is a partition of η. (2) G is the central product of the subgroups G(V ), V ∈ η. (3) If A(z(V )) = ∅ then G(V ) is transitive on η(V ) and D(V ) is a conjugacy class of 3-transpositions of G(V )∗ . (4) If A(z(V )) = ∅ then G(V )∗ ∼ = D12 has two orbits V G(V ) and V˜ G(V ) on η(V ) and {z(U ) : U ∈ η(V )} = {z(V )} ∪ A(z(V )) is of order 3.    Proof. By 3.1.25 there is a partition η = η0 η1 · · · ηr of η such that, setting Di = {dV : V ∈ ηi } and G∗i = Di , we have G∗ = G∗0 × G∗1 × · · · × G∗r ; for i = j, G∗i centralizes ηj ; D0 is a set of 3-transpositions of G∗0 ; and for i > 0, G∗i ∼ = D12 . Indeed when i > 0, {z(V ) : V ∈ ηi } = {z(V )} ∪ A(z(V )) is of order 3

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by 3.2.4. In particular η0 is partitioned by the sets V G = V G0 for V ∈ η0 , while for i > 0 and V ∈ ηi , we have ηi = η(V ). Therefore (1) follows. Let E1 , . . . , Es be the orbits of G∗0 on D0 and Hi∗ = Ei . By 4.1.4, G∗0 = ∗ ∗ ∗ H1 · · · H s is acentral product and Hi is transitive on Ei . Set ξi = {V : dV ∈ Ei }; then ξ1 · · · ξs is a partition of η0 , and by 4.2.9, for V ∈ ξi , ξi = V Hi = η(V ) and G(V )∗ = Hi∗ , so G(V ) is transitive on η(V ). Similarly we saw earlier that for j > 0 and V ∈ ηj , ηj = η(V ) and G(V )∗ = G∗j . This completes the proof of (3) and (4). Next for Vi ∈ η, i = 1, 2, such that η(V1 ) = η(V2 ), [dV1 , dV2 ] = 1, so dVi centralizes V3−i by 3.1.22; hence K(Vi ) centralizes V3−i . Then as K = K(V3−i ) is the unique member of Λ containing V3−i , K(Vi ) acts on K. Therefore G(Vi ) acts on K. We may choose K(Vj ) ∈ Ω for j = 1, 2, so [K(Vi ), K] = 1 by 3.1.5.1. Then as G(Vj ) = K(V )G(Vj ) : V ∈ η(Vj ) and K(V ) ∈ Ω , it follows that G(Vi ) centralizes K, and then also G(Vi ) centralizes G(V3−i ). Now (2) follows from 4.2.4.1, completing the proof.  Remark 4.2.11. Because of 4.2.10, in many situations we may assume either (1) μ ∼ = D12 and A(z) = ∅ for each z ∈ Z, so that G is described in 4.2.6, or (2) G is transitive on η and D is a conjugacy class of 3-transpositions of μ. Lemma 4.2.12. (1) Suppose ai ∈ D for i = 1, 2 such that a1 , a2 = ν ∼ = S3 . Then for each d ∈ D, d commutes with some member of D ∩ ν. (2) If D is a set of 3-transpositions of μ then μ satisfies Hypothesis 4.1.11. Proof. Assume the setup of (1) and let zi = ζ(ai ) and F = z1 , z2 . From 4.2.5, D ∩ ν = aν1 is of order 3 and F # = {ζ(d) : d ∈ D ∩ ν}. Let Z = Z . By 3.1.20.3, [Z, d] ≤ ζ(d) , so d induces a transvection on Z, and hence there exists f ∈ CF (d). Then by 4.2.8.1, d centralizes the member of D ∩ ν in ζ −1 (f ), proving (1). Part (2) follows from (1) and 4.1.14.  Lemma 4.2.13. Assume D is a set of 3-transpositions of μ and D(D) is disconnected. Let d ∈ D. Then G is transitive on η, μ is transitive on D, and one of the following holds: (1) μ ∼ = S3 and |Vd | = 1. (2) μ ∼ = S4 and |Vd | = 2. (3) μ is isomorphic to Weyl(D4 ) or Weyl(D4 )/Z(Weyl(D4 )) and |Vd | = 4. Proof. As D(D) is disconnected it follows from 4.2.10.2 that η = η(U ) for each U ∈ η, and then, from 4.2.10.3 that D is a conjugacy class of 3-transpositions of μ. Hence G is transitive on η by 4.2.8.1. By 4.2.12.2 and 4.1.18, μ/O2 (μ) ∼ = S3 . Let a = dV and set 2n−1 = |Va |. By 4.1.8, μ = di : 0 ≤ i ≤ n , where a = d1 , di ∈ a⊥ = Va for 1 ≤ i ≤ n, and d0 ∈ Aa . Let Vi ∈ η with di = dVi . By 3.1.20, [W, di ] ≤ Vi , so G acts on V0 V1 · · · Vn . Then as G is transitive on η, W = V0 V1 · · · Vn . Let Ui = ui be the subgroup of Vi of order 4, A = V1 · · · Vn , B = U1 · · · Un , zi = z(Vi ), Z = zi : 1 ≤ i ≤ n and A+ = A/Z. If n ≤ 2 then μ ∼ = S3 or S4 , using 4.1.5, and (1) or (2) holds. Thus we may assume n > 2. Therefore as |a⊥ | = 2n−1 , there exists X ∈ η with d = dX ∈ a⊥ − {ai : 1 ≤ i ≤ n}. Then K(X) centralizes A but not u ∈ X of order 4. As W = V0 · · · Vn , u = by with b ∈ A and y ∈ V0 . Now

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z = u2 = b2 y 2 , so y 2 = b−2 z ∈ Az ∩ V0 . But K(X) centralizes z, A but not z0 , so y 2 = 1. Thus y = z0 and b2 = z. Next [A, K(X)] = 1 = [u, K(X)], so y = z0 . As K(Vi ) centralizes Vj for i = j and 1 ≤ i, j ≤ n, A+ = V1+ × · · · × Vn+ , so as b2 = z, b = u1 · · · uk e for some ui , some k ≤ n, and some e ∈ Z. Thus u = z0 u1 · · · uk e. Let r ∈ a⊥ − {d}. Then u = ur = ζ(r)z0 ur1 · · · urk e = u or uζ(r), depending on whether r = di for some 1 ≤ i ≤ k or not. It follows that n = 3, so that |Va | = 2n−1 = 4. Moreover by 4.1.10, μ is ω = Weyl(D4 ) or ω/Z(ω), and hence (3) holds, completing the proof.  Lemma 4.2.14. If D is a set of 3-transpositions of μ then μ satisfies Hypothesis 4.1.12. Proof. By 4.2.12.2, μ satisfies Hypothesis 4.1.11. Then the lemma follows from 4.2.13 and 4.1.7.1.  4.3. The groups ω ¯ (Φ, m) Notation 4.3.1. Let Φ be a connected Coxeter diagram with no multiple bonds. Thus Φ is An , Dn , or En . Take n ≥ 2 and if Φ is Dn take n ≥ 3. Set I = {1, . . . , n} and write Weyl(Φ) for the Coxeter group defined by Φ. Let m ≥ 8 be a power of 2. ¯ = ω Define G ¯ (Φ, m) to be the group presented by generators k¯i , v¯i , i ∈ I, subject to the following relations: ¯ m/2 m/4 = 1 = [¯ vi , v¯j ], k¯i2 = v¯i , v¯i−1 = v¯iki for all i, j ∈ I. (1) v¯i vi , k¯j ] = v¯j . (2) If (i, j) is an edge in Φ then (k¯i k¯j )3 = 1 and [¯ (3) If i, j ∈ I are distinct and (i, j) is not an edge in Φ then [k¯i , k¯j ] = 1 = [k¯i , v¯j ]. ¯ = ¯ ¯ i = k¯i , v¯i . Set W vi : i ∈ I and for i ∈ I, set V¯i = ¯ vi , and K ∼ ¯ ¯ From 4.3.8.2, Ki = Qm ; set z¯i = z(Ki ). In 5.8.2 we find that if Φ = Dn then ¯ then we define ω(Dn , m) = G/¯ ¯ u . u ¯ = z¯n−1 z¯n ∈ Z(G); Lemma 4.3.2. Let (W, S) be a Coxeter system of type Φ. ¯ is an abelian normal subgroup of G. ¯ (1) W ∗ ¯ ¯ ¯ (2) Set G = G/W and S = (si : i ∈ I). Then the map si → k¯i∗ extends to a ¯ ∗. surjective homomorphism of W onto G ¯ is abelian. By 4.3.1.1, Proof. From 4.3.1.1, [¯ vi , v¯j ] = 1 for all i, j ∈ I, so W −2 ¯ ¯ ¯ ¯ , and v¯i = [¯ vi , ki ] ∈ W and for i = j in I, [¯ vi , kj ] = 1 or v¯j , so [¯ vi , k¯j ] is in W ¯ ¯ hence W  G, completing the proof of (1). m/4 ¯ we have k¯∗2 = 1. If (i, j) is an edge in Φ then (k¯i k¯j )3 = 1, so As k¯i2 = v¯i ∈W i ∗ ¯∗ 3 ¯ (ki kj ) = 1. If (i, j) is not an edge in Φ then [k¯i , k¯j ] = 1, so [k¯i∗ , k¯j∗ ] = 1. Therefore  (k¯i∗ : i ∈ I) satisfies the defining relations for W, so (2) follows. Lemma 4.3.3. Assume Hypothesis 4.2.1 and adopt Notation 4.2.3. Let (W, S) be a Coxeter system of type Φ with S = (si : i ∈ I) and assume there exists a surjective homomorphism α : W → G∗ = G/CG (W ) with Sα ⊆ D and ker(α) ≤ Z(W). For i ∈ I set si α = di = dVi for Vi ∈ η, Ki = K(Vi ), and ki ∈ Ki − Vi . (1) G = Ki : i ∈ I and W = Vi : i ∈ I . (2) We can choose generators vi for Vi such that the maps v¯i → vi and k¯i → ki , ¯ → G with π : K ¯ i → Ki an i ∈ I, extend to a surjective homomorphism π : G isomorphism.

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¯ π = W and if |W | ≥ (m/2)n then π : W ¯ → W is an isomorphism and (3) W ¯ = V¯1 × · · · × V¯n . W ¯ ∗ = G/ ¯ W ¯ → (4) If α is an isomorphism then π induces an isomorphism π ∗ : G ∗ ∗ ¯ → g¯πW , CG (W ) = W , and G ¯ ∼ G/W via π : g¯W = W. ¯ → G is an isomor(5) If α is an isomorphism and |W | ≥ (m/2)n then π : G phism, so G ∼ ¯ (Φ, m) with W = CG (W ) ∼ = Znm/2 and G/W ∼ = Weyl(Φ). =ω Proof. As α : W → G∗ is a surjection and W = S , we have G∗ = di : i ∈ I . Therefore as di = ki∗ , it follows that G = G0 CG (W ), where G0 = Ki : i ∈ I . As W is transitive on its set R of reflections, we have R = sW 1 , so Rα = D and ∗ G is transitive on D. Therefore as G = G0 CG (W ), we conclude from 4.2.8.1 that G0 is transitive on η. Then as Vi ≤ Ki ≤ G0 , also W = η ≤ G0 . Set H = CG (W ). Then [Ki , H] ≤ CKi (Vi ) = Vi ≤ W , so G0 centralizes H/W . Hence as G = G0 H with W ≤ G0 , we have G0  G. Thus as G = K G by 4.2.4.1, we conclude that G = G0 . Pick generators vi of Vi . As Ki ∼ = Qm and W is abelian, (ki , vi : i ∈ I) satisfies the relations in 4.3.1.1. If (i, j) is not an edge in Φ, then by 3.1.22, [Ki , Kj ] = 1, so the relations in 4.3.1.3 are satisfied. Finally as the graph of Φ is a tree, we conclude from parts (1) and (5) of 4.2.5 that we can choose the vi so that the relations in 4.3.1.2 are satisfied. This proves (2). ¯  G ¯ by 4.3.2.1 and as ¯ π = V¯i π : i ∈ I = Vi : i ∈ I ≤ W . As W Next W ¯ ¯ ¯ π : G → G is a surjection, we have W π  G. Then as V1 ≤ W π and G is transitive ¯ π, so W ¯ π = W , completing the proof of (1). on η, we conclude that W = η ≤ W ¯ ¯ is abelian by 4.3.2.1, it ¯ ¯ Further as W = Vi : i ∈ I with |Vi | ≤ m/2, and as W n ¯ ¯ ¯ ¯ follows that |W | ≤ (m/2) with W = V1 × · · · × Vn in case of equality. Then (3) follows. ¯ ∗ → G/W defined by ¯ π = W , π induces the homomorphism π ∗ : G Next as W ∗ ∗ ¯∗ → ¯ g π)W . Let ψ : G/W → G be the natural map; then ϕ = π ∗ ψ : G (¯ gW )π = (¯ ∗ G is a surjective homomorphism. Suppose that α is an isomorphism. Then G∗ ∼ = W, so as ϕ is a surjection, ∗ ¯ ∗ |, so all inequalities are equalities, and hence ¯ |G | ≥ |W|. But by 4.3.2.2, |W| ≥ |G ¯ ∗ → G∗ is an isomorphism. Therefore (4) in particular W = CG (W ) and π ∗ : G holds. Finally (3) and (4) imply (5).  Remark 4.3.4. Let (X, Γ) be a Coxeter system of type Dn with n ≥ 3; say Γ = {s1 , . . . , sn }. Set x0 = sn−1 sn , X0 = xX 0 , Γ0 = {s1 , . . . , sn−1 }, and L = Γ0 . Then (L, Γ0 ) is a Coxeter system of type An−1 , so L ∼ = Sn is a complement to X0 in X, X0 is the core of the permutation module for L (ie. the vertices of even weight in the permutation module) on an n-set, and x0 is a vector of weight 2 in the module X0 . Lemma 4.3.5. Assume Hypothesis 4.2.1 and adopt Notation 4.2.3. Assume D is a conjugacy class of 3-transpositions in μ and O2 (μ) ≤ Z(μ). Then (1) μ ∼ = Weyl(Dn ) for some n ≥ 3. (2) If n is even then Z(μ) = ι ∼ = Z2 and ι inverts W . Proof. By 4.2.14, μ, D satisfies Hypothesis 4.1.12, so as O2 (μ) ≤ Z(μ) and μ is transitive on D, we conclude from 4.1.22 that there exists an integer n ≥ 3 such that μ is W or W/Z(W), where W = Weyl(Dn ). Thus in proving (1) we may assume Z(W) = 1 and μ ∼ = W/Z(W), and it remains to derive a contradiction. As Z(W) = 1 it follows that n is even. On the other hand in proving (2), n is even by

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hypothesis. So, in any event, n is even and there exists a Coxeter system (W, S) and a surjection α : W → μ with Sα ⊆ D and ker(α) ≤ Z(W). As n is even, Z(W) = ι is of order 2. Observe that the hypotheses of 4.3.3 are satisfied, and adopt the notation of that lemma. In particular si α = di and Vi = vi ≤ W . Let x = sn−1 sn , Q = xW , and L = si : 1 ≤ i < n ≤ W. From Remark 4.3.4, Q is the core of the permutation module for L ∼ = Sn . Thus, as in section 5.1, we can identify Q with the set E of vectors of even weight in the power set of I under the operation of symmetric difference, and x = [Q, sn−1 ] = en−1,n . In particular Q  for s ∈ sL 1 , s is a transposition (i, j) on I and [Q, s] = ei,j , so s = {s, s }, where  ss = ei,j . Next CQ (L) = ι , where ι = eI = e1,2 e3,4 · · · en−1,n . For i < n − 1 odd, ei,i+1 = si si centralizes Vn as si α = di and si α = di centralize dn . Similarly en−1,n = sn−1 sn inverts Vn , as dn inverts Vn and dn−1 centralizes Vn . Thus ι inverts Vn , so as ι ∈ Z(W), ι inverts W = Vnμ . This shows ι = 1, proving (1), and completes the proof of (2).  Lemma 4.3.6. Assume Hypothesis 4.2.1 and adopt Notation 4.2.3. Assume μ/Z(μ) ∼ = Weyl(Φ)/Z(Weyl(Φ)) for Φ of type E7 or E8 . Then μ ∼ = Weyl(Φ). ∼ ∼ Proof. Let  W = Weyl(Φ) and assume μ = W/Z(W). Then μ = Sp6 (2) or Hence d∈D∩S d = 1, so for d ∈ D ∩ S, d ∈ Dd ∩ S . But d = dV for some V ∈ η and by 3.1.22, each a ∈ Dd centralizes V , a contradiction as d inverts V .  O8+ (2).

¯ W ¯ ∼ Lemma 4.3.7. (1) G/ = Weyl(Φ). ¯ ¯ ). ¯ ¯ (2) W = V1 × · · · × Vn = CG¯ (W ¯ G ¯ Ω ¯ =K ¯ 1 ∩ S, ¯ F¯ = FS¯ (G), ¯ and τ¯ = (F¯ , Ω). ¯ Then τ¯ is a (3) Let S¯ ∈ Syl2 (G), quaternion fusion packet. ¯ i }. ¯ K ¯ i )) = {K (4) Ω(z( ¯ (5) G is isomorphic to a member of M (τ ), where τ is the fusion packet of + + ˜ SL+ n+1 [m], Spin2n [m], E6 [m], E7 [m], E8 [m], for Φ of type An , Dn , E6 , E7 , E8 , respectively. Proof. We first prove (5), and in the process establish (1) and (2). We exhibit a group L of Lie type over Fq , where (q 2 − 1)2 = m, such that, writing τ (L) for the Lie packet of L, μ = μ(τ (L)) ∼ = Weyl(Φ) = W and, for W ∈ W (τ ), we have ¯ → G is an isomorphism for G ∈ M (τ (L)), |W | ≥ (m/2)n . Then by 4.3.3.5, π : G ∼ W = CG (W ), and G/W = W. ∼ Suppose that Φ = Dn and let L = Spin+ 2n (q). Then by 5.3.4.4, μ = W, while n from the proof of 5.3.5, |W | ≥ (m/2) . Next suppose Φ = An and let L = SLπn+1 (q), where q ≡ π mod 4. By 5.4.13.4, ∼ μ = Sn+1 = W, while from the proof of 5.4.13.5, |W | ≥ (m/2)n . Finally suppose Φ = En . Then μ ∼ = W by 5.5.8, while from the proof of 5.5.10, |W | ≥ (m/2)n . This completes the proof of (5). During the proof we showed that (1) and (2) hold. ¯ → G is an isomorphism with K ¯ i π = Ki , (3) holds. As L is a universal As π : G group, Ω(z(Ki )) = {Ki }, so applying the isomorphism π, we conclude that (4) holds. 

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Lemma 4.3.8. Assume Hypothesis 4.2.1 and adopt Notation 4.2.3. Assume D is a conjugacy class of 3-transpositions in μ. Then (1) μ ∼ = Weyl(Φ), where Φ is of type An , Dn , or En . ¯ =ω ¯ →G (2) Let G ¯ (Φ, m). Then there is a surjective homomorphism π : G ¯ i π ∈ Λ is an isomorphism, and ker(π) ≤ W ¯. ¯ π = W, π : K ¯ i → Ki = K with W ¯ (3) ker(π) ≤ Z(G). (4) W = CG (W ). Proof. By 4.2.14, the pair μ, D satisfies Hypothesis 4.1.12. Hence by 4.1.25, μ = W/Z for some Z ≤ Z(W), where W = Weyl(Φ) and Φ is of type An , Dn , or En . Claim Z = 1, so that (1) holds. If Z(W) = 1 this is trivial, so assume otherwise. Then Φ = Dn with n even, E7 , or E8 . Now 4.3.5.1 and 4.3.6 complete the proof of the claim. Thus (1) is established. ¯ → G with W ¯ π = W and π By (1) and 4.3.3 there is a surjection π : G ∗ ¯ ¯ ¯ induces isomorphisms Ki → Ki and π : G/W → G/W = μ. This last fact says ¯ , so (2) holds. ker(π) ≤ W ¯ i ∩ ker(π) = 1. Therefore K ¯ i centralizes ¯ i → Ki is an isomorphism, K As π : K ¯ ¯ ker(π) by 3.1.4, so G = Ki : i ∈ I centralizes ker(π), proving (3). It remains to prove (4), so assume W < CG (W ) and let Y be the preimage ¯ , so Y ∗ = 1 and [Y, W ¯ ] ≤ ker(π) ≤ Z(G) ¯ by (3). ¯ of CG (W ). Then Y ≤ W in G ∗ ∗ ¯ ¯ ¯ Therefore as W = CG¯ (W ) by 4.3.7.2, Y ≤ O2 (G ), so in particular O2 (W) = 1. Hence Φ = Dn , E7 or E8 . Suppose Φ is Dn . If n is even then Z(W) = ι is the unique minimal normal subgroup of W by Remark 4.3.4, so ι ∈ Y ∗ , contradicting 4.3.5.2. Thus n is odd, ¯ ∗ is irreducible on Q = O2 (G ¯ ∗ ) by Remark 4.3.4, and hence Y ∗ = Q. However so G x = dn−1 dn ∈ Q and x inverts Vn , a contradiction. ¯ ∗ ) = ι is of order 2, and hence Y ∗ = Therefore Φ is En for n ∈ {7,8}, so O2 (G ι . As in the proof of 4.3.6, ι = d∈D∩S d, so as D ∩ S − {d1 } ⊆ Dd1 ⊆ CD (v1 ), we conclude that v¯1ι = v¯1d1 = v1−1 , a contradiction that completes the proof of (4). 

CHAPTER 5

Some examples

Chapter 5 is devoted to our examples τ of quaternion fusion packets appearing as conclusions in Theorems 1 and 2. ˇ n . These Section 5.1 considers the groups and 2-fusion systems AEn and AE groups have an associated quaternion fusion packet τ by 5.1.6.3. The groups are essentially the Weyl groups of type Dn , obtained from their embedding in Ωn (q) ˆ n , defined in for suitable and q. The quaternion fusion packet of the group AE 5.1.16 and 5.1.17, is universal among coverings of τ by 5.1.21. In section 5.2 the 2-share of the order of groups of Lie type over a field Fq of odd order q is described from a point of view convenient for discussing their quaternion fusion packets. Sections 5.3; 5.4; and 5.5 discuss the quaternion fusion packets of orthogonal and spin groups; linear, unitary, and symplectic groups; and exceptional groups, respectively. Section 5.6 contains a proof that the 2-fusion system FS (G) of a simple group G of Lie type and odd characteristic (which is not a Goldschmidt group) is simple. This is used in Theorem 5.6.18 to show that if G is a known simple group that is not Goldschmidt, then the 2-fusion system of G is simple. As a consequence, the known simple 2-fusion systems are the 2-fusion systems of the simple groups other than Goldschmidt groups, together with the exotic Solomon systems. Section 5.7 considers the systems Lπn [m], their coverings, and their subsystems ω ¯ (An−1 , m). In particular 5.7.4 and 5.7.5 contain detailed information about ω ¯ (A3 , m). Similarly section 5.8 contains information on ω ¯ (Dn , m), particularly when n = 4. ω (Cn , m) of ω ¯ (Dn , m), found Section 5.9 considers extensions ω ¯ (Cn , m) and 2¯ in the Lie packets of Spin2n+1 (q) and Spin− (q) for suitable q. The factor group 2n+2 of these systems modulo CS (W ) is the Weyl group of type Cn . Finally section 5.10 studies the constrained systems L3 (2)/E8 , L3 (2)/E64 , and L3 (2)/23+6 and the coverings of such systems. We close this chapter with a section listing the most basic notions, notation, and results from the theory of quaternion fusion packets that we’ve generated in the first five chapters of this manuscript. Hopefully this will provide the reader with a convenient place to refer when the need to refresh his or her memory arises.

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5.1. AEn In this section B = Sym(I) is the symmetric group on I = {1, . . . , n} with n ≥ 5, and V is the power set of I regarded as an F2 -space via X +Y = (X ∪Y )−(X ∩Y ) the symmetric difference of subsets X and Y of I. We also write eX for X. Observe B ≤ GL(V ) and for b ∈ B, ebX = eXb . Indeed V is the n-dimensional permutation module for B over F2 . For X ⊂ V , write M ov(X) for the set of points in I moved by X. Next form the semidirect product H = BV and observe that H is the wreath product Z2 wr Sn ; in particular Z(H) = eI . Define E = ei,j : i, j ∈ I distinct , A = F ∗ (B), and G = O 2 (H). Then G = AE is the split extension of E ∼ = E2n−1 by A ∼ = An . If n is odd then V = E ⊕ eI is a B-invariant decomposition of V , so G has no center. If n is even then eI = Z(G). In any case set H ∗ = H/eI and H + = H/V . Denote the isomorphism type of G∗ by AEn , the isomorphism type of A∗ V ∗ by AVn , and the isomorphism type ˇ n. of G by AE Lemma 5.1.1. Let X be the stabilizer in A of n. Then XE ∼ = Z2 wr An−1 and X E∗ ∼ = Z2 wr An−1 or AVn−1 for n odd or even, respectively. ∗

Proof. Set vk = ek,n for 1 ≤ k < n. Then {vk : 1 ≤ k < n} is a basis for E such that vkx = vkx for x ∈ X. Thus XE ∼ = Z2 wr An−1 . Further σ = k vk =  eI + nen , so σ ∗ = 0 iff n is even, completing the proof. Lemma 5.1.2. (1) H 1 (A, E ∗ ) = 0, F2 for n odd, even, respectively. (2) If n is even then H 1 (A, V ) = 0. (3) If n is even and U is an indecomposable F2 A-module of dimension n with [U, A] A-isomorphic to E, then U is isomorphic to V as an A-module. Proof. See Exercise 6.3 in [Asc86] for (1), and the fact that V ∗ is the module U (A, E ∗ ) of section 17 of [Asc86]. Then by 17.3 in [Asc86], V ∗ is transitive on the complements to V ∗ in G∗ , so V is transitive on the complements to V in AV , and hence (2) holds. Assume the setup of (3) and let X be the stabilizer in A of n. By 5.1.1 and (1), U splits over [U, A] as an X-module, so U is an image of the permutation module for A on A/X; then (3) follows.  Definition 5.1.3. Write Aˆ = Aˆn for the universal 2-covering group of A; that is Aˆ is the largest perfect group Y such that Z(Y ) is a 2-group and Y /Z(Y ) ∼ = A. ˆ = σ is of order 2. Then (cf. 33.15 in [Asc86]), Z(A) ˆ for the semidirect product of Aˆ with E, such that σ ∈ Z(G) ˆ If n is even, write G ∼ ˆ ˆ and G/σ = G. As A is the universal 2-covering group of A, it follows from 5.1.2.1 ˆ is the largest 2-covering group Λ of AEn such that O2 (Λ) is elementary that G abelian. Lemma 5.1.4. Assume U is a normal elementary abelian 2-subgroup of a finite group X such that X/CU (X) ∼ = AEn and U = [U, X]. Then either X ∼ = AEn or n ˇ n or G/σe ˆ is even and X ∼ . = AE I

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Proof. As U = [U, X] it follows from 5.1.2.1 that Z = CU (X) = 0 if n is odd, ˆ In particular if while if n is even it follows from 5.1.3 that X is an image of G. ˆ n is odd then the lemma holds, so we may take n even and X = G/Σ for some ∼ ˆ ˇ ˆ Σ ≤ Z(G) = σ, eI . Then X = AEn , AEn for Σ = Z(G), σ , respectively, while Σ is not contained in eI as in that event X/U ∼ = Aˆ rather than A. Hence the lemma holds.  Notation 5.1.5. Set F = (1, 2)(3, 4), (1, 3)(2, 4) ≤ A and z = e1,2,3,4 . Set y = (1, 2, 3) ∈ A. Lemma 5.1.6. (1) F y is a normal subgroup of NA (F ) isomorphic to A4 . ˜ ˜ are the Q8 -subgroups of K K ˜ ∼ where K and K (2) O 2 (EF y ) = K Ky , = Q28 . ˜ Moreover K Ky  CG (z). (3) Let F ≤ S ∈ Syl2 (G), Ω = K G ∩ S, and F = FS (G). Then τ = (F, Ω) is ˜ a quaternion fusion packet with Ω(z) = {K, K}. (4) Set k = (n−1)/2, n/2 for n odd, even, respectively. Then μ(τ ) ∼ = Weyl(Dk ). Proof. Set Y = F y and α = {1, 2, 3, 4}. Observe that NA (F ) = (Y × L)λ , where L = Aα ∼ = An−4 , λ = 1 if n = 5, and λ = (1, 2)(5, 6) if n > 5. Hence (1) follows. Further [E, Y ] = ei,j : 1 ≤ i < j ≤ 4 ∼ = E8 , so Y0 = O 2 (Y E) = Y [E, Y ], 2 with O2 (Y0 ) = F [E, Y ] ∼ = Q8 . Hence all but the last remark in (2) follows, with ˜ = ae1,4 , be3,4 , a = (1, 2)(3, 4), and b = (1, 3)(2, 4). K = ae1,3 , be1,4 , K As CG (z) = ENA (F ), we have EY  CG (z) by (1), so Y0  CG (z), completing the proof of (2). ˜ so K ˜ ∈ Ω(z). By construction, Ω is an FObserve next that K e1,5 = K, invariant set of Q8 -subgroups of S, so τ satisfies (QFP1). By (2), CG (z) acts on ˜ so (QFP4) holds. Suppose J ∈ Ω−{K}. Then J + is a root 4-subgroup θ = {K, K}, + of G (ie. a conjugate of F + ) contained in S + , so either J + = K + or J + ∩ K + = 1. In the former case z(J) ∈ z G ∩ Φ(KE) = {z}, so J ∈ θ as CG (z) acts on θ. In the latter case, |K ∩ J| ≤ 2; therefore (QFP2) is satisfied. Similarly if x ∈ CJ (z) ˜ and since it is of order 4 then as CG (z) acts on θ, either x acts on K or K x = K, remains only to verify (QFP3), we may assume the latter. Next x2 acts on K and ˜ |α ∩ β| = 0 or 2. But then up x2 = eβ for some 4-subset β of I, so as K e1,5 = K, to conjugacy under CG (z), x = (5, 6)(7, 8)e5,7 or (1, 2)(5, 6)e1,5 , and in either case x acts on K, completing the proof of (3). It remains to prove (4). If n = 5 we check that μ = μ(τ ) ∼ = Weyl(D2 ); = E4 ∼ G ˜ thus we may take n > 5. If A(z) = ∅ then by 3.2.4.3, z ∩ K K = {z}, whereas ˜ ∩ E of weight 4. Thus by 4.2.10.3, D is a set of z is the unique member of K K 3-transpositions of μ. Set Σ = {{2i, 2i − 1} : 1 ≤ i ≤ k}, a partition of I  = {1, . . . , 2k}. We may assume that S acts on Σ. For i ≤ k, set ti = (2i, 2i − 1), and for i = j set ˜ i,j for the conjugate of K, K ˜ moving M ov(ai,j ), and zi,j ai,j = ti tj . Write Ki,j , K ˜ i,j of order 4 for the involution in Ki,j . Pick elements vi,j ∈ Ki,j and v˜i,j ∈ K + + / Ω then Δ(zi,j ) = {Vi,j , V˜i,j , where Vi,j , V˜i,j are with vi,j = ai,j = v˜i,j . If Ki,j ∈ generated by vi,j , v˜i,j , respectively. Further CK (vi,j ) = V1,2 for |{i, j} ∩ {1, 2}| = 1. Thus η = {Vi,j , V˜i,j : i, j} ⊆ η0 for some η0 ∈ η(τ ). Let W = η ; then {zi,j : ˜ i,j : i, j is transitive on the i, j} = z G ∩ CG (W ), so η = η0 . Also M = Ki,j , K ˜ i,j : i, j}, so μ is transitive on D. Thus D is a class of 3-transpositions. If {Ki,j , K

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˜ in μ then d˜ ∈ Vd , so by 4.1.7 and 4.3.5, μ ∼ d, d˜ are the images of K, K = Weyl(Dl ) for some l. Then as NG (W ) = NG (Σ), NG (W )/W E ∼ = Sk , so l = k, completing the proof of (4).  Lemma 5.1.7. Assume X is a finite group, T ∈ Syl2 (X), J ≤ T , and Γ = J X ∩ T such that ρ = (FT (X), Γ) is a quaternion fusion packet and X = J X . Suppose β : X → G∗ is a surjective homomorphism with ker(β) ≤ Z(X) ∩ O2 (X) ˜ =K ˜ ∗ , and ˜j = z(J). ˜ Set f = j ˜j. and Jβ = K ∗ . Let j = z(J), J˜ ∈ Γ with Jβ ∼ ˇ Then f = Φ(O2 (X)) and X/f = AEn or AEn . Proof. Assume otherwise and let X be a minimal counter example. Set Q = O2 (X) and Q0 = j X , so that Q0 ≤ Q. ˜ Then J x centralizes J J˜ by 3.1.5.1, so Suppose x ∈ X with J x ∈ Γ − {J, J}. x ∗ ˜∗ J β ∈ Ω centralizes K K , and hence J x β ≤ K ∗ E ∗ , so J x ≤ JO2 (X). Thus ˜ = Γ ∩ JO2 (X). Hence for t ∈ j X , J t = J or J, ˜ so [t, j] ∈ f . As f ∈ Z(X) {J, J} we conclude that f = Φ(Q0 ). Then by minimality of X, f = 1. Therefore ˜j = j and as E = z G , we have Q = Q0 Z(X) with Φ(Q0 ) = 1, so Φ(Q) = Φ(Z(X)). Suppose Φ(Q) = 1. Then Z(X/Φ(Q)) = 1, so by minimality of X, X/Φ(Q) ∼ = ˇ n with n even. But then Q/Φ(Q) = (jΦ(Q))X , so Q = Q0 , Φ(Q) = Q0 , so AE Φ(Q) = 1. ˆ ˆ → X Suppose X = G/Z, where Z = σeI . Then ξβ = ζ, where ξ : G ∗ ˆ and ζ : G → G are the natural maps ξ : g → gZ and ζ : g → gΣ and Σ = ˜ = F D where D = e1,3 , e1,4 , z . Therefore σ, eI . From the proof of 5.1.6, K K −1 ∗ ˜∗ ˆ U = ζ (K K ) = F DeI , where Q8 ∼ = Fˆ ≤ Aˆ and Φ(Fˆ ) = σ . Then U0 = ˜ J JZ(X) = U ξ = F0 D, where F0 = Fˆ ξ and Φ(F0 ) = σξ = Z(X). This is a ˜ ˜ = j . contradiction as Φ(U0 ) = Φ(J J)Φ(Z(X)) = Φ(J J) ˆ Therefore X is not G/σeI , so by 5.1.4 it suffices to show Q = [Q, X], so assume otherwise. Claim Q = Q0 . As X = J X and J ≤ O 2 (X), we have X = O 2 (X). Thus ˆ However Φ(J) = j , so JQ0 /Q0 ∼ if Q = Q0 then X/Q0 ∼ = A. = A4 , where as the ˆ ˆ preimage F of F in A is isomorphic to Q8 . This completes the proof of the claim. As Q0 = Q = [Q, X], j ∈ Q − [Q, X]. But as j = ˜j, [J J˜ ∩ Q, J] = z , contradicting j ∈ / [Q, X] and completing the proof.  Lemma 5.1.8. Let g ∈ G, J = K g , X = K, J , t = z(J), α = {1, 2, 3, 4}, and β = αg. Then one of the following holds: ˜ (1) α = β, J + = K + , and J ∈ {K, K}. + ∼ t ˜ (2) |α ∩ β| = 3, X = A5 , and K = K. + ∼ (3) |α ∩ β| = 2, X = S4 , and NJ (K) = CJ (z) ∼ = Z4 . ˜ (4) |α ∩ β| = 1, X + ∼ = A7 , and K t = K. ˜ (5) α ∩ β = ∅, X + ∼ = E16 , and J centralizes K K. Proof. The structure of X + is a consequence of 3.2.1 in [Asc08b]. We saw ˜ iff |α ∩ β| is odd. during the proof of 5.1.6 that K t = K  Lemma 5.1.9. Let g ∈ G, α = {1, 2, 3, 4}, t = z g , J = K g and β = αg. (1) t ∈ D(z) iff α ∩ β = ∅. (2) t ∈ D ∗ (z) − D(z) iff |α ∩ β| = 2. (3) If n > 8 then D(τ ) is connected. (4) If n > 5 then D ∗ (τ ) is connected.

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˜ t] = 1 iff α ∩ β = ∅ by 5.1.8. Proof. Observe that by 3.1.14, t ∈ D(z) iff [K K, Thus (1) holds. Similarly by 3.1.15, t ∈ D ∗ (z) − D(z) iff NJ (K) ∼ = Z4 iff |α ∩ β| = 2 by 5.1.8. Thus (2) holds. Then (1) implies (3) and (2) implies (4).  Lemma 5.1.10. Assume n = 6 or 7, Y = G or G∗ , and X = K X ≤ Y with ∼ X = AE5 . Let ξ ∈ Aut(Y ) be of odd order centralizing X. Then ξ = 1. Proof. Let Q = O2 (Y ) and Y ! = Y /Q. Then ξ induces an automorphism ξ ! of odd order on Y ! centralizing X ! , so as |CAut(Y ! ) (X ! )| ≤ 2, we conclude ξ ! = 1. Thus [Y, ξ] ≤ Q, so as Q = O2 (X)Y , ξ also centralizes Q. Then as ξ is of odd order, ξ = 1.  Lemma 5.1.11. Assume n = 8 and set γ = {5, 6, 7, 8}. Then (1) z ∗ = e∗γ , so Ω∗ = Ω(z ∗ ) is of order 4 and O(z ∗ ) ∼ = Q48 . (2) D∗ (z ∗ ) ⊆ O(z ∗ ). Proof. Let α = {1, 2, 3, 4}. As n = 8, eα + eγ = eI , so (1) holds. Then (1) and 5.1.9.2 imply (2).  Y0∗ ,

˜ Lemma 5.1.12. Let X = G or G∗ , u = z or z ∗ , Y0 = K Ky , and Y = Y0 or respectively. Then (1) Either Y  CX (u) or X ∼ = AE8 . ˇ 8 and NX (Ω) is maximal in X. (2) Either CX (u) is maximal in X or X ∼ = AE

Proof. By 5.1.6.2, Y0  CG (z), while CG (z)∗ = CG∗ (z ∗ ) unless n = 8. Thus (1) holds. Part (2) is well known; cf. Exercise 5.6.2 in [Asc86].  Lemma 5.1.13. Let M be the F2 -permutation module for A on the set Γ of 2-subsets of I, and set M0 = ei,j + ei,k + ej,k : {i, j, k} is a 3-subset of I . Then M/M0 is isomorphic to E as an F2 A-module. Proof. Set M ! = M/M0 . First observe that E = eγ : γ ∈ Γ and for each 3-subset {i, j, k} of I, ei,j + ei,k + ej,k = 0 in E. Therefore E is an image of M ! , so it suffices to show that dim(M ! ) ≤ n − 1. But for i, j > 1 distinct, e!i,j = e!1,i + e!1,j by definition of M ! , so M ! = e!1,i : i > 1 is indeed of dimension at most n − 1.  Lemma 5.1.14. Let U be an irreducible F2 A-module, set Y = y , and let X be the global stabilizer in A of {1, 2}. Assume (1) there exists 0 = u ∈ CU (X), and (2) dim(uY ) = 2. Then U is the natural module E ∗ for A. Proof. Adopt the notation of 5.1.13. As A is irreducible on U , U = uA , so there is an F2 A-surjection ϕ : M → U with uϕ = e1,2 . Further eY1,2 = {e1,2 , e1,3 , e2,3 }, so by (2), e1,2 + e1,3 + e2,3 ∈ ker(ϕ). Therefore M0 ≤ ker(ϕ), so the lemma follows from 5.1.13.  Lemma 5.1.15. Let τ ∗ = (FS ∗ (G∗ ), Ω∗ ) and O = O(τ ∗ ). Assume T is a 2-group acting on G∗ and centralizing O, and set Q = O2 (G∗ T ). Then |T Q : Q| ≤ 2.

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Proof. Write n = 4r + a with 0 ≤ a ≤ 2, and for 1 ≤ i ≤ r let Ii = ˜ i : 1 ≤ i ≤ r}, where {4i − j : 0 ≤ j < 4} and zi = eIi . We may take Ω = {Ki , K ˜ ˜ K z(Ki ) = z(Ki ) = zi and M ov(Ki i ) = Ii . Next AutT G∗ (G∗+ ) ≤ H + ∼ = Sn and O + b+ is a Sylow 2-subgroup of CH + (O + ), where b+ = 1 if a ∈ {0, 1} and b+ = (4r+1, 4r+2) otherwise. Therefore T ≤ OQb , so T ∩OQ is of index at most 2 in T . Therefore it suffices to show that COQ (O) ≤ Q. Next G∗ is irreducible on E ∗ , so E ∗ ≤ Z(Q). Then by 5.1.2, |Q : CQ (G∗ )E ∗ | ≤ 2 with AutQ (G∗ ) = V ∗ . Passing to G∗ T /CQ (G∗ ), we may assume CQ (G∗ ) = 1 and then take V ∗ = Q. So it suffices to show Z(OV ∗ ) ≤ V ∗ . Let Vi = Ii , so that V = V1 ⊕· · ·⊕Vr ⊕V0 , where V0 = e4r+j : 1 ≤ j ≤ a . Let P be the preimage in AE ˜ i . Then P V = P1 V1 ×· · ·×Pr Vr ×V0 , of O, so that P = P1 ×· · ·×Pr with Pi = Ki K ∗ ∗ and for x ∈ P V with x ∈ Z(OV ), we have x = x1 · · · xr x0 with xi ∈ Pi Vi for 1 ≤ i ≤ r and x0 ∈ V0 . Then for 1 ≤ i ≤ r, x∗i centralizes Pi∗ ∼ = Pi , so xi ∈ zi ,  and hence x∗ ∈ V ∗ = Q∗ , completing the proof. Remark 5.1.16. Suppose n is odd; then, up to isomorphism, there exists a ˆ n largest group X satisfying the hypotheses of 5.1.7. We denote this group by AE 1+2w and observe that ρ = (FT (X), Γ) is a quaternion fusion packet, Q = O2 (X) ∼ 2 = is extraspecial, where w = (n − 1)/2, f = Z(Q) = Z(X) with f = j ˜j and j, ˜j ∈ ZT (ρ) with |Γ(j)| = 1, and X/Z(X) ∼ = AEn . Also the induced quadratic form on Q/Z(Q) is the form q of Exercise 7.7 in [Asc86]. Namely we can find a subpacket ρ = (FT (X), Γ) of the Lie packet ξ = (F, Γ) of Spinn (q) and a homomorphism β : X → AEn satisfying the hypothesis of 5.1.7. In ξ, Γ(j) = {J} is of order 1 for j ∈ ZS (ξ), so from 5.1.7, f = j ˜j is an involution, Γ(j) = {J} is of order 1, and setting Q = O2 (X) and n = 2w + 1, Q ∼ = 21+2w is ∼ extraspecial with f = Z(Q) = Z(X) and X/Z(X) = AEn . As E is an absolutely irreducible F2 A-module, A preserves a unique quadratic form q on E ∼ = Q/Z(Q) (cf Exercise 9.1 in [Asc86]), so from Exercise 7.7 in [Asc86], q is the form described in that exercise. Thus it remains to show there is a largest group satisfying the hypotheses of 5.1.7, so assume β2 : X2 → G∗ is such a group with X2 not isomorphic to G or X. Let M be the universal 2-covering group of G and X1 = X. Recall ˆ in Definition 5.1.3 and let Zˆ = Z(A) ˆ = Z(G) ˆ = σ . By 8.17 the definition of G ∗ ˆ ˆ ˆ ˆ in [Asc94], F (M ) = Q = Z × [Q, M ] with [Q, M ] = Q. Thus Xi = M/Zi with Zi generated by σ or f σ for i = 1, 2. Now there exist surjections α1 : M → X, α : ˆ and γ : G ˆ → G with α1 β = αγ. Therefore by 6.1 in [Asc94], M ≤ G ˆ ×X M → G, ˜ ∩ A, u ∈ G, ˆ and x ∈ X with is the fiber product of this diagram. Let y ∈ Y = K K xβ = y = uγ. Then xu is in the fiber product M with (xu)2 = x2 u2 = f σ as the ˆ and X of Y is isomorphic to Q8 . But for i = 1 or 2, Zi = σf , so preimage in G the image in Xi = M/Zi of xu is an involution, a contradiction as, by 5.1.7, the ˜ is isomorphic to Q8 × Q8 . This contradiction establishes the preimage in Xi of K K uniqueness of X. Definition 5.1.17. Suppose n = 2w is even and set n0 = n + 1. Form I0 = {1, . . . , n0 }, G0 , E0 , etc. as at the beginning of the section, but with respect to n0 . Form the group X0 of 5.1.16 with respect to n0 . Regard I as a subset of I0 invariant under T0 ∈ Syl2 (X0 ) and let v ∈ Q0 with vβ0 = eI ∈ E0 . Set X = CX0 (v) ˆ n for X/v . ˆ n for X. If w is even, write H AE and write AE

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Lemma 5.1.18. Assume n = 2w is even and adopt the notation in 5.1.17. Set Q = O2 (X), T = CT0 (v), Γ = Γ0 , and ρ = (FT (X), Γ). (1) ρ is a quaternion fusion packet with |Γ(j)| = 1 for j ∈ ZT (ρ). (2) Q = CQ0 (v) = v P where P ∼ = 21+2(w−1) is extraspecial. (3) Φ(Q) = f0 ∼ = Z2 , Z(X) = Z(Q) = v, f0 , and Q = [Q, X]. ∼ ˇ (4) X/Z(X) ∼ AE = n and X/f0 = AEn . (5) If w is odd then Z(X) = v ∼ = E4 . = Z4 , while if w is even then Z(X) ∼ Proof. As vβ0 = eI , CX0 (v)/CQ0 (v) ∼ = CA0 (eI ) ∼ = An and E0 ∼ = V as a A . As Q is extraspecial CA0 (eI )-module by 5.1.1. Thus Q = CQ0 (v) and X/Q ∼ = n 0 of order 21+2w , part (2) follows. By (2), Φ(Q) = Φ(Q0 ) = f0 . As E0 ∼ = V as a CA0 (eI )-module, it follow that Q/f0 = [Q0 /f0 , X] is the unique X-invariant hyperplane E of V . Thus (3) and (4) hold. From 5.1.16, the induced form q on Q0 /Z(Q0 ) is the one described in Exercise 7.7 in [Asc86], so vZ(Q0 ) is nonsingular if w is odd and singular if w is even. Thus |v| = 4 if w is odd while |v| = 2 if w is even, so (5) holds. Similarly from the description of q in the exercise, v commutes with j = z(J) for each J ∈ Γ0 , so as Γ0 (j) = {J}, it follows that v acts on J. But as eI is T0 -invariant, [v, T0 ] ≤ f0 , so [v, J] ≤ j ∩ f0 = 1. Thus v centralizes Γ0 , so (1) follows from 2.5.2.  ˆ n is the largest group X satisfying the hypothesis of 5.1.7. Lemma 5.1.19. AE Proof. If n is odd this follows from 5.1.16, so we may assume n = 2w is ˆ n of 5.1.17. By 5.1.17 the natural map β : even, and form the group X = AE ∗ satisfies the hypothesis of 5.1.7, so it remains to show that if X → X/Z(X) ∼ G = β2 : X2 → G∗ satisfies that hypothesis then there is a homomorphism α : X → X2 with αβ2 = β. The proof is similar to that in 5.1.16. ˆ of Definition Let M be the universal 2-covering of G∗ and recall the group G ˆ ˆ ˆ = 5.1.3 and set Z = σ = Z(A). The proof of 8.17 in [Asc94] shows that Q ∗ ˆ ˆ ˆ O2 (M ) = Z × Q and M ≤ G × X is the fiber product of the diagram X → G and β : X → G∗ . (We need two observations to implement that proof in our situation: ˇ be the dual of E as an F2 A-module; then by 5.1.2, H 1 (A, E) ˇ = 0. First, let E Second, E is an indecomposable F2 A-module that admits a unique nonzero Ainvariant bilinear form; this replaces condition (a) in the hypothesis of 8.17 in [Asc94].) Now there is a homomorphism α : M → X2 with ker(α) ≤ Z(M ) = σ ×Z(X) = σ, v, f0 , and to complete the proof it suffices to show that σ ∈ ker(α). By 5.1.7, |Z(X2 )| ≤ 4, so ker(α) = 1. If f0 ∈ ker(α) then U = O2 (X2 ) is / ker(α). elementary abelian, so σ ∈ ker(α) by 5.1.4. Thus we may assume f0 ∈ Similarly if σf0 ∈ ker(α), an argument in the proof of 5.1.16 supplies a contra/ ker(α), so ker(α) ∩ f0 , σ = 1 and hence | ker(α)| = 2. diction. Hence f0 σ ∈ Choose y, x, u as in the proof of 5.1.16; as | ker(α)| = 2 it follows from 5.1.7 that  f0 σ = x2 u2 = (xu)2 ∈ f0 ker(α), so σ ∈ ker(α), completing the proof. ˆ be the universal group AE ˆ n of 5.1.19. Definition 5.1.20. Let X Pick ˆ ˆ ˆ ˆ ˆ ˆ S ∈ Syl2 (X), set F = FSˆ (X), and denote F by AEn also. By construction in ˆ Ω) ˆ is a ˆ of Q8 -subgroups of Sˆ such that τˆ = (F, 5.1.16 and 5.1.18, there is a set Ω quaternion fusion packet. ˇ n, Similarly if X = G∗ or G then, from the start of the section, X is AEn or AE respectively, and by 5.1.6 there is S ∈ Syl2 (X) and a set Ω of Q8 -subgroups of S

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such that τ = (FS (X), Ω) is a quaternion fusion packet. Denote FS (X) by AEn or ˇ n , respectively. AE ˆ ˆ n and, Finally if n ≡ 0 mod 4 then from 5.1.17, X/v is denoted by H AE + + + ˆ+ ˆ ˆ ˆ setting F = F /v , we have τ = (F , Ω ) a quaternion fusion packet. Denote ˆ n also. the fusion system Fˆ + by H AE Lemma 5.1.21. Assume τ = (F, Ω) is a quaternion fusion packet on S with F = F ◦ and F + = F/Z(F) ∼ = AEn . Let K ∈ Ω and z = z(K). (1) F is constrained, so F has a model X. ˆ = S, Ωπ ˆ = Ω, and (2) There is a surjective morphism π : Fˆ → F such that Sπ ˆ ker(π) ≤ Z(F). (3) Either |Ω(z)| ≤ 2 or F ∼ = AE8 , and |Ω(z)| = 4. ˆ n. ˆ n or 8 = n ≡ 0 mod 4 and F ∼ (4) |Ω(z)| = 1 iff either F ∼ = H AE = AE (5) |Ω(z)| = 2 iff either (a) F ∼ = AEn with n = 8, or ˇ n. (b) n is even and F ∼ = AE Proof. As F + ∼ = AEn , F + is constrained, so as F + = F/Z(F), (1) holds. Then (2) follows from (1) and 5.1.19. ˇ n , G∗ = AEn , and taking α = {1, 2, 3, 4} and z0 = eα , from Recall G = AE 5.1.6.3 there is a quaternion fusion packet (FT (G), Γ) such that |Γ(z0 )| = 2. Now for z0 = eβ ∈ z0G , z0∗ = e∗β iff n = 8 and β = I − α, so |Γ∗ (z0∗ )| = 2 unless n = 8 where |Γ∗ (z0∗ )| = 8. Therefore as F + is AEn and |Ω+ (z + )| ≥ |Ω(z), (3) follows once we show |Ω(z)| ≤ 2 if n = 8 and F = F + , which we do during the proof of (4) and (5). ˆ and f = Φ(Q). By 5.1.16 and 5.1.18, f = zˆzˆ1 for distinct zˆ Let Q = O2 (X) / ker(π). and zˆ1 in ZSˆ . Thus if |Ω(z)| = 1 then f ∈ But suppose n is not divisible by 4; then from 5.1.16 and 5.1.18, f is the ˆ It follows that in that event, |Ω(z)| = 1 iff ker(π) = 1 unique involution in Z(X). ∼ ˆ = f so F ∼ ˆ iff F = AEn . So suppose f ∈ ker(π). When n is odd, Z(F) = AEn . ∼ ˆ ˇ When n is even, Z(F ) = v = Z4 by 5.1.18.5, so F is AEn or AEn . Therefore (4) and (5) hold in this case, so we may assume n ≡ 0 mod 4. ˇ 8 for each involution x ∈ Z(F), ˆ If n = 8 then Fˆ /x ∼ completing the proof = AE ˆ = f, v ∼ of the lemma in this case; thus we may take n > 8. By 5.1.18, Z(F) = E4 ∼ ˆ ˇ with F /f = AEn , so we may assume ker(π) is generated by v or f v. Then as f v ˆ acting on Sˆ and Ω, ˆ it follows that is conjugate to v under an automorphism of X ˆ F is H AEn , completing the proof. 

5.2. The 2-share of the order of some groups In this section q is an odd prime power. Recall that if n is a positive integer then n2 denotes the 2-share of n: the highest power of 2 dividing n. Given positive integers n and k, write n ∼ k to indicate that n2 = k2 . Set m = (q 2 − 1)2 and define n  q i − 1. Π(n, q) = i=1

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Set Π+ (n, q) = Π(n, q) and Π− (n, q) =

n 

q i − (−1)i .

i=1

Lemma 5.2.1. (1) m ≥ 8. (2) If = ±1 ≡ q mod 4 then q + ∼ 2 and q − ∼ m/2. (3) q 2 + 1 ∼ 2. Proof. For each odd integer n, n2 ≡ 1 mod 8, so (1) and (3) hold. Assume the setup of (2). Then q + ≡ 2 ≡ 2 mod 4, so q + ∼ 2. Then q − =  (q 2 − 1)/(q + ) ∼ m/2, completing the proof of (2). Lemma 5.2.2. If n is odd then q n − 1 ∼ q − 1 and q n + 1 ∼ q + 1. Proof. Recall q n − 1 = (q − 1)(q n−1 + · · · + q + 1). As n and q are odd, so is + · · · + q + 1, so the first statement in the lemma holds. Similarly the second q holds.  n−1

Lemma 5.2.3. Let n = 2a k with a ≥ 0 and k odd. Then (1) If q ≡ 1 mod 4 then q n − 1 ∼ 2a m/2. (2) If a > 0 then q n − 1 ∼ 2a m/2. (3) If q ≡ 1 mod 4 then q i − 1 ∼ im/2 for each positive integer i. (4) q 2i − 1 ∼ im for each positive integer i. Proof. We first prove (1). Replacing q by q k and appealing to 5.2.2, we may assume k = 1. We induct on a; the case a = 0 follows from 5.2.1.2. Suppose a > 0. a a−1 a−1 a−1 Then q 2 − 1 = (q 2 − 1)(q 2 + 1) ∼ 2(q 2 − 1) by 5.2.1. By induction on a, a−1 2 a−1 q −1∼2 m/2, completing the proof of (1). Observe (1) implies (3). Suppose a > 0 and set q0 = q 2 and m0 = (q02 − 1)2 . By 5.2.1, m0 = 2m, so as q0 ≡ 1 mod 4, it follows from (1) that a−1

q n − 1 = q02

k

− 1 ∼ 2a−1 m0 /2 = 2a m/2, 

proving (2). Then (2) implies (4).

Lemma 5.2.4. (1) If q ≡ 1 mod 4 then Π(n, q) ∼ n!(m/2) . (2) Π(n, q 2 ) ∼ n!mn . (3) If q ≡ mod 4 with = ±1 then Π (n, q) = n!(m/2)n . (4) If q ≡ − mod 4 then Π (n, q) = r! · mr · 2r+a , where n = 2r + a and a ∈ {0, 1}. n

Proof. Part (1) follows from 5.2.3.3, while (2) follows from 5.2.3.4. Suppose n = 2r. Then r  q 2i−1 − ∼ r!mr · (q − )r Π (n, q) = Π(r, q 2 ) · i=1

by (2) and 5.2.2. Hence if q ≡ mod 4 then by 5.2.1.2 we have Π (n, q) ∼ r!mr · (m/2)r ∼ r!2r · (m/2)n ∼ n! · (m/2)n , establishing (3) when n is even. On the other hand if q ≡ − mod 4 then by 5.2.1.2, Π (n, q) ∼ r!mr · 2r , establishing (4) when n is even.

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Finally assume n = 2r + 1 is odd. Then Π (n, q) = (q n − n ) · Π (n − 1, q) ∼ (q − ) · Π (n − 1, q) by 5.2.2. Then if q ≡ mod 4, we have Π (n, q) ∼ (m/2) · Π (n − 1, q) ∼ n!(m/2)n by 5.2.1.2 and (3) applied to 2r = n − 1. Finally if q ≡ − mod 4 then Π (n, q) ∼ 2·Π (n−1, q) ∼ r!mr ·2r+1 by 5.2.1.2 and (4) applied at 2r = n−1. This completes the proof of (3) and (4), and hence completes the proof of the lemma.  Lemma 5.2.5. (1) If q ≡ mod 4 then |SLn (q)| ∼ n!(m/2)n−1 . (2) If q ≡ − mod 4 and n = 2r + a with a ∈ {0, 1} then |SLn (q)| ∼ r!mr · 2r+a−1 . Proof. From Table 16.1 in [Asc86], x = |SLn (q)| ∼ Π (n, q)/(q − ). Therefore if q ≡ mod 4 then x ∼ Π (n, q)/(m/2) ∼ n!(m/2)n−1 by 5.2.1.2 and 5.2.4.3, so that (1) holds. Similarly if q ≡ − mod 4 then x ∼ Π (n, q)/2 ∼ r!mr · 2r+a−1 by 5.2.1.2 and 5.2.4.4, establishing (2).  Lemma 5.2.6. |Sp2r (q)| ∼ r!mr . Proof. From Table 16.1 in [Asc86], |Sp2r (q)| ∼ Π(r, q 2 ), so the lemma follows from 5.2.4.2.  Lemma 5.2.7. (1) If r ≥ 4 is even or r ≥ 5 is odd and q ≡ 1 mod 4, then r r−1 . |Spin+ 2r (q)| ∼ r! · (m/2) · 2 r−1 r (2) If r ≥ 5 is odd and q ≡ −1 mod 4 then |Spin+ ·2 . 2r (q)| ∼ (r −1)!·(m/2) r r r (3) If r ≥ 2 then |Spin2r+1 (q)| ∼ r!m ∼ r! · (m/2) · 2 . (4) If r ≥ 4 is even or r ≥ 5 is odd and q ≡ 1 mod 4 then |Spin− 2r (q)| ∼ (r − 1)! · (m/2)r−1 · 2r . r r−1 (5) If r ≥ 5 is odd and q ≡ −1 mod 4 then |Spin− . 2r (q)| ∼ r! · (m/2) · 2 r 2 Proof. From Table 16.1 in [Asc86], x = |Spin+ 2r (q)| ∼ (q − 1) · Π(r − 1, q ). r Suppose first that r is odd. Then q − 1 ∼ q − 1 by 5.2.2, so by 5.2.1.2 and 5.2.4.2, r−1 x = |Spin+ or 2 · (r − 1)! · mr−1 for q congruent to 2r (q)| ∼ (m/2) · (r − 1)! · m 1, −1 modulo 4, respectively. Therefore (2) holds, and when r is odd and q ≡ 1 mod 4, (1) also holds. So assume r is even. Then q r − 1 = (q 2 )r/2 − 1 ∼ rm/2 by 5.2.3.4, so by 5.2.4.2, x ∼ r · (m/2) · (r − 1)! · mr−1 = r! · (m/2)r · 2r−1 , completing the proof of (1). Next from Table 16.1 in [Asc86], |Spin2r+1 (q)| ∼ Π(r, q 2 ) ∼ r!mr = r!·(m/2)r · r 2 by 5.2.4.2, establishing (3). r 2 Finally set z = |Spin− 2r (q)|. By Table 16.1 in [Asc86], z ∼ (q + 1)Π(r − 1, q ). r r−1 or Suppose r is odd. Then q +1 ∼ q +1 by 5.2.2, so by 5.2.4.2, z ∼ 2·(r −1)!·m (m/2) · (r − 1)! · mr−1 for q congruent to 1 or −1 modulo 4, respectively. Therefore (5) holds, as does (4) in this case when q ≡ 1 mod 4. Thus we may assume r is even, so that q r + 1 = (q 2 )r/2 + 1 ≡ 2 mod 4 by 5.2.1.3. Therefore x ∼ 2 · (r − 1)! · mr−1, so that (4) holds, completing the proof. 

Lemma 5.2.8. |F4 (q)| ∼ (m/2)4 · 27 .

4 Proof. From Table 16.1 in [Asc86], x = |F4 (q)| ∼ i=1 q 2di − 1, where the di are 1, 3, 4, 6. Hence by 5.2.3.4, x ∼ 1 · 3 · 4 · 6 · m4 ∼ (m/2)4 · 27 . 

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Lemma 5.2.9. |E6 (q)| ∼ e(q, ), where (1) e(q, ) = (m/2)6 · 27 if q ≡ mod 4, and (2) e(q, ) = (m/2)4 · 29 if q ≡ − mod 4. Proof. From Table 16.1 in [Asc86], y = |E6 (q)| ∼ xz where z = (q 9 − )(q 5 − ) and x = |F4 (q)|. By 5.2.8, x ∼ (m/2)4 · 27 . By 5.2.2, z ∼ (q − )2 . If q ≡ mod 4 then by 5.2.1.2, z ∼ (m/2)2 , so (1) holds. Thus we may assume q ≡ − mod 4, so z ∼ 4 by 5.2.1.2, and hence (2) holds.  ˜7 (q)| ∼ (m/2)7 · 210 . Lemma 5.2.10. (1) |E 8 (2) |E8 (q)| ∼ (m/2) · 214 .

˜7 (q)| ∼ 7 q 2di −1, where the di Proof. From Table 16.1 in [Asc86], u = |E i=1 are 1, 3, 4, 5, 6, 7, 9. Therefore by 5.2.3.4, u ∼ 4 · 6 · m7 , so (1) holds. Similarly v = 8 |E8 (q)| ∼ i=1 q 2di −1, where the di are 1, 4, 6, 7, 9, 10, 12, 15, so v ∼ 4·6·10·12·m8 , so (2) holds. 

5.3. Orthogonal groups and packets In this section we assume the following hypothesis: Hypothesis 5.3.1. Assume q is an odd prime power and let F = Fq . Set π = ±1 where q ≡ π mod 4. Let U be an orthogonal space over F of dimension 4r + b with r ≥ 1, 0 ≤ b ≤ 4, and quadratic form q. Let = sgn(U ) and assume = +1 if b = 0 and = −1 if b = 4. Set m = (q 2 − 1)2 . Notation 5.3.2. Write U = U1 ⊥ · · · ⊥Ur ⊥Ur+1 , where Ui is nondegenerate of dimension 4 and sign +1 for 1 ≤ i ≤ r. Thus dim(Ur+1 ) = b and if b = 0 then Ur+1 is nondegenerate, and of sign when b is even. Let L = Ω(U, q) be the subgroup of the orthogonal group O(U, q) generated by ¯ = Spin (q) be the universal Chevalley root elements. Thus L ∼ = Ω4r+b (q). Let L 4r+b group covering L. If b > 2 or b = 2 and = π, set n = 2r + 1 and a = b − 2. Otherwise set n = 2r and a = b. Thus dim(U ) = 2n + a with 0 ≤ a ≤ 2 and = π n if a = 0 and = −1 if a = 2. For 1 ≤ i ≤ r, write Ui = Ui,1 ⊥Ui,2 with Ui,j of sign π. If n is odd then Ur+1 contains a subspace Ur+1,1 of dimension 2 and sign π. For 1 ≤ i < r, set U2i−1 = Ui,1 + Ui,2 = Ui and U2i = Ui,2 + Ui+1,1 . Set U2r−1 = Ur,1 + Ur,2 = Ur , and if n is odd set Un−1 = U2r = Ur,2 + Ur+1,1 . Set Γr = {Ui,j : 1 ≤ i ≤ r, j = 1, 2}. If n is even set Γ = Γr , while if n is odd set Γ = Γr ∪ {Ur+1,1 }. Thus |Γ| = n and the members Ui,j of Γ are nondegenerate lines of sign π. Let M = NL (Γ) be the global stabilizer in L of Γ. Write W for the kernel of the action on M on Γ. If π = +1 then each member of Γ has exactly two singular points and we write Γs for the set of singular points in some member of Γ; thus |Γs | = 2n. Lemma 5.3.3. Let d = dim(U ). Then one of the following holds: ¯∼ (1) d = 2n is even, n = 2r is even, and L = Spin+ 2n (q). ¯∼ (2) d = 2n is even, n = 2r + 1 is odd, and L = Spinπ2n (q).

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¯∼ (3) d = 2n + 1 is odd and L = Spin2n+1 (q). ¯∼ (4) d = 2n + 2 is even and L = Spin−π 2n+2 (q). Proof. First assume that d is even. Then as d = 4r + b, also b is even so b = 0, 2, or 4. Thus from 5.3.2, one of the following holds: (i) b = 0, n = 2r, d = 2n, and = +1. (ii) b = 2, n = 2r + 1, d = 2n, and = π. (iii) b = 2, n = 2r, d = 2n + 2, and = −π. (iv) b = 4, n = 2r + 1, d = 2n + 2, and = −π. Now (1) holds in case (i), (2) holds in case (ii), and (4) holds in cases (iii) and (iv). Thus we may assume that d is odd, so d = 4r + b with b = 1 or 3. If b = 1 then  n = 2r by 5.3.2, while if b = 3 then n = 2r + 1, and in either case (3) holds. Lemma 5.3.4. (1) For each 2-subset {A, B} of Γ there exist unique fundamental subgroups LA+B and LA+B of L such that A + B = [LA+B , U ] = [LA+B , U ]. Moreover z(LA+B ) = z(LA+B ) and A + B = [U, z(LA+B )]. Set L2i−1 = LU2i−1 and L2i−1 = LU2i−1 . (2) There exists a Sylow 2-group SL of L contained in M such that, setting   = L2i−1 ∩ SL , and ΩL = {K2i−1 , K2i−1 : FL = FSL (L), K2i−1 = L2i−1 ∩ SL , K2i−1 1 ≤ i ≤ r}, we have τL = (FL , ΩL ) is a Lie packet for L.  (3) Set VA+B = LA+B ∩ SL ∩ W, VA+B = LA+B ∩ SL ∩ W, and ηL =  {VA+B , VA+B : A, B ∈ Γ}. Then ηL ∈ τL , and if n > 2 or m > 8 then ηL is the unique member of η(τL ). (4) FL is transitive on ΩL , ΩL (z(K1 )) = {K1 , K1 } is of order 2, and μ(τL ) ∼ = W eyl(Dn ). (5) Set W = ηL . Then NL (W ) = M. Proof. Part (1) is well known; see for example 15.8 in [Asc77] and the discussion following that lemma. For 1 ≤ i ≤ r, set Yi = L2i−1 L2i−1 . By (1), U2i−1 = [Yi , U ] and CU (Yi ) = ⊥  U2i−1 , so [Yi , Yj ] = 1 for distinct i, j. Pick Sylow 2-groups K2i−1 and K2i−1 of   L2i−1 and L2i−1 . Then Oi = K2i−1 K2i−1 is the central product of quaternion subgroups, and Oi is irreducible on U2i−1 . We conclude that O = O1 · · · Or is a maximal commuting product of such subgroups, so (using 3.1.5.1) τ = (FS (L), ΩL ) is a Lie packet for L for any Sylow 2-group S of L containing ΩL . In particular FL is transitive on ΩL . Define Δ = Δ(τ ) as in Definition 3.1.9.  as in (3). Let O ≤ SL ∈ Syl2 (M) and choose SL ≤ S. Define VA+B and VA+B Then VA+B is Sylow in the stabilizer in LA+B of A and B, so VA+B is cyclic of order m/2. Observe that by Witt’s Lemma, M induces Sym(Γ) on Γ. Hence for  are in Δ. Thus ηL each A, B with A + B ∈ / {U2i−1 : 1 ≤ i ≤ r}, VA+B and VA+B is contained in some η ∈ η(τ ). Then as η centralizes ηL , as Γs is the set of singular points that are weight spaces for W = ηL when π = +1, and as Γ is the set of weight spaces for W that are definite lines when π = −1, it follows that (5) holds and η ⊆ W. But each V ∈ η is fused into O1 , so [U, V ] is of dimension 4 and sign +1, so as V ≤ W it follows that V ∈ ηL . Therefore η = ηL , so as S acts on η, we have S = SL , establishing (2). Indeed by 3.3.14, if n > 2 or m > 8 then ηL is the unique member of η(τ ), so (3) also holds.  ) with From (1), zi = z(K2i1 ) = z(K2i−1  U2i−1 = [U, zi ], so ΩL (zi ) = {K2i−1 , K2i−1 }

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107

is of order 2. For {A, B}, {C, D} 2-subsets of Γ, VA+B centralizes LC+D iff {A, B}∩ {C, D} = ∅, so μ(τL ) is transitive on it set DηL of reflections. Then by 4.1.7.1, O2 (μ(τL )) ≤ Z(μ(τL )). Now by 4.3.5.1 and as |Γ| = n, μ(τL ) ∼ = W eyl(Dn ), completing the proof of (4) and the lemma.  

Lemma 5.3.5. Let p = char(F ) and for H ≤ L set θ(H) = O p (H). ¯ u , where u ¯ is an involution. (1) L = L/¯ ¯ ∈ Z(L) (2) Let X be a fundamental subgroup of L, X  the second member of Ω(z(X)), ¯ X ¯  in L ¯ respectively. Then X ¯ = θ(X) ¯ × ¯ ¯ X ¯  the preimages of X, u with and X,  ∼ ¯ ¯ ¯ ¯ ¯. θ(X) = SL2 (q) a fundamental subgroup of L, and z(θ(X))z(θ(X )) = u ¯ Ω ¯ L = {θ(X) ¯ ∩ S¯L : X ∩ SL ∈ ΩL }, (3) Let S¯L be the preimage of SL in L, ¯ ¯ FL¯ = FS¯L (L), and τL¯ = (FL¯ , ΩL ). Then τL¯ is a quaternion fusion packet with ¯ = {K} ¯ for K ¯ ∈Ω ¯ L. z (K)) ΩL¯ (¯ ¯ = M (τL¯ ). Then M = M ¯ /¯ ¯ ∼ (4) Set M = M (τL ) and M u , M ¯ (Dn , m), and =ω M∼ = ω(Dn , m). (5) |¯ ω (Dn , m)| = n!mn /2. Proof. Parts (1) and (2) are well known; cf. Table 6.1.2 in [GLS98]. Then ¯ /¯ (3) follows from (1) and (2). Similarly by (1) and (2), M ∼ u . =M ¯ Let W be the preimage of W = ηL . For 1 ≤ i < n, set Vi = VUi , Vi = VU i , zi = z(Vi ), and Wi = V1 · · · Vi . Observe [U, Wi ] = Ai , where Ai = U1 + · · · + Ui . Also Wi = V1 × · · · × Vi is a direct product. This follows by induction on i and the fact that [U, zi ] = Ui ≤ Ai−1 . In particular, |Wn−1 | = (m/2)n−1 . Further     = zn−1 and Vn−1 ∩ Wn−1 = Vn−1 ∩ Vn−1 , so as Wn−1 Vn−1 ≤ W, Vn−1 ∩ Vn−1 n n ¯ we conclude that |W | ≥ (m/2) /2. Therefore |W | ≥ (m/2) . But by 5.3.4.4, ¯ /CM¯ (W ¯ )W ¯ ∼ M = W eyl(Dn ), which is of order n! · 2n−1 by 4.3.4. Therefore we have n ¯ |M | ≥ n!m /2, and by 4.3.2 the universal group ω ¯ (Dn , m) is at most of that order, so (4) and (5) follow from 4.3.3.2.  ¯ L , FL , FL¯ , τL , τL¯ , M = M (τL ), M ¯ = Notation 5.3.6. Define SL , S¯L , ΩL , Ω M (τL¯ ) as in 5.3.4 and 5.3.5. From 5.3.4, ηL ∈ η(τL ); set W = ηL ∈ W (τL ). Then ¯ = ηL¯ ∈ W (τL¯ ). Let GW and G ¯ W be models from 5.3.5 there is ηL¯ ∈ η(τL¯ ); set W ¯ ), respectively, and notice that GW = G ¯ W /¯ for NFL (W ) and NFL¯ (W u . Lemma 5.3.7. (1) CL (W ) = O(M)CS (W ). (2) GW = M SL with M/O(CL (W )) ∼ = GW . (3) If n > 2 then AutL (S) = Inn(S). Proof. From the proof of 5.3.4, CL (W ) is the kernel of the action of M on Γ, Γs for π = −1, 1, respectively, with O 2 (CL (W )) = O(CL (W )), so (1) holds. Set MK = K1M O(M); thus MK  M with MK /O(M) ∼ = M . Next as M/W ∼ = W eyl(Dn ) by 5.3.5.4, we conclude that either O 2 (Aut(M/W )) = O 2 (M/W ) or n = 4 and there is an outer automorphism of M/W of order 3. But if O 2 (M) ≤ MK then (2) follows from (1) and the fact that SL ≤ GW . Therefore we may assume that n = 4 and some x of order 3 in M induces an outer automorphism on MK /CL (W ). However we may choose x to act on K1 and have a cycle of length 3 on Ω − {K1 }. This contradicts NM (K1 ) ≤ NM (K1 ). Hence (2) is established. It remains to prove (3), so assume n > 2. Hence W is strongly closed in S with respect to L by 5.3.4.3, so NL (S) = NM (S) by 5.3.4.5. By (2), M∗ = M/CL (W ) ∼ = GW , so NM∗ (S ∗ ) = S ∗ . Hence NL (S) = NM (S) = S(CL (W ) ∩ NL (S)). Finally  [NO(M) (S), S] ≤ O(M) ∩ S = 1, so AutL (S) = Inn(S) by (1).

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¯ is Spin+ (q) with n even or L ¯ is Spinπ2n (q) Lemma 5.3.8. Suppose n > 2 and L 2n ¯ and M contains the Sylow ¯ contains the Sylow group S¯L of L with n odd. Then M ¯ =G ¯ W and M = GW . group SL , so M ¯W = M ¯ S¯L , and it remains to show Proof. By 5.3.7.2, GW = M SL , so also G ¯ ¯ ¯ ¯ By parts (4) and (5) of M contains SL ; thus it suffices to show that |M | ∼ |L|. n ¯ 5.3.5, |M | ∼ n!m /2. ¯ ∼ n!mn /2, Suppose n is even or n ≥ 5 is odd and π = +1. Then by 5.2.7.1, |L| π ∼ ¯ so the lemma holds in this case. Further if n = 3 then L = SL4 (q), so this time ¯ ∼ |M ¯ by 5.2.5.1. Thus we can assume n is odd and π = −1. Then if n ≥ 5, the |L| lemma follows from 5.2.7.5, while if n = 3 it follows from 5.2.5.1.  ¯ is Spin2n+1 (q) with n > 2. Then Lemma 5.3.9. Suppose L ¯W : M ¯ | = |GW : M | = 2. (1) |G ¯W ∼ ¯ (Cn , m) and GW ∼ (2) G =ω = ω(Cn , m). Proof. As GW = SL M , (1) follows from 5.2.7.3 and parts (4) and (5) of 5.3.5. Let A = Γ and B = A⊥ . Then B is a nonsingular point. Set E = B + U1,1 and Y = CL (E ⊥ ). Then E is nondegenerate of dimension 3 and Y acts faithfully as Ω(E) on E. Further SY = SL ∩ Y is Sylow in Y and isomorphic to Dm/2 . Next CSY (B) = W ∩ Y = x is of index 2 in SY , acts on U1,1 , and an involution ¯. By (1), GW = M r . r ∈ SY − x acts with eigenvalue −1 on B. Also r¯2 = u Next (cf. 15.8 in [Asc77]) K1r = K1 and r centralizes LQ+R for Q, R ∈ Γ − {U1,1 }, so (2) follows from 5.9.2 and 5.3.5.4 and its proof, which say that ¯ satisfies the relations in 4.3.1. M  ¯ is Spin−π (q). Then Lemma 5.3.10. Suppose L 2n+2 (1) |GW : M | = 4. ω (Cn , m). (2) GW ∼ = 2¯ ¯ ∼ 4|M ¯ |, Proof. As in the proof of 5.3.8, to prove (1) it suffices to show that |L| n ¯ and |M | ∼ n!m /2. Now appeal to parts (2) and (4) of 5.2.7 and to 5.2.5.2 when n = 2. Define A, B, E, Y , and SY as in the proof of 5.3.9. This time E is of dimension 2 ∼ 4 and sign −π, Y ∼ = Ω− 4 (q) = L2 (q ), and SY is dihedral of order m. Also CSY (B) = y is of order m/2 and centralizes W and LQ+R for Q, R ∈ Γ − {U1,1 }. From (1), GW = M y, r . Let v generate VU1 , k ∈ K1 − VU1 , and x = [r, v]. As W acts on E, x ∈ SY . As K1r = K1 , x = v −r v is of order m/4, so as |y| = m/2, x generates y 2 . Next −r kx = kv v = kv = kv 2 . Then from the computation in the last paragraph of the proof of 5.7.5, we may choose y so that y 2 = x and ky = kv. Now (2) follows from 5.9.3.  Lemma 5.3.11. Assume τ = (F, Ω) is the Lie packet τL . (1) Suppose F ∼ = Ω7 [m], Ω8 [m], Ω9 [m], or Ω− 10 [m]. Then F contains no sub− ◦ ∼ − Ω [m] and Y = Y system Y = Y1 × Y2 with Y1 ∼ = 4 2 2 = Ω6 [m]. − (2) Suppose F ∼ = Ω2n+2 [m] or Ω2n+1 [m] with d ≥ 6, and F contains a subsystem Y = Y1 × · · · × Yr with r = |ZS | ≥ 1 and Yi = Yi◦ ∼ = Ω− 6 [m]. Then r = 1 so n = 2 or 3. Proof. We regard F as FS (L) with L = Ωd (q) and U the defining orthogonal space for L. We first prove (1).

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As Y2 = Y2◦ , we have Y2 = FT2 (M2 ) where V2 = [U, M2 ] is nondegenerate of dimension 6 and sign −π. Then Y = FT (M ) where M = M1 × M2 , Y1 = FT1 (M1 ), and M1 ∼ = Ω− 4 (q). As M2 is irreducible on V2 we have M1 ≤ CL (V2 ), so V1 = [V, M1 ] ≤ V2⊥ . Now dim(V1 ) ≥ 4 so V = V1 + V2 is of dimension at least 10. Thus V = U , d = 10, and dim(V1 ) = 4. As M1 is faithful on V1 , V1 is of sign −1, so U is of sign (−1)(−π) = π, contrary to the hypothesis of (1). This completes the proof of (1). The proof of (2) is similar. This time Y = FT (M ) where M = M1 × · · · × Mr with Mi ∼ = Ω−π 6 (q) and Vi = [V, Mi ] of dimension 6 and sign −π. Then V = Vi : 1 ≤ i ≤ r is of dimension 6r. If d = 2n + 2 then from 5.3.2 and 5.3.3, n = 2r, so 6r ≤ d = 2n + 2 = 4r + 2, and hence r = 1. If d = 2n + 1 then n = 2r + 1, so 6r ≤ d = 2n + 1 = 4r + 3, and again r = 1. 

5.4. Linear, unitary, and symplectic groups and packets In this section we assume the following hypothesis: Hypothesis 5.4.1. Assume q is an odd prime power and set π = ±1 where q ≡ π mod 4. Set m = (q 2 − 1)2 . Let U be an finite dimensional vector space over a field F , and assume one of the following holds: (I) F = Fq and G = SL(U ). (II) F = Fq2 and G = SU (U ) is an isometry group of a unitary form (·, ·) on U. (III) F = Fq and G = Sp(U ) is an isometry group of a symplectic form (·, ·) on U . Notation 5.4.2. Write U = U1 ⊥ · · · ⊥Ur ⊥Ur+1 , where dim(Ui ) = 2 for 1 ≤ i ≤ r and dim(Ur+1 ) = a ∈ {0, 1}, with a = 0 in (III). Let Σ = {Ui : 1 ≤ i ≤ r + 1} and N = NG (Σ). For 1 ≤ i ≤ r, set Ui = Uj : j = i . Li = CG (Ui ) ∩ NG (Ui ), and Ki ∈ Syl2 (Li ). Set Ω = {Ki : 1 ≤ i ≤ r}. Lemma 5.4.3. Let U be the set of pairs u = (U, U ) such that U = U ⊕ U , dim(U) = 2, and in (II) and (III) we have U nondegenerate and U = U⊥ . Then (1) for each u ∈ U, L = L(u) = CG (U ) ∩ NG (U) is a fundamental subgroup of G. (2) U = [U, L] = [U, z(L)] and U = CU (L) = CU (z(L)). (3) L acts naturally on U as SL(U), SU (U), Sp(U) in (I), (II), (III), respectively. Proof. A fundamental subgroup X of G is of the form X = Z, Z − ∼ = SL2 (q), with Z ∼ = Eq the center of a long root subgroup of G and Z − an opposite of Z. In particular Z is a group of transvections on U with center U (Z) and axis A(Z), U (Z − ) is a complement to A(Z) in U , and A(Z − ) is a complement to U (Z) in U . Therefore [U, X] = U (Z) ⊕ U (Z − ) is a line in U , and CU (X) = A(Z) ∩ A(Z − ) is a complement to [U, X], with [U, X] a natural module for X. Moreover in (II) and  (III), A(Z) = U (Z)⊥ . Now the lemma follows from Witt’s Lemma. Lemma 5.4.4. Let X1 be a fundamental subgroup of G, U1 = [U, X1 ], and z1 = z(X1 ). Suppose z2 ∈ z1G with U1 = [U, z2 ], and in (I) assume [z1 , z2 ] = 1. Then z1 = z2 .

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Proof. Claim z2 acts on A = CU (z1 ). If (I) holds then by hypothesis, [z1 , z2 ] = 1, so the claim is trivial. In (II) and (III), A = U1⊥ by 5.4.3, so again the claim holds. By hypothesis, [U, z1 ] = U1 = [U1 , z2 ]. Then as z2 acts on A by the claim, it  follows that A = CU (z2 ), so z1 = z2 . Lemma 5.4.5. Let Xi , i = 1, 2, be fundamental subgroups of G, Ui = [U, Xi ], Si ∈ Syl2 (Xi ), and zi = z(Xi ). Then the following are equivalent: (1) [X1 , X2 ] = 1. (2) [S1 , S2 ] = 1. (3) z1 centralizes S2 and z1 = z2 . (4) Ui ≤ CU (z3−i ) for i = 1, 2. Proof. Trivially, (1) implies (2). Suppose (2) holds. Then [z1 , S2 ] ≤ [S1 , S2 ] = 1. Also S1 is irreducible on U1 , so as [S1 , S2 ] = 1, it follows that z2 ∈ [S2 , S2 ] ≤ CG (U1 ), and hence z1 = z2 . That is (2) implies (3). Suppose (3) holds. As S2 is irreducible on U2 , either U2 = [U2 , z1 ] or U2 ≤ CU (z1 ) = A, and the latter holds by 5.4.4. Therefore U1 ∩ U2 = 0, so as [z1 , z2 ] = 1, it follows that (4) holds. Finally assume (4) holds. Then [Xi , U ] = Ui ≤ CU (X3−i ) = A3−i , so X3−i acts on Ui and on Ai . Then by 5.4.3, X3−i acts on Xi , so [X1 , X2 ] ≤ X1 ∩ X2 = 1, as X1 centralizes A1 and X2 is faithful on U2 ≤ A1 . Thus (4) implies (1).  Lemma 5.4.6. (1) Li : 1 ≤ i ≤ r = L1 × · · · × Lr . (2) O = Ki : 1 ≤ i ≤ r = K1 × · · · × Kr . (3) Let S ∈ Syl2 (G) with O ≤ S and set F = FS (G), Ω = {Ki : 1 ≤ i ≤ r}, and τ = (F, Ω). Then τ is a Lie packet of G. (4) F is transitive on Ω and Ω(z(L1 )) = {K1 }. Proof. Set L0 = Li : 1 ≤ i ≤ r . By 5.4.5, L0 is a central product of the Li . Set Li = Lj : j = i . Then Li ∩ Li ≤ CLi (Ui ) = 1, so L0 is a direct product of the Li , proving (1). Then (1) implies (2). Set Ω0 = K1G ∩ S and τ0 = (F, Ω0 ); then τ0 is a Lie packet of G by 5.4.3. By construction, Ω ⊆ Ω0 , so to complete the proof of (2) we may assume K ∈ Ω0 − Ω, and it remains to produce a contradiction. By 3.1.5, K centralizes O, so by 5.4.5, W = U1 + · · · + Ur ≤ CU (K). But U = [U, K] ⊕ CU (K), so d = dim(U ) ≥ dim(W ) + dim([U, K]) = 2r + 2 > d, for our contradiction. Thus (3) holds. As G is transitive on it fundamental subgroups, F is transitive on Ω. By (2),  Ω(z(Ki )) = {Ki }. Notation 5.4.7. Define S, F = FS (G), and τ = (F, Ω) as in 5.4.6. Set Z = Z(τ ), ZS = ZS (τ ), and zi = z(Li ). For H ≤ NG (O), write H Ω for the image of H in Sym(Ω) under the conjugation map. Lemma 5.4.8. (1) If t ∈ z1G ∩ NG (O) with tΩ = 1 then tΩ = (Ki , Kj ) is a transposition and t centralizes each member of Ω − {Ki , Kj }. (2) For each i = j there exists t ∈ z1G ∩ NG (O) with tΩ = (Ki , Kj ). (3) NG (O)Ω = Sym(Ω).

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Proof. Assume the setup of (1). Then for some distinct i, j, tΩ has a cycle (Ki , Kj ). Therefore t interchanges Ui and Uj , so setting W = Ui + Uj , we have dim([W, t]) = 2. Then as dim([U, t]) = 2, we have [U, t] = [W, t] ≤ W and hence t / {i, j}. Therefore t also centralizes Kk , proving (1). centralizes Uk for k ∈ Assume the setup of (2). There exists an involution t ∈ N interchanging Ui and Uj and centralizing Uk for k ∈ / {i, j}. Then [U, t] = [W, t] is a line, so t ∈ z1G , and hence (2) holds. Finally (2) implies (3).  Lemma 5.4.9. Assume G = Sp(U ). (1) CZ (z1 ) = z1⊥ = CZ (K1 ). (2) CG (K1 ) = z1 × CG (U1 ). (3) CG (O) = ZS . (4) CS (z1 ) = K1 CS (K1 ). (5) Let X be the kernel of the action of NG (O) on Ω. Then either X = O or m = 8 and X = X1 × · · · × Xr , where SL2 (3) ∼ = Xi = NLi (Ki ). (6) If m > 8 then NG (S) = S. Proof. By 3.3.4, CZ (K1 ) = z1⊥ . Assume z ∈ CZ (z1 ) − z1⊥ . Then 0 ≤ e = dim([U1 , z]) ≤ 2. By 5.4.4, e = 2. As [z, K1 ] = 1 we conclude from 5.4.5 that e = 0. Therefore e = 1, so U1 = F u + F v where z centralizes u and vz = −v. Then 0 = (u, v) = (uz, vz) = (u, −v) = −(u, v), contradicting z ∈ G = Sp(U ). This proves (1). Let g ∈ CG (K1 ). As K1 is absolutely irreducible on U1 , g acts on U1 via scalar multiplication by some a ∈ F . As g|U1 ∈ Sp(U1 ), we have a = ±1, so gz1 ∈ CG (U1 ) for some ∈ {0, 1}, establishing (2). Then (2) implies (3). Next CS (z1 ) induces a subgroup of Sp(U1 ) = L1 on U1 , so as K1 is Sylow in L1 , (4) follows. Define X as in (5). Suppose m > 8. Then Aut(K1 ) is a 2-group, so X = K1 CX (K1 ) by (4), and then X = O by (3). Therefore we may assume that m = 8, so Xi = NLi (Ki ) ∼ = SL2 (3). In particular AutXi (Ki ) = O 2 (Aut(Ki )), so (5) follows from (3) and (4). Let Y = NG (S) and assume m > 8. As Ω = K1G ∩ S by 5.4.6, we have O  Y . By (5), X = O, so by 5.4.8.3, NG (O)/O = NG (O)Ω = Sym(Ω). Then as  NSym(Ω) (S Ω ) = S Ω , we have Y Ω = S Ω , so Y = S, establishing (6). Lemma 5.4.10. Assume G = SL−π (U ). (1) CS (K1 ) = z1 × CS (U1 ). (2) Δ(τ ) = Ω. (3) CS (O) = ZS . (4) For each i > 1 there exists s1,i ∈ Z inducing an outer automorphism on L1 in P GL2 (q) with CK1 (s1,i ) = z1 and dim([U1 , s1,i ] = dim([Ui , s1,i ] = 1. Therefore / {1, i}, and if i ≤ r then CKi (s1,i ) = zi . s1,i centralizes Kj for j ∈ (5) Let L0 = L1 · · · Lr and LO = NL0 (O). If m > 8 then LO = O, while if m = 8 then LO = X1 × · · · × Xr , where SL2 (3) ∼ = Xi = NLi (Ki ). (6) Let X be the kernel of the action of NG (O) on Ω. Then X = O(X)L0 XS , where XS = s1,i : 1 < i ≤ r + 1 and X/O(X)LO ∼ = E2r−1+a . (7) AutG (S) = Inn(S). Proof. Set Y1 = CS (K1 ). As K1 is irreducible on U1 , Y¯1 = AutY1 (U1 ) is cyclic and as G = SL−π (U ), we have Y¯1 ≤ Z(GL−π (U1 ) ∼ = Zq+π . But q ≡ π mod 4, so (q + π)2 = 2, and hence Y¯1 = ¯ z1 , so (1) holds. Then (1) implies (3).

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Suppose (2) fails. Then there is V ∈ Δ(τ ) such that V is cyclic of order m/2, and we may take [V, K1 ] = 1. Then by 4.2.5, T = V K1 is the wreath product of V by Z2 , so Z(T ) = x is of order m/2 with z1 ∈ x . But then x ¯ ∈ Y¯1 is of order m/2, contrary to (1). This proves (2). Let i > 1, W1 a point of U1 , and Wi a point of Ui . Choose W1 and Wi nondegenerate if G is SU (U ) and set W = W1 + Wi . Replacing W by a suitable conjugate under L0 , there exists s ∈ z1G ∩ S with W = [U, s]. In particular as S / {1, i}. As acts on O, s acts on U1 and Ui , and then s centralizes Uj for each j ∈ W1 = [U1 , s] is a point and L1 induces SL−π (U1 ) on U1 , we conclude that s induces an outer automorphism in P GL2 (q) on L1 with CK1 (s) = z1 . We now conclude that (4) holds. The proof of (5) is essentially the same as that of 5.4.9.5. Moreover AutX (U1 ) = ¯ s AutX∩L1 (U1 ), so it follows from (3) and (4) that X = O(X)LO XS , and then that (6) holds. Set Y = NG (S). Arguing as in the proof of 5.4.9.6, Y = SX. Then as [S, O(X)] ≤ S ∩ O(X) = 1, it follows from (6) that AutG (S) = Inn(S)AutX (S) = Inn(S)AutLO (S). Hence (7) holds if L0 is a 2-group, so from (5) we may assume m = 8. Now for xi of order 3 in Xi , xi is inverted by some s1,i , so NLO (S) = O. This completes the proof of (7) and the lemma.  In the remainder of the section we assume the following hypothesis: Hypothesis 5.4.11. Hypothesis 5.4.1 holds and G = SLπ (U ) with dim(U ) ≥ 3. Notation 5.4.12. Let {Vi : 1 ≤ i ≤ r} be an S-invariant collection of cyclic subgroups with Vi of index 2 in Ki . As Li induces SLπ (Ui ) on Ui , Vi has two weight spaces {Ui,1 , Ui,2 } on Ui , each of dimension 1, with Ui,j nondegenerate if π = −1. Set Γ0 = {Ui,j : 1 ≤ i ≤ r, j = 1, 2}, and take Γ = Γ0 if a = 0 and Γ = Γ0 ∪ {Ur+1 } if a = 1. Let λ ∈ F # of order m/2. Then for distinct A, B ∈ Γ, there is vA,B ∈ S with eigenvalue λ, λ−1 on A, B, respectively, and centralizing each member of Γ − {A, B}. Set VA,B = vA,B , η = {VA,B : A, B ∈ Γ distinct}, and W = η . Set M = NG (Γ) and d = dim(U ). Lemma 5.4.13. (1) η is the unique member of η(τ ). (2) M = NG (W ). (3) CG (W ) = W O(CG (W )) is the kernel of the action of M on Γ. (4) μ(τ ) ∼ = Sym(Γ) ∼ = Sd . (5) M/O(M) ∼ ¯ (Ad−1 , m). =ω (6) AutG (S) = Inn(S). Proof. By construction, S acts on V = {Vi : 1 ≤ i ≤ r}, and hence also on the set Γ of weight spaces for V . For pairs (A, B), (C, D) of distinct members of Γ, VA,B and VC,D are in the kernel of the action of M on Γ, and by construction the induced actions of the subgroups on each member of Γ commute. Therefore W is an abelian 2-group. Then as S acts on W and S ∈ Syl2 (G), it follows that W ≤ S. From 3.3.10.1, η0 = η − {V1 , . . . , Vr } ⊆ η1 ∈ η(τ ). As d ≥ 3 by 5.4.11, Γ is the set of weight spaces of W0 = η0 . By 3.1.11.2, W1 = η1 is abelian, so W1 acts on each member of the set Γ of weight spaces for W0 . Therefore for V ∈ η1 , V = VA,B for some A, B ∈ Γ, so η = η1 . Thus η ∈ η(τ ), and indeed the proof shows

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η is the unique member of η(τ ), proving (1). As Γ is the set of weight spaces of W , (2) holds. Further the proof shows that the kernel M0 of the action of M on Γ is abelian with W ∈ Syl2 (M0 ), so M0 = W O(CG (W )), establishing (3). For k ∈ Ki − Vi , kΓ = (Ui,1 , Ui,2 ) is a transposition. Similarly for distinct A, B ∈ Γ, a Sylow 2-subgroup KA,B of NG ({A, B}) ∩ CG (Γ − {A, B} ) is in K1G , so MΓ contains the transposition (A, B). This proves (4). ¯ (Ad−1 , m). By Set M∗ = M/O(M). By (4) and 4.3.3.2, M∗ is an image of ω (3) and (4), M/O(MW ) ∼ = Sd . Let W2i−1 = Vi for 1 ≤ i ≤ r, W2i = VUi,2 ,Ui+1,1 for 1 ≤ i < r, and if a = 1 let W2r = VUr,2 ,Ur+1 . Then W = W1 × · · · W2r−1 × W0 , where W0 = 1 if a = 0 and W0 = W2r if a = 1. Therefore W is the direct sum of d − 1 copies of Zm/2 , so (5) follows from 4.3.3.5. As W is weakly closed in S with respect to G, we have NG (S) ≤ NG (W ) = M. As usual [NO(M) (S), S] ≤ O(M) ∩ S = 1, so from (3), AutG (S) = AutM∗ (S ∗ ). But by (4), NM∗ /W ∗ (S ∗ /W ∗ ) = S ∗ /W ∗ , so AutG (S) = Inn(S), proving (6). 

5.5. Exceptional groups and packets In this section G is a 2-simply connected exceptional group of Lie type over Fq with q ≡ π mod 4, where π = ±1, and G is of Lie rank at least 4. Thus G is F4 (q), ˜7 (q), or E8 (q). Let S ∈ Syl2 (G), F = FS (G), and τ = (F, Ω) a Lie packet E6 (q), E for G. We will see that there is a unique η ∈ η(τ ). Set W = η and M = NG (W ). Pick K ∈ Ω and set z = z(K), m = (q 2 − 1)2 , and O = O(τ ). Set MK = K M . C (z) Let K2 ∈ Ω − {K} and set z2 = z(K2 ), u = zz2 , and X = K2 G . Among other things, we wish to update proofs of some of the results in [Asc80], so we will prove:

Theorem Sym(Ω). (2) If G is (3) If G is

5.5.1. (1) If G is F4 (q) or E6 (q) then |Ω| = 4 and NG (O)Ω = ˜7 (q) then |Ω| = 7 and NG (O)Ω = L3 (2). E E8 (q) then |Ω| = 8 and NG (O)Ω = E8 L3 (2) is 3-transitive.

Theorem 5.5.2. (1) If G is F4 (q) or E6−π (q) then μ(τ ) ∼ = W eyl(D4 ) and ¯ (D4 , m). MK /O(MK ) ∼ =ω (2) If G is E6π (q) then μ(τ ) ∼ ¯ (E6 , m). = W eyl(E6 ) ∼ = O6− (2) and MK /O(MK ) ∼ =ω ∼ ˜ (3) If G is E7 (q) then μ(τ ) = W eyl(E7 ) ∼ = Z2 × Sp6 (2) and MK /O(MK ) ∼ = ω ¯ (E7 , m). (4) If G is E8 (q) then μ(τ ) ∼ = W eyl(E8 ) ∼ = O8+ (2)/Z2 and MK /O(MK ) ∼ = ω ¯ (E8 , m). ˜ Lemma 5.5.3. (1) X ∼ = Sp6 (q), SL6 (q), Spin+ 12 (q), E7 (q) for G equal to F4 (q), ˜7 (q), E8 (q), respectively. E (2) z ∈ X. ˜7 (q), or E8 (q) and let l be 6, 7, or 8, respectively. (3) Assume G is E6π (q), E Then |W | ≥ (m/2)l . E6 (q),

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∼ L  CG (z). Proof. As τ is a Lie packet for G, K is Sylow in SL2 (q) = Inspecting Table 4.5.1 in [GLS98] for such a centralizer, we find that (1) and (2) hold. Assume the setup of (3). Let K ∈ F f , τX = (X, Ω − {K}), ηX ∈ η(τX ), and WX = ηX . Then ηX ⊆ η so WX ≤ W . Claim |WX | ≥ (m/2)l−1 . If l = 6 this follows from (1) and the proof of 4.2.13.5. If l is 7 or 8 it follows from (1) and induction on l. Let V ∈ η with [K, V ] = 1. Then [K, z(V )] = 1, so V ∩ WX = 1  and hence |W | ≥ |WX ||V | = (m/2)L , completing the proof of (3). Recall that μ = μ(τ ) = M/CM (W ) is generated by the set D = {dJ : J ∈ K M }, where dJ is the involutory automorphism induced by J on W . Moreover D is a set of 3-transpositions of μ, and we adopt the notation from section 4.1 in discussing this 3-transposition group. Lemma 5.5.4. (1) If G is F4 (q) or E6−π (q) then |Ω| = 4 and DdK ∼ = E8 . (2) If G is E6π (q) then |Ω| = 4 and DdK ∼ = S6 . ˜7 (q) then |Ω| = 7 and Dd ∼ (3) If G is E = W eyl(D6 ) ∼ = S6 /E32 . K (4) If G is E8 (q) then |Ω| = 8 and DdK ∼ = W eyl(E7 ) ∼ = Z2 × Sp6 (2). Proof. Let ξ = (E, Γ) be the fusion packet for X and write d for dK , and observe that ν = Dd ∼ = μ(ξ). However we know μ(ξ) when X is in the first three cases of 5.5.3, and we know μ(ξ) by induction on the order of G in the last case ˜7 (q). In particular we know |Γ| = |Ω| − 1, so the lemma follows. when X is E  Observe that we can now prove Theorem 5.5.1. Namely from 5.5.4, |Ω| is as indicated in Theorem 5.5.1. Further setting Y = NG (O), Y Ω is transitive and NY (K)Ω contains NX (O)Ω as a normal subgroup. In case (1) of 5.5.1, NX (O)Γ = Sym(Γ), so Y Ω = Sym(Ω). In case (2) of 5.5.1, NX (O)Ω acts transitively as S4 on the six points of Γ, so Y Ω is L3 (2) on the seven points of the projective plane Ω. Finally in case (3) of 5.5.1, by induction NX (O)Ω acts 2-transitively as L3 (2) on the projective plane Γ, so Y Ω is the 3-transitive group E8 L3 (2). This completes the proof of Theorem 5.5.1. ˜7 (q) then u ∈ Lemma 5.5.5. (1) If G is not E / zG. ˜7 (q) then u = az3 with z3 ∈ ZS − {z, z2 } and Z(G) = a . (2) If G is E Proof. Let H = NG (O) and observe the actions of H on Ω and ZS are equivalent. But if u ∈ z G then Hz,z2 fixes a third member u of ZS , so H Ω is not ˜7 (q) by 5.5.1. This proves (1), so it remains to prove 3-transitive, and hence G is E ˜7 (q). (2), and hence we may take G to be E Set Z(G) = a and G∗ = G/Z(G). From 5.5.3, X ∗ is an image of Spin+ 12 (q) with z ∗ = Z(X ∗ ) and there is a pairing {v, v¯} on V = ZS − {z} such that v ∗ v¯∗ = z ∗ . Then setting E = ZS , E ∗ ∼ = E8 and from 5.5.1, AutH (E ∗ ) = GL(E ∗ ). Therefore u = zz2 ∈ z3 a . Finally if u = z3 then 1 = zz2 z3 in G. But by 5.5.3, in B = E8 (q) we may take a = z(L) for some fundamental subgroup L centralizing G, and by 5.5.1, A = NB (OB ) is 3-transitive on ΩB , whereas Az,z2 fixes z3 , a  contradiction. Thus u = az3 , proving (2).

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˜7 (q). Then Lemma 5.5.6. Assume G is not E (1) G has two classes of involutions with representatives z and u.  (2) O p (CG (u)) is isomorphic to Spin9 (q), Spin10 (q), or HSpin+ 16 (q) for G equal to F4 (q), E6 (q), or E8 (q), respectively. (3) u ∈ Z(S). Proof. From Table 4.5.1 in [GLS98], G has two classes of involutions, and then by 5.5.5.1, z and u are representatives for the two classes, so (1) holds. Moreover (2) follows from the description of the involution centralizers in Table 4.5.1 of [GLS98] together with 5.5.3. By 5.5.1, |Ω| is even, so z ∈ / Z(S), and hence (3) holds.  Lemma 5.5.7. Assume G is F4 (q) or E6−π (q). Then (1) μ(τ ) ∼ = W eyl(D4 ). (2) η(τ ) has a unique member η. ¯ (D4 , m), (3) Set E = CW (MK ). Then u ∈ E ∼ = E4 with MK /O(MK ) ∼ = ω + p ∼ O (CG (E)) = Spin8 (q), and AutG (E) = GL(E). (4) CG (W ) = CS (W )O(M) with M/CG (W ) ∼ = Weyl(F4 ). (5) AutF (S) = Inn(S). Proof. By 5.5.6.3, u ∈ Z(S), so ν = (FS (CG (u)), Ω) is a quaternion fusion  packet with ν ≤ μ = μ(τ ). By 5.5.6.2, Gu = O p (CG (u)) ∼ = Spin9 (q) or Spin−π 10 (q), so ν ∼ of μ = D is = Weyl(D4 ). On the other hand by 5.5.4.1 the subgroup d⊥ K abelian and |Ω| = 4, so D(D) is disconnected, and hence (1) follows from 4.2.13.1 and 4.3.5.1. Then (1) and 3.3.14 imply (2). Indeed as μ ∼ = ν it follows that u ∈ ¯ (D4 , m). Hence Z(MK ). Then by 5.3.5.4 and 5.3.7 applied to Gu , MK /O(MK ) ∼ =ω  by 5.8.4, E = CW (MK ) ∼ E . From the structure of G , G = O p (CG (E)) = = 4 u E  O p (CGu (E)) ∼ = Spin+ 8 (q). Then by 5.5.1.1, AutG (E) = GL(E), completing the proof of (3). From Table 4.5.1 in [GLS98], u is Sylow in CG (Gu ) and |CG (u) : Gu CG (Gu )| ≤ 2, so CG (W ) = CCG (u) (W ) = O(M)CS (W ) by 5.3.7.1. As MK  M, we have M ≤ NG (E) by (3). As W is weakly closed in S with respect to G by (2) and W is abelian, AutG (E) = AutM (E), so M/CM (E) = GL(E) by (3). Then as ¯ (D4 , m), it follows that M/CG (W ) ∼ MK /O(MK ) ∼ =ω = Weyl(F4 ), establishing (4). As W is weakly closed in S, NG (S) ≤ M, so NG (S) = NM (S). Set M∗ = M/CG (W ). By (4), NM∗ (S ∗ ) = S ∗ , so NG (S) = S(NG (S) ∩ CG (W )). Then as  NO(M) (S) centralizes S, (5) follows from the first statement in (4). Lemma 5.5.8. (1) μ is transitive on D. (2) If G is not F4 (q) or E6−π (q) then O2 (μ) ≤ Z(μ) ≥ O3 (μ). (3) If G is E6π (q) then μ ∼ = Weyl(E6 ) ∼ = O6− (2). ∼ ∼ ˜ (4) If G is E7 (q) then μ = Weyl(E7 ) = Z2 × Sp6 (2). (5) If G is E8 (q) then μ ∼ = Weyl(E8 ) ∼ = O8+ (2)/Z2 . Proof. If G is F4 (q) or E6−π (q) then (1) follows from 5.5.7.1, so assume G is neither group. Let d = dK ∈ μ. Then by 5.5.4, ν = Dd is transitive on Dd , ν is nonabelian, and for b ∈ Dd , Dd ∩ Db = ∅. Now (1) follows from these observations. To prove (2), it suffices by 4.1.7 to show Vd = Wd = {d}. But this follows from the description of ν in 5.5.4 together with (1).

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Next by (1) and (2) and 4.1.24, μ/Z(μ) is isomorphic to Sn for some n ≥ 5, O6− (2), Sp6 (2), or O8+ (2). Then from the structure of ν in 5.5.4, it follows that ˜7 (q), and O + (2) if G is E8 (q). μ/Z(μ) ∼ = O6− (2) or S8 if G is E6π (q), Sp6 (2) if G is E 8 Suppose G is E6π (q). Then μ/Z(μ) is the Weyl group of A7 or E6 , so μ is also that Weyl group. Thus (3) holds unless μ is S8 . But ξ = (FS (CG (u)), Ω) is a subpacket of τ by 5.5.6.3 and μ(ξ) is Weyl(D5 ) ∼ = S5 /E24 by 5.3.4.4, whereas S8 has no such D-subgroup. This proves (3). ˜7 (q) or E8 (q) then (4) or (5) holds by 4.3.6.  Similarly if G is E Lemma 5.5.9. CG (W ) = O(M)CS (W ). Proof. If G is F4 (q) or E6−π (q), this follows from 5.5.7.4, so assume otherwise. ˜7 (q) then u ∈ Z(S), so we can form the quaternion fusion packet If G is not E ˜7 (q) then |Ω| ξ = (FS (CG (u)), Ω). In this case set W1 = W (ξ) and w = u. If G is E is odd by 5.5.1.2, so we can pick z ∈ Z(S), so ξ = (FS (CG (z)), Ω) is a quaternion fusion packet. Here set w = z and let W1 be the member of W (ξ) containing K ∩W . Now in any event, w ∈ W1 ≤ W , so C = CG (W ) ≤ CG (W1 ) = C1 ≤ CG (w).  Set Gw = O p (CG (w)). From Table 4.5.1 in [GLS98], CG (Gw ) is cyclic, as is + π CG (w)/Gw CG (Gw ). From 5.5.6.2, Gw is Spin+ 10 (q), HSpin16 (q) for G = E6 (q), E8 (q), respectively, so by 5.3.7.1, CG (W1 ) = O(NGw (W1 ))CS∩Gw (W1 ). Then as C ≤ C1 ≤ CG (w), the lemma holds in these two cases. On the other hand ˜7 (q) then Gw is SL2 (q) ∗ Spin+ (q), so again by 5.3.7.1, CG (W1 ) = if G is E w 12 O(NGw (W1 ))CS∩Gw (W1 ), and the lemma holds in this case too.  Lemma 5.5.10. Assume G is not F4 (q) or E6−π (q). Let l = 6, 7, 8 for G equal ˜7 (q), E8 (q), respectively. Let Φ = El . to E6π (q), E  (1) S ≤ MK , so MK = O 2 (M). (2) M = O(M)MK . (3) M/O(M) ∼ ¯ (Φ, m). =ω (4) W0 = Ω1 (W ) is the l-dimensional orthogonal module for μ/Z(μ) ∼ = Ol (2). Proof. By 5.5.8, μ ∼ = Weyl(Φ) and by 5.5.3.3, |W | ≥ (m/2)l . Therefore by 4.3.3.5, M ∈ M (τ ) is isomorphic to ω ¯ (Φ, m) with |W | = (m/2)l and M/W ∼ = l μ. Thus |S| ≥ |M |2 ∼ (m/2) |μ|. However by 5.2.9 and 5.2.10, |G| ∼ (m/2)l · |Weyl(Φ)|, so S ≤ MK and S ≤ M . Then as MK = K M , (1) holds. Indeed as S ≤ M and CM (W ) = W by 4.3.3.4, we conclude from 5.5.9 that CG (W ) = O(M)W . Set M∗K = MK /O(MK ). As CG (W ) = O(M)W , we have ¯ (Φ, m). Therefore (2) implies (3), so it remains W ∗ = CM∗K (W ∗ ), so M∗K ∼ =M ∼ =ω to prove (2) and (4). ∼ ∼ ∼ Set M+ = M/CM (W ). Then as CG (W ) = O(MW ), M+ K = M/W = μ = Weyl(Φ). By 3.1.20.3, dK induces a transvection on W0 with center z, so as dim(W0 ) = l and μ/Z(μ) ∼ = Ol (2), (4) follows. + + ∼ As MK  M, we have M+ K  M . But as μ = Weyl(Φ), Out(μ) = 1, so M = + + + + + + MK CM+ (MK ). By (1), Z = Z(MK ) is Sylow in CM+ (MK ), so CM+ (MK ) = O(M+ )Z + . By (4), M+ K is absolutely irreducible on W0 = Ω1 (W ), unless l = 7 where M+ is absolutely irreducible on W0 /Z(G). Therefore O(M+ K K ) centralizes W0 , and hence also centralizes W by 24.8 in [Asc86]. Hence if M0 is the preimage 2 ∗ of O(M+  K ) then O (M0 ) centralizes MK , so (2) follows.

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We are now in a position to establish Theorem 5.5.2. If G is F4 (q) or E6−π (q) then Theorem 5.5.2 follows from 5.5.7. In the remaining cases the theorem is a consequence of 5.5.10. Theorem 5.5.11. AutF (S) = Inn(S). Proof. If G is F4 (q) or E6−π (q) this is 5.5.7.5, so we may assume otherwise. As W is weakly closed in S with respect to G, we have NG (S) = NM (S). Set M∗ = M/O(M). By 5.5.10.3, NM∗ (S ∗ ) = S ∗ , so NG (S) = SNO(M) (S). As usual NO(M) (S) centralizes S, so the theorem holds.  5.6. FS (G) is simple In this section G is a simple group of Lie type in odd characteristic which is not a Goldschmidt group. (Recall a Goldschmidt group is a nonabelian simple group that either has an abelian Sylow 2-subgroup or is a rank one group of Lie type and characteristic 2. Thus G is neither L2 (q) with q ≡ ±3 mod 8 nor 2 G2 (q). The main result of this section is: Theorem 5.6.1. Let G be simple of Lie type in odd characteristic such that G is not Goldschmidt. Let S ∈ Syl2 (G) and F = FS (G) be the 2-fusion system of G. Then F is simple. Until the proof of Theorem 5.6.1 is complete, assume G is a counter example. Lemma 5.6.2. There is no nontrivial proper subgroup of S that is strongly closed in S with respect to G. Proof. As G is not Goldschmidt, this follows from a theorem of Foote in [Foo97].  

Notation 5.6.3. For P ≤ S set B(P ) = O 2 (AutF (P )). Set B = B(P ) : P ≤ S S ≤ F, and Aut0F (S) = α ∈ AutF (S) : α|P ∈ homB (P, S) for some P ∈ F c . Lemma 5.6.4. AutF (S) = Aut0F (S). Proof. As G is a counter example to Theorem 5.6.1, it follows that F is not simple. Then the lemma follows from 5.6.2 and part (4) of Theorem 8 in [Asc11].  Lemma 5.6.5. (1) Inn(S) ≤ Aut0F (S). (2) AutG (S) = Inn(S). Proof. For each α ∈ Inn(S), α = cs ∈ Inn(S) for some s ∈ S. Then α ∈ B(S) and S ∈ F c , so α ∈ Aut0F (S), establishing (1). Then 5.6.4 and (1) imply (2).  Lemma 5.6.6. G is not L2 (q).

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Proof. Assume otherwise. As G is not Goldschmidt, q ≡ ±1 mod 8, so S is dihedral of order at least 8. Therefore Aut(S) is a 2-group, contrary to 5.6.5.2.  Notation 5.6.7. By 5.6.6, G is not L2 (q), and as G is not Goldschmidt, G is not 2 G2 (q). Therefore G has a Lie packet τ = (F, Ω). Let K ∈ Ω and z = z(K). Lemma 5.6.8. G is not G2 (q) or 3 D4 (q). Proof. Assume otherwise; then z = Z(S), so NG (S) ≤ CG (z) = H. Now H = (L1 ∗ L2 )s , where Li ∼ = SL2 (qi ) and s ∈ S induces an outer automorphism in P GL2 (qi ) on Li . Set Si = S ∩ Li and Xi = NLi (Si ). By 5.6.5.2, NG (S) = S, so NG (Si ) = S for i = 1, 2, and hence Xi = Si for some i. It follows that q ≡ ±3 mod 8 and Xi ∼ = SL2 (3) for i = 1, 2. Indeed O 2 (NG (S1 S2 )) = X1 X2 . But setting ∗ NG (S1 S2 ) = NG (S1 S2 )/S1 S2 , we have that s∗ inverts Xi∗ for i = 1, 2, so NG (S) = S, a contradiction.  Lemma 5.6.9. (1) G is classical. (2) G is not P Ωd (q) for d ≥ 7. Proof. Assume (1) fails; then G is exceptional and, by 5.6.8, G is of Lie rank at least 4, so the hypotheses of section 5.5 are satisfied. But now 5.6.5.2 and Theorem 5.5.11 supply a contradiction. This proves (1). Part (2) follows from 5.6.5.2 and 5.3.7.3.  ¯ ¯ where G ¯ = SL (U ) or Sp(U ) satisfies Hypothesis 5.4.1. By 5.6.9, G = G/Z( G) ¯ map onto S under the projection of G ¯ onto G. By 5.6.5.2: Let S¯ ∈ Syl2 (G) ¯ = Inn(S). ¯ Lemma 5.6.10. AutG¯ (S) Lemma 5.6.11. G ∼ = P Sp2r (q) for some r ≥ 2 and some q with m = (q 2 −1)2 = 8. ¯ = Inn(S), ¯ so by 5.4.10.7 and 5.4.13.6, G ¯ = Sp(U ). Proof. By 5.6.10, AutG¯ (S) Thus G is P Sp2r (q). By 5.6.6, r ≥ 2. By 5.4.9.6, m = 8.  Notation 5.6.12. Adopt Notation 5.4.2 and, as in 5.4.6, set O = Ki : 1 ≤ i ≤ r . By 5.4.6.3, we may choose Ω = {Ki : 1 ≤ i ≤ r} and O ≤ S. Take K = K1 . Let X be the kernel of the action of H = NG¯ (O) on Ω. By 5.4.9.5, X = X1 × · · · × Xr , where Xi = NLi (Ki ) ∼ = SL2 (3). By 5.4.8.3, H/X ∼ = H Ω = Sym(Ω). Let Yi ∈ Syl3 (Xi ) and Y = Y1 · · · Yr ; then Y ∈ Syl3 (X), so by a Frattini argument, H = XNH (Y ). Set H ∗ = H/O, so that Y ∗ = X ∗  H ∗ . Write YS for the preimage in Y of CY ∗ (S ∗ ) and let yi ∈ Yi# . Lemma 5.6.13. NG (S) = SYS . Proof. As O is weakly closed in S with respect to G, we have NG (S) = NH (S). As H Ω = Sym(Ω), we have NH (S)Ω = NH Ω (S Ω ) = S Ω , so NH (S) = SNX (S). Next NX (S)∗ = NX ∗ (S ∗ ) = NY ∗ (S ∗ ) = CY ∗ (S ∗ ) = YS∗ , so the lemma holds.  Notation 5.6.14. For y ∈ YS , let cy ∈ Aut(S) be conjugation by s. From the construction in the proof of 5.4.8, for 1 ≤ i < r we can choose Yi and ti ∈ z G ∩ H ti / {i, i+1}, with tΩ i = (Ki , Ki+1 ) and Yi = Yi+1 . By 5.4.8.1, ti centralizes Kj for j ∈

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so ti acts on Y . Therefore Λ = ti : 1 ≤ i < r ≤ NH (Y ) and Λ2 = ti : 1 < i < r centralizes K. Therefore: Lemma 5.6.15. We may choose our generators yi so that (1) Λ acts on Y = {yi : 1 ≤ i ≤ r}, so Y = Y is the permutation module for Λ∗ = Sym(Y).  (2) For y = i yiai ∈ Y set a(Y ) = i ai ∈ F3 , and set Y0 = {y ∈ Y : a(Y ) =  0}. Then Y0 = [Λ, Y ] ≤ O 2 (H).  (3) Let Y1 , . . . , Yk be the orbits of NS (Y ) on Y and bi = y∈Yi y. Then YS = b1 , . . . , bk . (4) Set YS,0 = YS ∩ Y0 . Then |YS : YS,0 | = 3 with YS = bj YS,0 for each 1 ≤ j ≤ k. Lemma 5.6.16. AutYS,0 (S) ≤ Aut0F (S). 

Proof. Observe that O ∈ F c with Y0 ≤ O 2 (H) by 5.6.15.2. Thus AutYS,0 (O) ≤ B(O), so the lemma follows from 5.6.3.  Lemma 5.6.17. Let b = bj for some 1 ≤ j ≤ k such that l = |Yj | > 1. Then cb ∈ Aut0F (S). Proof. Choose notation so that Yj = {y1 , . . . , yl } and set θ = {K1 , . . . , Kl }. As l > 1 there exist conjugates s1 , . . . , sl/2 of t1 under Λ such that {sΩ i : 1 ≤ i ≤ l/2} is the set of transpositions in S Ω nontrivial on θ. Choose notation so that sΩ i = (K2i−1 , K2i ). ∼ Set Wi = U2i−1 + U2i and Ji = CG¯ (W⊥ i ). Then Ji acts faithfully as Sp(Wi ) = ∼ Sp4 (q) on Wi . Set Pi = CK2i−1 K2i (si )si , zi and observe that Pi = Q8 D8 with NJi (Pi )/Pi acting faithfully as Ω− 4 (2) on Pi /Z(Pi ). Moreover Si = S ∩ Ji = K2i−1 K2i si is Sylow in Ji and xi = y2i−1 y2i generates a Sylow 3-subgroup  of NJi (Si ) with xi ∈ NJi (Pi ) = O 2 (NJi (Pi )). Set P = P0 P1 · · · Pl/2 , where  P0 = Kj : j > 2i . Then P ∈ F c and b = x1 · · · xl/2 ∈ O 2 (NG¯ (P )), so cb|P ∈ B(P ) and hence cb ∈ Aut0F (S) by 5.6.3.  Choose j as in 5.6.17. By 5.6.15.4, YS = b YS,0 and by 5.6.16 and 5.6.17, AutYS,0 (S) and cbj are in Aut0F (S). Therefore AutYS (S) ≤ Aut0F (S). Then by 5.6.13, AutF (S) = AutSYS (S) = Aut0F (S), contrary to 5.6.5.2. This contradiction completes the proof of Theorem 5.6.1. Theorem 5.6.18. Let G ∈ K be a known simple group that is not Goldschmidt. Then the 2-fusion system of G is simple. Proof. By 16.3, 16.5, and 16.8 in [Asc11], either the theorem holds or G is of Lie type and odd characteristic. Hence the theorem follows from Theorem 5.6.1.  Remark 5.6.19. Given a saturated 2-fusion system F on S, set B∗ (F) =  O 2 (NF (R)) : R ∈ F f c . The proof of Theorem 5.6.1 shows that if F is the 2fusion system of a quasisimple group of Lie type of odd characteristic that is not Goldschmidt, then (a0 ) Each proper subgroup of S strongly closed in S with respect to F is contained in Z(F), and (b) F = B∗ (F).

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More generally if τ = (F, Ω) is a quaternion fusion packet with F = F ◦ transitive on Ω and τ appears as a conclusion in Theorem 1, then, inspecting the examples in Theorem 1 that are not Lie packets, and using Theorem 2, we conclude that (a) and (b) hold, where (a) S is the only subgroup of S strongly closed in S with respect to F and containing Ω.

¯ (Ad−1 , m) 5.7. Lπd [m] and ω In this section we assume Hypothesis 5.4.1 with G = SLπ (U ); that is G = SL(U ) if π = 1 and G = SU (U ) if π = −1. Adopt Notation 5.4.2 and 5.4.12. ˆ = GLπ (U ); that is G ˆ = GL(U ) if π = 1 and G ˆ = Notation 5.7.1. Let G # q−π GU (U ) if π = −1. Set Fπ = {b ∈ F : b = 1}. For A ∈ Γ and b ∈ Fπ ˆ have eigenvalue b on A and centralize each member of Γ − {A}. let hA (b) ∈ G Set hA = hA (λ), HA = {hA (b) : b ∈ Fπ }, VA = hA , H = {hA : A ∈ Γ}, and ˆ = ˆ η . For A ∈ Γ pick uA ∈ A# H = HA : A ∈ Γ . Set ηˆ = {VA : A ∈ Γ} and W with (uA , uA ) = 1 if π = −1. Set B = {uA : A ∈ Γ} and observe that B is a basis ˆ via its action on the basis B. Set for U . Set S = Sym(B) and embed S in G d = dim(U ).  Lemma 5.7.2. (1) NGˆ (Γ) = HS with H = AıΓ HA the kernel of the action of NGˆ (Γ) on Γ. ˆ S) with W ˆ = (2) NGˆ (Γ) = O(H)(W A∈Γ VA . Indeed for s ∈ S and A ∈ Γ, hsA = hAs , so S acts faithfully as Sym(H) on H. (3) For distinct A, B ∈ Γ, vA,B = hA h−1 B and kA,B = (A, B)zA and vA,B generate KA,B ∈ K1G , where (A, B) denotes the transposition in S and zA is the involution in VA .  ˆ set a(w) = kA . Then W = {w ∈ W ˆ : a(w) ≡ 0 (4) For w = A hkAA ∈ W A mod m/2}. ˆS∩G ∼ ¯ ¯ = KA,B : A, B ∈ Γ distinct = W ¯ (Ad−1 , m) with W  M (5) M =ω ¯ and M = M O(M).  ¯ ) = f k is cyclic of (6) Let f = h∈H h and k = (m/2)/(m/2, d). Then Z(M order (m/2, d). Proof. The kernel of the action of N = NGˆ (Γ) on Γ is the diagonal subgroup ˆ with Γ the set of weight spaces for H. By construction, S ≤ G ˆ acts as H of G, Sym(B) on B, and hence also acts faithfully as the symmetric group on Γ and H via As = uA s and hA s = hAs . In particular (1) holds. Also H = O(H)O2 (H) with ˆ , so (2) holds. O2 (H) = W Next for distinct A, B ∈ Γ, KA,B is a Sylow 2-group of LA,B the fundamental subgroup L(A+B, (A+B)⊥ ) of 5.4.3. Thus A+B = [U, KA,B ] and KA,B centralizes (A + B)⊥ . Further KA,B ∩ W = vA,B , where vA,B is defined in 5.4.12, and in particular from that definition, vA,B = hA h−1 B . Further k ∈ KA,B − VA,B induces the transposition (A, B) on Γ, so we can choose KA,B to contain (A, B)zA . Thus (3) holds. ˆ : det(w) = 1}, (4) holds. As W = {w ∈ W

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By construction, [W, kA,B ] = VA,B ≤ W , so KA,B acts on W with KA,B W = ¯ /W ∼ ¯ =W ˆ S ∩ G. Then from 5.4.13, M ¯ ∼ ¯ (Ad−1 , m), kA,B W , so M = S with M =ω establishing (5). ˆ S) = C ˆ (S) = f . Then By (2), S acts faithfully as Sym(H) on H, so Z(W W i i by (4), CW (S) consists of those f such that 0 ≡ a(f ) = di mod m/2. But m/2 divides di iff (m/2)/(d, m/2) divides i, so (6) holds.  Lemma 5.7.3. Let W0 = Ω1 (W ). Then W0 is the core of the permutation ¯ /W ∼ module for M = Sd on a d-set, with ZΔ the set of vectors of = M/CM (W ) ∼ weight 2. ˆ ). By 5.7.2.2, W ˆ =  VA , so W ˆ 0 is the permutation ˆ 0 = Ω1 ( W Proof. Let W A module for S on I = {zA : A ∈ Γ}, where zA is the involution in VA . By 5.7.2.4, ˆ 0 : a(w) ≡ 0 mod 2}, so W0 is the core of W ˆ 0 . By 5.7.2.3, the W0 = {w ∈ W ˆ 0. involution zA,B in VA,B is zA zB , so the zA,B are the vectors of weight 2 in W  Finally ZΔ consists of the zA,B . Lemma 5.7.4. Assume Hypothesis 4.2.1 and adopt Notation 4.2.3, except write ¯=ω ¯ → M the M for the model for F. Assume μ ∼ ¯ (A3 , m) and π : G = S4 and let G ¯ homomorphism supplied by 4.3.8.2. Write Vi for Vi π, vi for v¯i π, etc. Then (1) W = V1 V2 V3 and M/W ∼ =μ∼ = S4 . (2) Z(M ) = x is cyclic of order q ≤ 4, where x ¯ = (¯ v1 v¯22 v¯3−1 )m/8 . / V1 V2 , and q = 1 if z3 = z1 and (3) q = 4 if z3 = z1 , q = 2 if z3 = z1 and z2 ∈ z2 ∈ V1 V1 . If q = 4 then W = V1 × V2 × V3 . (4) If q ≤ 2 then M splits over W , while if q = 4 then M does not split, but M = LW where L ∼ = GL2 (3) and L ∩ W = Z(L) ≤ Z(M ). (5) M is determined up to isomorphism by q. (6) If m = 8 and q = 1, 2 then M splits over each A ∈ A(O2 (M )), and m(A) = 4, 5, respectively. Proof. By 4.3.7.2, W = V1 V2 V3 and by 4.3.8.4, W = CM (W ), so M/W ∼ = μ. Thus (1) holds. ¯ is isomorphic to the group M ¯ of that lemma when d = 4. Thus By 5.7.2.5, G m/8 ¯ is of order 4. Now f = h1 h2 h3 h4 = by 5.7.2.6, Z(G) = ¯ x where x ¯ = f m/2 −1 −1 2 −1 3 4 (h1 h2 )(h2 h3 ) (h3 h4 ) h4 . Therefore x ¯ = (¯ v1 v¯22 v¯33 )m/8 h4 = (¯ v1 v¯22 v¯3−1 )m/8 as x . Arguing as in the proof |hi | = m/2. Therefore x ∈ Z(M ). By 4.3.8.3, ker(π) ≤ ¯ 2 ¯ ¯ we have CW (O 2 (M )) = x , = ¯ x , so as ker(π) ≤ Z(G) of 5.7.2.6, CW ¯ (O (G)) completing the proof of (2). Observe that by (2), ker(π) is determined by q, so (5) also holds. ¯ = ¯ Next by 4.3.8.3, ker(π) ≤ Z(G) x , so q = 4 iff π is an isomorphism. In that event W = V1 × V2 × V3 by 4.3.7.2, so z3 = z1 . Thus we may assume q ≤ 2, m/2 so 1 = x2 = (v1 v3−1 )m/4 v2 = z1 z3 . Finally q = 1 iff 1 = x = w1 w3−1 z2 , where m/8 is of order 4, iff z2 ∈ V1 V3 . This completes the proof of (3). wi = vi We can represent M on I = {1, 2, 3, 4} as S4 with kernel W , so that d1 = (1, 2), d2 = (2, 3), and d3 = (3, 4). Let Mi,j = Ki , Kj , z = z1 , t = z2 , and k ∈ K1 − V1 . Now a = kt is an involution, so by the Baer-Suzuki Theorem (cf. 39.6 in [Asc86]), a inverts Y of order 3 in M1,2 . Then aY contains b with bW = d2 . Thus b = il for some i ∈ K2 −V2 and l ∈ V1 V2 , and 1 = b2 = i2 li l = tli l, so li l = t. However z i z = t and z i z = li l iff i inverts lz iff lz ∈ V2 iff l ∈ V2 z, so we may choose i ∈ K2 − V2 so that b = iz.

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Similarly iz3 inverts Y1 of order 3 in M2,3 and iz3 Y1 contains c with cW = d3 . Arguing as in the previous paragraph, we may choose j ∈ K3 − V3 so that c = jt. Now ca = (jt)kt = j t tkt = jz3 tz = czz3 . In particular if z = z3 then [c, a] = 1, so L = a, b, c is the Coxeter group S4 , and hence a complement to W in M . By (3), this holds precisely when q ≤ 2. Thus (4) holds when q ≤ 2. So assume q = 4. Then from (2) and (3), zz3 is the involution in Z(M ) and L/zz3 ∼ = S4 , so L ∼ = GL2 (3) as ac ∈ O2 (L) is of order 4. Thus (4) holds in this case too. Suppose m = 8 and q ≤ 2. Then by (5), M is the normalizer in H = AE6 or ˇ 6 of η ∈ η(τH ), since μ(τH ) ∼ AE = S4 and qH = q from the discussion in paragraph one of section 5.1. Now (6) can be checked in H.  ∼ ¯ = ω Lemma 5.7.5. Let G ¯ (A3 , m) be the universal group of 4.3.1 with μ = ¯ ¯ k k ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ S . Set z ¯ = z( K ), u ¯ = z ¯ z ¯ , k = k , k = k , K = K k W eyl(A3 ) ∼ = 4 i i 1 3 1 3 4 4 2 2, ¯ = k¯1 k¯2 . Set M = G/¯ ¯ u , let π : G ¯ → M be the natural v1 v¯2 v¯3 )−1 , and h v¯4 = (¯ ¯ 3 . Then ¯=K ¯ 1K ¯ i π, etc. Set t = z2 and O map, and set zi = z¯i π, Ki =K ¯ −1 k (1) z = z1 = z3 , so Ω = Ω(z) = {K1 , K3 }. Also v¯2 = v¯4 . ¯ = ¯ x0 , z¯1 , where x ¯0 = v¯1 v¯22 v¯3 . (2) CW ¯ (O) (3) CW (O) = x0 , z . ¯ is of order 3 with O ¯ 4 and ¯ ¯ h¯ 2 = K ¯ 2K ¯ h¯ ∩ O ¯ h¯ 2 . (4) h x0 = O (5) Assume ρ = (E, Ω) is a quaternion fusion packet with Sylow group P containing S, μ(ρ) ∼ = S4 , and H is a model for NE (CP (W )) such that M = K H . Assume further that r is an involution in P centralizing O and inverting x0 , and that s = kr ∈ z E . Then (a) K2r = K4 , v2r = v4 , s ∈ D ∗ (t) ∪ Do (t), and s inverts W . v2 = rx−1 (b) x0 , r is dihedral of order m/2 and [v2 , r] = x−1 0 , so r 0 . ∗ (c) Suppose s ∈ D (t). Then s = j3 j4 c for some ji ∈ Ki − vi , i = 2, 4, and 2 some c ∈ CP (O h ). Further, replacing k2 by j2 , k2r = k4 . Hence the action of r on M is determined up to an automorphism of M centralizing vi for 1 ≤ i ≤ 3 and k1 and k3 , and acting on K2 and K4 . (d) Suppose y ∈ P centralizes OW with y 2 = x0 . Then k2y = k2 v2 t for some ∈ {0, 1} and, replacing y by yb if necessary, where b is the involution in x0 , k2y = k2 v2 . ¯ = ¯ so z = z1 = z3 . As μ = μ(¯ τ) ∼ Proof. By 5.7.4.2, u ¯ ∈ Z(G), = S4 , Ω ¯ 1, K ¯ 3 } is of order 2, so Ω = Ω(z) = {K1 , K3 }. {K ¯ in 4.3.1, Next from the defining relations for G ¯ ¯ ¯ ¯ k k v2 v¯1 )k3 = v¯2 v¯1 v¯3 = v¯4−1 , (*) v¯2k = v¯2 1 3 = (¯ so (1) holds. ¯ v2 , k¯1 ] = v¯22 v¯1 , so k¯1 Next k¯1 inverts v¯1 and centralizes v¯3 , while v¯2 v¯2k1 = v¯22 [¯ 2 2 ¯ ¯ centralizes v¯2 v¯1 . Therefore C¯v1 ,¯v2 (k1 ) = ¯ v2 v¯1 , so CW v22 v¯1 , v¯3 . Similarly ¯ (k1 ) = ¯ 2 a b c ¯ ¯ v1 , v¯2 v¯3 . Hence by 4.3.7.2, O centralizes v¯1 v2 v¯3 iff 2a = b = 2c modulo CW ¯ (k3 ) = ¯ m/2, so (2) holds. ¯ , k¯1 ] = ¯ ¯ k¯1 ] = u ¯, impossible as [W v1 does not If w ∈ CW (k1 ) − x0 then [w, contain u ¯. Thus (2) implies (3). ¯ is of order 3. Set G ¯ W ¯ , so ¯ in 4.3.1, h ¯ ∗ = G/ From the defining relation for G ∗ ¯ ∗ ∗ ∗ ∗ ∗ ∗ h ¯ , {k¯ , k¯ , k¯ }) is a Coxeter system of type A3 , so k¯ ¯ = μ and (G = k¯∗ that G 1

2

3

2

1

5.8. ω ¯ (Dn , m)

123

¯ 4 )h¯ = K ¯ k¯1 k¯2 = K ¯ k¯1 k¯3 k¯1 k¯2 = K ¯ k¯3 k¯2 = ¯ h¯ = K ¯ 1 . Then (K ¯ 2K ¯ 1K ¯ 1K ¯ 1K and hence K 2 4 2 2 2 ¯ ¯ 1K ¯ 3 = O, ¯ 4 . Also by (2), k¯1 centralizes x ¯ and therefore O ¯h = K ¯ 2K K ¯0 , so ¯ ¯

¯

¯ x ¯h0 = x ¯0k1 k2 = x ¯0k2 = (¯ v1 v¯22 v¯3 )k2 = v¯1 v¯2 · v¯2−2 · v¯3 v¯2 = v¯1 v¯3 ∈ O, ¯

¯

and then

¯2 ¯ ¯ ¯ ¯ ¯ v1 v¯3 )h = (¯ v1 v¯3 )k1 k2 = (¯ v1−1 v¯3 )k2 = v¯1−1 v¯3 ∈ O. x ¯0h = (¯ 2 2 ¯ ¯ ¯ ¯ ¯3 ∩ ¯ so x ¯h ∩ O ¯ h . Conversely O ¯∩O ¯ h¯ 2 = K ¯ 1K ¯0h are in O, ¯0 ∈ O That is x ¯h0 and x −1 ¯ ¯ K2 K4 = ¯ v1 , v¯3 ∩ ¯ v2 , v¯4 = ¯ v1 v¯3 as v¯1 v¯3 = (¯ v2 v¯4 ) . Thus as |¯ v1 v¯3 | = m/2 = |¯ x0 |, we conclude that (4) holds. 2 Assume the hypothesis of (5). As O h = K2 K4 and Ω(z) = {K1 , K3 }, we have Ω(t) = {K2 , K4 }. Then xr0 = x−1 = (v1 v22 v3 )−1 , while as r centralizes O, xr = 0 2 r r 2 r 2 (v1 v2 v3 ) = v1 v3 (v2 ) , so (v2 ) = (v1 v2 v3 )−2 = v42 , and hence as Ω(t) = {K2 , K4 } we conclude that v2r = v4 t for some ∈ {0, 1}. In particular K2r = K4 . But also K2k = K4 , so s = rk acts on K2 . Hence as μ(ρ) ∼ = S4 and s ∈ z E , it follows that o ∗ s ∈ D (t) ∪ D (t). Also s acts on K2 ∩ W = V2 . Next as r centralizes O and k inverts v1 , v3 , s also inverts v1 , v3 . Now s acts on K1 and K2 and hence also on M1 = K1 , K2 . Further s inverts V1 and acts on V2 , so s acts on W1 = V1 V2 , and centralizes E = z, t . By 4.2.5, there is X of order 3 in M1 with W1 = [W1 , X]. Then X acts on F = W1 s = CM1 s (E), and F = W1 · CF (X). Now s = f w with f ∈ CF (X) and w ∈ W , and f inverts V1 , so f inverts W1 = V1X . Thus s also inverts W1 , so s inverts W = W1 V3 , completing the proof of (5a). As r is an involution inverting x0 of order m/4, x0 , r is dihedral of order m/2. By (5a), [v2 , r] = v2−1 v2r = v2−1 v4 = x−1 0 , so (5b) holds. Suppose s ∈ D∗ (t). By (5a), s inverts W so s inverts v2 , v4 . Therefore 2 s = j2 j4 c with ji ∈ Ki − vi and c ∈ CP (O h ). By 4.2.5.5, the presentation for G is still satisfied if we replace k2 by j2 , so we may assume that j2 = k2 . Thus k2 = k2s = k2kr = k4r by (5a), so k2r = k4 . Therefore the action of r on the generators ki , vi , 1 ≤ i ≤ 3 is determined, so the action of r on G is determined. This completes the proof of (5c). Assume the setup of (5d). Observe that xk02 = (v1 v22 v3 )k2 = v1 v2 · v3 v2 · v2−2 = k2 −1 x0 −1 −1 −2 −1 2 v1 v3 , so (k2−1 )x0 = x−1 0 x0 k2 = x0 v1 v3 k2 = v2 k2 , and hence k2 = k2 v2 . As y a 2 y centralizes W and acts on M , y acts on K2 , so k2 = k2 v2 . Therefore as y = x0 and y centralizes W , we have k2y = k2 v2 t . Further as k2x0 = k2 v22 and |x0 | = m/4, [k2 , b] = t, so k2yb = (k2 v2 t )b = k2 v2 t+1 , so replacing y by yb if necessary, we may  assume k2y = k2 v2 . This completes the proof of (5d) and the lemma.

5.8. ω ¯ (Dn , m) ¯ is the group ω In this section we assume G ¯ (Dn , m) for some n ≥ 3 defined in Notation 4.3.1, and adopt the notation found there. For i ∈ I write di for the image ¯ W ¯ , and set Q = (dn−1 dn )μ ≤ μ and L0 = di : i ∈ I − {n} ≤ μ. of k¯i in μ = G/ μ ¯ i ). Let D = d1 be the set of reflections in the Weyl group μ. Set z¯i = z(K Lemma 5.8.1. L0 ∼ = Sn is a complement to Q in μ, Q is the core of the permutation module for L0 on I, and x = dn−1 dn is of weight 2 in Q.

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Proof. This is remark 4.3.4.



¯ and for each d ∈ D, u ¯ ∈ Z(G) ¯ = Lemma 5.8.2. Let u ¯ = z¯n−1 z¯n . Then u   Q ζ(d)ζ(d ), where {d, d } = d . ¯ = K ¯ i : i ∈ I and K ¯ i centralizes u Proof. First G ¯ for i = n − 2. Let ¯ ¯ ¯ ¯ ¯ M = Kn−2 , Kn−1 , Kn . Then M is an image of the universal group ω ¯ (D3 , m), so as the diagrams A3 and D3 are some same, modulo ordering of the vertices, it ¯ ¯ ), so that u ¯ n−2 and hence u ¯ ∈ Z(G). follows from 5.7.4.2 that u ¯ ∈ Z(M ¯ centralizes K   ¯. Next dd is fused to dn−1 dn in μ, so by 4.2.8.2, ζ(d)ζ(d ) is fused to z¯n−1 z¯n = u ¯ we have u Then as u ¯ ∈ Z(G), ¯ = ζ(d)ζ(d ).  ¯ 0 = Ω1 ( W ¯ ) and G ¯ ! = G/¯ ¯ u . Lemma 5.8.3. Set W ¯ 0 ). (1) Z(μ) = Cμ (W ¯ 0 with center u ¯ ! is the core (2) Q/Z(μ) is a group of transvections on W ¯ and W 0 of the permutation module for μ/Q. zn! ) ∼ (3) Cμ/Q (¯ = Z2 ×Sn−2 is the centralizer in μ/Q of the transposition dn Q/Q. Proof. For d, a ∈ D, [ζ(a), d] = 1, ζ(d) for a ∈ Dd , a ∈ Ad , respectively. Thus ¯ 0 , so as L0 is irreducible on Q/Z(μ) by 5.8.1, d and x = dn−1 dn are faithful on W ¯ we conclude that CQ (W0 ) ≤ Z(μ). If n is odd then Z(μ) = 1, so (1) holds in this ¯ , so ι case. If n is even then by 4.3.5.2, Z(μ) = ι is of order 2 and ι inverts W ¯ centralizes W0 , completing the proof of (1). ¯ 0 . From the ¯ = V¯1 × · · · × V¯n , so {¯ z1 , . . . , z¯n } is a basis for W By 4.3.7.2, W ¯ previous paragraph, for j ∈ {n − 1, n}, dj is a transvection on W0 with axis A = {¯ zi : i = n−2 and center z¯j , so x = dn−1 dn is a transvection with axis A and center ¯ 0 with center u u ¯. Then as Q = xμ , Q/Z(μ) is a group of transvections on W ¯, so ! ! ∼ ¯ ¯ Q centralizes W0 . Now by 5.8.1, μ/Q = L0 acts on W0 . As D0 = L0 ∩ D is the set of transpositions in L0 , for d ∈ D0 we have CL0 (d) ∼ = Z2 × Sn−1 . Then by 4.2.8.2, CL0 (ζ(d)) = CL0 (d) and the action of L0 on Γ = {ζ(d) : d ∈ D0 } is equivalent to ¯ 0 = ¯ its action on the 2-subsets of I. As W zi : i ∈ I and z¯n−1 z¯n = u ¯, we conclude ! ! ! ! a ¯ ¯ zi : i ∈ I − {n} , so W = ζ(d) : d ∈ D . Also ζ(d)ζ(a)ζ(d )=1 that W0 = ¯ 0 0  ! ¯ for a ∈ Ad by 4.2.5, and i 1. (2) Ω(zn ) = {Kn−1 , Kn }. (3) u ¯ ∈ ker(π). Proof. If K ∈ Ω(zn ) and V = K ∩ W , then dV ∈ Vdn by 3.1.6. Then by 4.1.7, O (μ) dV ∈ dn 2 . As n = 4, O2 (μ) = Q, so Vdn = dQ n = {dn−1 , dn } and hence V = Vi for i ∈ {n − 1, n}. Thus if (1) holds then so does (2) and zn = zn−1 , so that (3) also holds. Visibly if u ¯ ∈ ker(π) then zn−1 = zn , so (2) holds. Trivially, (2) implies (1).  ∼ Lemma 5.8.6. Assume Hypothesis 4.2.1 and Notation 4.2.3. Assume μ = ¯ → G be the homomorphism of 4.3.8. Set Ki = K ¯ i π, W eyl(Dn ) and let π : G u=u ¯π, etc. Assume u = 1 and let wi ∈ Vi be of order 4 and set w = wn−1 wn and W1 = wG . Then Q centralizes W1 and W1 is the n-dimensional permutation module for μ/Q ∼ = Sn with Cμ (w) = Q, d1 , . . . , dn−2 . Moreover for each d ∈ D with [d, w] = 1, w ∈ O(ζ(d)). Proof. Set μ ˜ = μ/Q; we may regard μ ˜ as L0 = Sym(I) with d˜n = (n − 1, n). Now w ∈ Vn−1 Vn ≤ O(zn ), so w centralizes O(ζ(d)) for dn−1 = d ∈ Ddn and [w, dn ] = zn = [w, dn−1 ]. Further by 5.7.4.2 applied to M = Kn−2 , Kn−1 , Kn , w centralizes O(z) for some z ∈ znM . Then, replacing w by wzn if necessary, we may assume w centralizes O(zn−2 ), so that [w, dn−2 ] = 1. Therefore the maximal subgroup Lw = d1 , . . . , dn−2 ∼ = Sn−1 of L0 centralizes w, so Lw = CL0 (w). Further, setting D0 = D ∩ L0 and observing that ζ(D0 ) = ζ(D), L0 is transitive on D0 − CD0 (w), so as w ∈ O(zn ), we conclude that w ∈ O(ζ(d)) for each d ∈ D with [d, w] = 1 and xG ⊆ CG (w), so Q = xG centralizes W1 . As [w, dn ] = zn , W0 ≤ W1 , so as w ∈ / W0 it follows that dim(W1 ) ≥ n by 5.8.3.2. But as |L0 : Lw | = n, W1 is ˜ on I, so W1 is that module. This an image of the permutation module for L0 = μ completes the proof of the lemma.  ¯ = {K ¯ 1, K ¯,K ¯ 3, K ¯ 4 }. Lemma 5.8.7. Assume n = 4 and let Ω 1 ¯ (1) S = Sym(I)2 is faithfully represented as a group of automorphisms of G via k¯is = k¯is and v¯is = v¯is . ¯ = (¯ (2) Z = Z(G) z3 z¯4 , z¯1 z¯4 .

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∼ Weyl(D4 ) and let (3) Assume Hypothesis 4.2.1 and Notation 4.2.3 with μ = ¯ i π and zi = z(Ki ). Then ¯ → G be the homomorphism of 4.3.8. Set Ki = K π:G (i) if | ker(π)| = 2 then |Ω(z4 )| = 2 and G ∼ = ω(D4 , m), while (ii) if ker(π) = Z then Ω = Ω(z4 ) is of order 4 and E = z2 , z4  G. Also for ui ∈ Vi and u1 ∈ V1 of order 4, u1 u1 u3 u4 ∈ E. (4) Let wi ∈ V¯i , i = 3, 4, be of order 4 and set M = CG¯ (w3 w4 ) . Then ¯ 1, K ¯ 1 , K ¯ 2 ∼ M = K ¯ (A3 , m). Further w3 w4 ∈ ZM . =ω (5) Assume ker(π) = Z and m = 8, and set G+ = G/E. Then for Y ∈ Syl3 (G), CS + (Y + ) = Z(G+ ) ∼ = E8 . τ ). (6) z¯2 ∈ O(¯  (7) Assume ker(π) = Z and let Y ∈ Syl3 (G), R = g∈G O(z4 )g , P = R ∩ W , and G− = G/P . Then W − = [W − , Y ] ∼ = E4 . ¯ in 4.3.1, so (1) follows. Proof. Visibly S preserves the defining relations for G By 5.8.4, Z ∼ E , and by 5.8.2, u ¯ = z ¯ z ¯ ∈ Z. Now s = (1, 3, 4) ∈ S with = 4 3 4 u ¯s = z¯1 z¯4 , so (2) follows from (1). ¯ u , so (3i) follows from (1) and (2). By definition in 4.3.1, ω(D4 , m) = G/¯ Suppose ker(π) = Z. Then by (2), Ω = Ω(z4 ) is of order 4. Also [K2 , z4 ] = z2 and [Ki , z2 ] = z4 for i ∈ I − {2}, so E  G. The final statement of (3ii) follows from (6), completing the proof of (3). ¯ and H = V¯1 V¯1 V¯3 V¯4 . Then g centralizes V¯1 V¯3 V¯4 and inverts V¯1 ¯ 1 − W Let g ∈ K 4 ¯ : H| ≤ 2, so as z¯2 ∈ Φ(W ¯ ), (6) holds. so |H| ≥ m /32. Thus |W By 5.7.4.2, there are elements wi of order 4 in V¯i , i = 3, 4, such that w0 = ¯ 2 . Thus w = w3 w4 = w0 z¯2 also centralizes K ¯ 2 , and hence w w3 w4 z¯2 centralizes K ¯ 1 , K ¯ 2 ) . Observe that M ∗ ∼ ¯ 1, K centralizes M = K = S4 is a maximal D-subgroup ¯ W ¯ . Thus M ∗ = CD (w) and M = CG¯ (w) . Moreover all products ¯ ∗ = G/ of G ¯ 4 , so (4) holds, modulo the ¯ 3K w = w3 w4 with wi of order 4 in V¯i are conjugate in K claim that w ∈ ZM . But all products u1 u1 u3 u4 with ui ∈ V¯i and u1 ∈ V¯1 of order ¯ ¯ O2 (G) 4 are conjugate in O(¯ τ ), so from (6), ww1 w1 = z(J) for some J ∈ K and some 2 z2 . Finally w1 and w1 . Therefore as w1 and w1 are in M , w ∈ gM , where g = z(J)¯ ¯ by (2), completing the proof of (4). z(J)¯ z2 ∈ Z(G) Assume the setup of (5). Observe Z(μ) = c W/W , where c ∈ O(z4 ). As Z ≤ ker(π), Ω = Ω(z4 ) by (3). Then as m = 8, E = Φ(W ), so W + ∼ = E16 . Similarly Φ(O(z4 )) = z4 , so O(z4 )+ ∼ = E28 and c+ is an involution. As c inverts W , X + = c+ , W + ∼ = E32 and X +  G+ . Now O(z4 )+ centralizes the hyperplane + + c , (O(z4 ) ∩ W ) of X + , so G+ = O(z4 )+ , O(z2 )+ centralizes a subgroup X0+ of index 4, and X0+ = Z(G+ ), proving (5). ¯ P¯ . By 4.2.5.5 we may Assume the setup of (7) and observe that G− ∼ = G/ ¯ ¯ take Y = y where y¯ = k1 k2 . From 10.1.3.5 (whose proof uses 5.8.7 but not ¯ ) ≤ P¯ . Therefore W − is an image of 5.8.7.7), |W : O(z4 ) ∩ W | = 2, so Φ(W ˜ ¯ ¯ ˜ ˜ W = W /Φ(W ) = V1 · · · V4 . Now y¯ has cycles (˜ v1 , v˜1 v˜2 , v˜2 ) and (˜ vi , v˜2 v˜i , v˜1 v˜2 v˜i ) ˜ , y¯] = ˜ for i = 3, 4, so [W v1 , v˜2 and CW (¯ y ) = ˜ v v ˜ , v ˜ v ˜ . But O(z4 ) contains ˜ 1 3 1 4 v1 v3 , v1 v4 , so Φ(W )v1 v3 , v1 v4 ≤ O(z4 ), and then as Y is transitive on z2 , z4 # ,  we conclude P = Φ(W )v1 v3 , v1 v4 , so (7) holds.

5.9. ω ¯ (Cn , m) AND 2¯ ω (Cn , m)

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5.9. ω ¯ (Cn , m) and 2¯ ω (Cn , m) ¯ Definition 5.9.1. Let I = {1, . . . , n} and write ω ¯ (Cn , m) for the group H ¯ presented by the generators {ki , v¯i , r : i ∈ I} and the following relations: ¯ (Dn , m). (1) The relations on {k¯i , v¯i : i ∈ I} from 4.3.1 defining ω r r (2) r 2 = u ¯, r centralizes k¯i , v¯i for 1 ≤ i ≤ n − 2, k¯n−1 = k¯n , and v¯n−1 = v¯n . ¯ ¯ ¯ i ) is As in 4.3.1, set Ki = ki , v¯i . Recall from section 5.8 that z¯i = z(K ¯ the involution in Vi = ¯ vi and that u ¯ = z¯n−1 z¯n is in the center of ω ¯ (Dn , m). ¯ Set ω(Cn , m) = ¯, we have r centralizing u ¯, so u ¯ ∈ Z(H). Observe that as r 2 = u u . ω ¯ (Cn , m)/¯ ¯ =ω Lemma 5.9.2. Let H ¯ (Cn , m). Then ¯ is normal in H ¯ and isomorphic to the (1) The subgroup k¯i , v¯i : i ∈ I of H ¯ = k¯i , v¯i : i ∈ I . universal group ω ¯ (Dn , m). Set G ¯ = Gr . ¯ (2) H ¯ Set H ¯ ∗ = H/ ¯ W ¯. ¯ = ¯ (3) The subgroup W vi : i ∈ I is normal in H. ¯ ∗ , {k¯∗ , r ∗ : 1 ≤ i < n}) is a Coxeter system of type Cn . (4) (H i (5) ω ¯ (Cn , m) is also presented as Grp(k¯i , v¯i , r : 1 ≤ i < n}) subject to the defining relations from 4.3.1 on {k¯i , v¯i : 1 ≤ i < n} for ω ¯ (An−1 , m), together with the relations k¯n−1 , v¯n−1 r commutes with k¯n−1 , v¯n−1 , r centralizes {k¯i , v¯i : 1 ≤ i ≤ n − 2}, and r 2 = [r, v¯n−1 ]m/4 . ¯ for the subgroup of H ¯ defined in (1); we first prove that Proof. Write G ¯  H. ¯ By definition, H ¯ = G, ¯ r . Further r centralizes K ¯ i : 1 ≤ i ≤ n − 2 and G r ¯ n−1 ¯ n , so r acts on G. ¯ Thus G ¯  H ¯ and (2) follows. K =K Next the action of r on the generators of L = ω ¯ (Dn , m) preserves the defining relations for that group, so r induces an automorphism of L of order 2. Let s of order 4 induce that automorphism and form the semidirect product D of L by s . ¯ with L+ ∼ us2 . Then D+ satisfies the defining relations for H Set D+ = D/¯ = L. + ∼ ¯ ¯ Then as D is an image of H, it follows that G = L, completing the proof of (1). ¯ = V¯1 × · · · × V¯n as it permutes {¯ vi : i ∈ I}. Thus By construction, r acts on W ¯ ¯ as W  G by 4.3.2.1, (3) follows from (2). From 4.3.2, {k¯i∗ : 1 ≤ i < n} satisfies the Coxeter relations of type An−1 , and as r centralizes k¯i for i ≤ n − 2, r ∗ centralizes k¯i∗ . Thus to prove (4), it remains ∗ r r ∗ )4 = 1. But (k¯n−1 r)2 = u ¯k¯n−1 k¯n−1 =u ¯k¯n−1 k¯n , which squares to to show (k¯n−1 ¯ . Thus (4) holds. u ¯∈W ¯ n, H ¯ is generated by {k¯i , v¯i , r : 1 ≤ i < n}. From ¯r = K Observe that as K n−1 ¯ n , r centralizes K ¯ i for 1 ≤ i ≤ n − 2, and r 2 = u ¯ n−1 commutes with K ¯= 5.9.1, K −1 m/4 ¯ , so the relations in (5) are satisfied in H. (¯ vn v¯n−1 ) Conversely let H = Grp(ki , vi , r : 1 ≤ i < n) subject to the relations in (5). Set r r vn = vn−1 , kn = kn−1 , and Kn = kn , vn ; we must show that the relations in 5.9.1 are satisfied in H. By definition the relations among the Ki = ki , vi are satisfied for 1 ≤ i < n, Kn commutes with Kn−1 , and r centralizes Ki for 1 ≤ i ≤ n − 2. r . Similarly Therefore as Kn−1 commutes with Kj for j < n − 2, so does Kn = Kn−1 3 r r 3 3 r 1 = (kn−2 kn−1 ) , so 1 = (kn−2 kn−1 ) = (kn−2 kn ) , and vn−2 = vn−2 commutes r = vn , so the relations defining ω ¯ (Dn , m) in 5.9.1.1 are satisfied. In with vn−1 −1 m/4 m/4 particular u = (vn vn−1 ) = [r, vn−1 ] = r 2 , establishing the last relation in ¯ H and completing the proof of (5). 

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Definition 5.9.3. Write 2¯ ω (Cn , m) for the group Y¯ presented by the generators {k¯i , v¯i , y, r : 1 ≤ i < n} together with the following relations: (a) The relations on k¯i , v¯i , r for 1 ≤ i ≤ n for ω ¯ (Cn , m) appearing in 5.9.1, r r and v¯n = v¯n−1 . where k¯n = k¯n−1 (b) y centralizes k¯i , v¯i for 1 ≤ i ≤ n−2 and v¯n−1 , r inverts y, y 2 = x = [r, v¯n−1 ], and [k¯n−1 , y] = v¯n−1 . u . We will see in a moment that u ¯ = r 2 ∈ Z(Y¯ ), and we set 2ω(Cn , m) = Y¯ /¯ Lemma 5.9.4. Let Y¯ = 2¯ ω (Cn , m). Then (1) The subgroup k¯i , v¯i , r : 1 ≤ i < n of Y¯ is normal in Y¯ and isomorphic to ¯ = k¯i , v¯i , r : 1 ≤ i < n . the universal group ω ¯ (Cn , m). Set H ¯ ¯ ¯ ¯ (2) Y = Hy and |Y : H| = 2. ¯ y . Then W ¯ Y = V¯1 × · × V¯n−1 × y is normal in Y¯ with ¯Y = W (3) Set W ∗ ¯ ¯ ¯ |Y | = m. Set Y = Y /WY . (4) (Y¯ ∗ , {k¯i∗ , r ∗ : 1 ≤ i < n}) is a Coxeter system of type Cn . ¯ for the subgroup of Y¯ defined in (1) and set K ¯ i = k¯i , v¯i Proof. Write H r ¯ n−1 ¯ i for ¯n = K . By definition in 5.9.3, y commutes with K for 1 ≤ i < n and K ¯ ¯ i ≤ n − 2 and with v¯n−1 , and [kn−1 , y] = v¯n−1 . Therefore y acts on Kn−1 . Then r ¯ n , y also acts on K ¯ n . Finally as r inverts y, we have ¯ n−1 =K as r inverts y and K 2 ¯ ¯ ¯  Y¯ = H, ¯ y and (2) holds. [r, y] = y = [r, v¯n−1 ] ∈ H, so y acts on H. Thus H − ∼ Let ρ = (L, Γ) be the Lie packet of L = Spin2n+2 (q) for (q − 1)2 = m/2, let ηL ∈ η(ρ), WL = ηL , and GL a model for NL (WL ). Then from the proof of 5.3.10, ω (Cn , m) and |GL | = 4|¯ ω (Dn , m)|. But by (2) GL satisfies the relations defining 2¯ ¯ ∼ and 5.9.2, |Y¯ | ≤ 4|¯ ω (Dn , m)|, so we conclude that H ¯ (Cn , m), completing the =ω proof of (1). ¯ = V¯1 × · · · × V¯n . We’ve seen that y centralizes V¯i for 1 ≤ i < n, By 4.3.7.2, W r ¯ . Thus = V¯n , it follows that y centralizes W so as y is inverted by r and V¯n−1 ¯ ¯ ¯ ¯ ¯ WY = W y is abelian. Then by 5.9.2.3, W is normal in Hy = Y . Also y ¯ , so also [k¯n , y −1 ] = [k¯r , y r ] centralizes k¯i for i ≤ n − 2 and [k¯n−1 , y] = v¯n−1 ∈ W n−1 2 ¯ ¯0 = 1 ¯ ¯ is in W , and hence WY  Y . Finally y = x = [r, v¯n−1 ] = v¯n−1 v¯n−1 , so y 2 ∩ W ¯ ¯ ¯ ¯ ¯ and W = W0 × x , where W0 = V1 × · · · × Vn−1 . This completes the proof of (3). Part (4) follows from (3) and 5.9.2.4. 

5.10. Some constrained examples In this section we assume: Hypothesis 5.10.1. τ = (F, Ω) is a quaternion fusion packet such that F = F ◦ is constrained with Sylow group S and model G with G/Z(G) ∼ = L3 (2)/E8 , L3 (2)/E64 , or L3 (2)/23+6 . In three instances in this section we appeal to results from section 6.2. We observe that section 6.2 does not depend upon this section. Notation 5.10.2. Set Z = Z(G), Q = O2 (G), G∗ = G/Q, and G+ = G/Z. Let z ∈ ZS , K ∈ Ω(z), and E = z G . Notice that z ∈ / Z as O(z) is not normal in F. Set F + = F/Z and τ + = (F + , Ω+ ). By 3.3.2.2, τ + is a quaternion fusion packet with model G+ .

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Lemma 5.10.3. G = O(τ )G . Proof. This follows as F = F ◦ .



Lemma 5.10.4. (1) K is the unique member of Ω in KZ. (2) NG (K + ) = NG (K) ≤ CG (z). (3) Φ(E) = 1. (4) If Ω = Ω(z) then E8 ∼ = E ≤ Z(Q) and E is the natural module for G∗ . Proof. Part (1) follows from 3.3.2.3. Then (1) implies (2). From 6.2.8, E ≤ NG (K + ), so E centralizes z by (2), and then as E = z G , part (3) follows. Suppose Ω = Ω(z). Then as S acts on Ω, z ∈ Z(S), so E ≤ Z(Q). Let g ∈ G − CG (z); by 6.2.10.2, z g ∈ D ∗ (z) − z ⊥ , so by 4.2.5, zz g ∈ z G . Therefore  E # = z G , so as E + ∼ = E8 is the natural module for G∗ , part (4) follows. Lemma 5.10.5. G = O 2 (G). Proof. By 5.10.3, G = O(τ )G , so it suffices to show that K ≤ O 2 (NG (K)). But K + ≤ O 2 (NG+ (K + )) (cf. the proof of 6.2.14), so K ≤ O 2 (NG (K)) by 5.10.4.2.  Lemma 5.10.6. If G/Z(G) ∼ = L3 (2)/E8 then Z(G) = 1. Proof. Assume otherwise. As E + = Q+ and [K + , E + ] ≤ z + , we have Q ≤ NG (K + ), so K  KQ and z ∈ Z(Q) by 5.10.4.2. Indeed [K, Q] ≤ z ≤ E, so Q/E ≤ Z(G/E) by 5.10.3. But K ∗ is a 4-subgroup of G∗ ∼ = = L3 (2) and KE/E ∼ K/z ∼ = E4 , so G/E is not SL2 (7). Therefore as G = O 2 (G) by 5.10.5, it follows that E = Q and E = [E, G]. Then as H 1 (G∗ , E + ) ∼ = F2 , we have m(E) = 4. ˆ be the dual of E as a G∗ -module; then Zˆ is a hyperplane of E ˆ and S ∗ Let E ˆ − Z, ˆ so for s∗ an involution in S ∗ we have C ˆ (s∗ ) = [E, ˆ s∗ ] ∼ is regular on E = E4 . E ∗ ∗ ∗ E . Therefore for t an involution in S −K , [z, t] = 1. Hence also CE (s) = [E, s] ∼ = 4  But t∗ acts on K + , so t ∈ CG (z) by 5.10.4.2, a contradiction. Lemma 5.10.7. (1) If Ω = Ω(z) then Z(G) = 1. (2) If G/Z ∼ = Z2 for s ∈ S − CS (z). = L3 (2)/E64 and Z = 1 then Z = zz s ∼ (3) If G/Z ∼ = L3 (2)/23+6 then Φ(Z) = 1, m(Z) ≤ 2, and Z = zz s : s ∈ S . Further |Ω(z)| = 4/|Z|. Proof. We first prove (1), so assume Ω = Ω(z) but Z = 1. By 5.10.4.4, E ∩ Z = 1. By 5.10.6 we may assume Q+ is E64 or 23+6 . Assume first that Q+ ∼ = E64 . From the proof of 6.2.16, Q+ is the core of the permutation module for G on G/SM , where M ∈ M (τ ), and E ∩ O(z) = z . Set G! = G/E and F = O(z) ∩ Q. Then as Ω = Ω(z) is of order 2, we have E8 ∼ = F  S with F ∩ ZE = z , so F0! = Z(S ! ) ∩ F ! = 1. Therefore Q!0 = F0!G ≤ Z(Q! ), and by 5.10.5, Q!0 = Q! . As in the proof of 5.10.6, m(Q! ) = 4 and for s∗ an involution in S ∗ , CQ! (s∗ )/Z ! = [Q! , s]/Z ! . This is a contradiction as K ∗ centralizes F ! ∼ = E4 . Therefore (1) holds when Q+ ∼ = E64 , so assume instead that Q+ ∼ = 23+6 . From the proof of 6.2.19, for k ∈ K − z , k+ E + is contained in a complement H + /E + + + + to Q+ /E + in G+ /E + ; H + has three irreducibles Q+ i /E , 1 ≤ i ≤ 3, on Q /E ; Qi H Qi and Ωi = K , we have Si = S ∩ Hi ∈ Syl2 (Hi ) with and setting Hi = K τi = (Fi , Ωi ) a quaternion fusion packet with model Hi , where Fi = FSi (Hi ) and Hi /Z(Hi ) ∼ = L3 (2)/E64 . Therefore from the previous paragraph, Z(Hi ) = 1, so

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∼ E64 , so by 5.10.5, we have Z(Hi! ) = 1. Therefore [Q! , H] = O2 (H1! )O2 (H2! ) = Z ! = 1, contradicting 1 = Z and Z ∩ E = 1. This finally completes the proof of (1). Assume the setup of (2). Then Ω(z) = {K} by (1), so Ω = {K1 , K2 } where z1 = z2 and zi = z(Ki ). Hence as Ω(z + ) = Ω+ , we have u = z1 z2 ∈ Z. Set G− = G/u . Then Ω(z − ) = Ω− , so Z(G− ) = 1 by (1). Therefore Z = u , establishing (2). Finally assume the setup of (3) and let U = zz s : s ∈ S . As μ(τ ) ∼ = W eyl(D4 ), U is a subgroup of E4 by 5.8.7.2. Then (3) follows as in the previous paragraph.  Lemma 5.10.8. Assume y is an element of order 3 acting on G and centralizing O(z). Assume |Ω(z)| > 1. Then y centralizes G. Proof. Assume otherwise and set O = O(τ ). Now y acts on KQ ≥ O, so y acts on O. Then as |Ω| is 1, 2, or 4, either y acts on each J ∈ Ω or y acts on a unique member of Ω. As y centralizes O(z) and |Ω(z)| > 1, the latter is impossible, so y acts on each J ∈ Ω. Then as y centralizes K ∗ = J ∗ it follows that y centralizes J so y centralizes O. Suppose [G+ , y] = 1. Then G = CG (y)Z , so y centralizes O CG (y) = O G = G by 5.10.3, contrary to assumption. Therefore [G+ , y] = 1, so replacing G by G+ , we may assume Z = 1. Next G∗ ∼ = L3 (2) and y centralizes K ∗ ∼ = E4 , so y centralizes G∗ . As F ∗ (G) = Q, y does not centralize Q. As G∗ is irreducible on E and y centralizes z ∈ E, we conclude that y centralizes E. Therefore E = Q. Set G! = G/E. Suppose Q ∼ = E64 . As we saw, O ∩ Q ≤ E and G∗ is irreducible on Q! , so y ! centralizes Q . But then y centralizes Q, a contradiction. This leaves Q ∼ = 23+6 . Now, as we saw during the proof of 5.10.7, G∗ has three irreducibles Q!i , 1 ≤ i ≤ 3, on Q! , and [Qi , O] ≤ Qi ∩ O ≤ E, so as usual y centralizes Qi . But then y centralizes Q = Q1 Q2 for our final contradiction.  Lemma 5.10.9. (1) Z ≤ E = Z ≤ O 2 (CG (z)). (2) G = O(z)g : z g ∈ O 2 (CG (z)) . Proof. By 5.10.6 and 5.10.7, Z ≤ E = Z . Let y of order 3 in CG (z). As O = O(τ )  CG (z), y acts on O. Then as |Ω| = 1, 2 or 4, either y acts on each J ∈ Ω or |Ω| = 4 and y acts on a unique K ∈ Ω. The latter does not hold as for each Ki ∈ Ω − {K}, {K, Ki } = Ωi and y ∈ Hi , as in the proof of 5.10.7. Therefore y acts on each J ∈ Ω, so as J ∗ = K ∗ = [K ∗ , y], we conclude that J = [J, y]. Hence O = [O, y] ≤ O 2 (CG (z)). Then as Z ≤ O by 5.10.6 and 5.10.7 and E = Z[E, Ky ], we have E ≤ O 2 (CG (z)), completing the proof of (1). Then (1) and 5.10.3 imply (2).  ˆ and ˆ be the covering group of Sp6 (2), Sˆ ∈ Syl2 (L), Definition 5.10.10. Let L ˆ Then there exists Ω ˆ ⊆ Sˆ with ρ = (G, Ω) ˆ a quaternion fusion packet. G = FSˆ (L). ˆ≤L ˜∼ ˜ so the Lie packet for L ˜ This can be seen as L = Spin7 (3) with Sˆ ∈ Syl2 (L), ˆ ˆ induces the packet ρ. Next let P be the preimage in L of the maximal parabolic ˆ L) ˆ isomorphic to L3 (2)/E64 , and set Fˆ = F ˆ (Pˆ ); by the same argument, of L.Z( S ˆ ˆ is a quaternion fusion packet. Denote both Pˆ and Fˆ by L ˆ 3 (2)/E64 . τˆ = (F , Ω)

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∼ ˆ Ω) ˆ be a quaternion fusion packet with Fˆ = Lemma 5.10.11. Let τˆ = (F, ˆ L3 (2)/E64 . Then ˆ ∼ (1) O2 (F) = D83 . (2) τˆ is the largest quaternion fusion packet (Y, Γ) with Y = Y ◦ and Y/Z(Y) ∼ = L3 (2)/E64 . ˆ Proof. Part (1) follows from the embedding of Pˆ in L. Suppose for i = 1, 2, (Yi , Ωi ) are quaternion fusion packets with Yi = Yi◦ and Yi /Ui ∼ = L3 (2)/E64 , where Ui = Z(Yi ) = 1. Let Li be a model for Yi . Then by 5.10.7.2, Ui ∼ = Z2 . As P = P ∞ also Li = L∞ i , so Li = M/Vi for some Vi ≤ Z(M ) = V , and M the universal covering group of P . Then |V : Vi | = 2 and we may assume V1 = V2 , so setting V0 = V1 ∩ V2 , we have V /V0 ∼ = E4 . Set L = M/V0 and Q = O2 (L); we abuse notation and regard V0 = 1. Let J/V ∈ Ω and z ∈ Q with zV = z(J/V ). Set Y = O 2 (CL (zV ); then Y = XJR where X ∼ = Z3 with J/V = [J/V, X] and R = [Q, JX] of index 2 in Q. As Y centralizes zV and V and Y = O 2 (Y ), it follows that Y centralizes U = V z . Let j ∈ J − U ; then j 2 ∈ zV so we may take z = j 2 . Then as X centralizes U and is transitive on (J/U )# , also z = i2 for each i ∈ J − U . Further for i ∈ J − U j , z = (ij)2 = zj i j, so i inverts j and hence i, j ∼ = Q8 . Therefore J = K × V where K = [J, X] ∼ = Q8 with z ∈ K ∼ Q . Next KV /V  Y /V , so [K, JR] ≤ V Z ∩ V Z = Z, and = 8 i i i 1 2 hence K  XJR = Y . Now either Y = CL (z) or CL (z) = Y t where t inverts X, and in the latter case t acts on [J, X] = K. Thus in any event, K  CL (z). Let K ≤ S ∈ Syl2 (L), F = FS (L), and Γ = K F . As K  CL (z), ζ = (F, Γ) is a quaternion fusion packet with V = Z(F) ∼ = L3 (4)/E64 , contrary to = E4 and F/V ∼ 5.10.7.2. This contradiction completes the proof of (2).  ˆ 3 (2)/23+6 a 2-fusion system Fˆ such that Definition 5.10.12. We denote by L ˆ ⊆ Sˆ with (Fˆ , Ω) ˆ a quaternion fusion packet such that Fˆ = Fˆ ◦ , there exists Ω ∼ ∼ ˆ ˆ ˆ Z(F) = E4 , and F/Z(F ) = L3 (2)/23+6 . Such a system exists. Namely, arguing as ˆ for the covering group of Ω+ (2), we have L ˆ ≤ Spin+ (3), so in 5.10.10, writing L 8 8 ˆ with G the 2-fusion system of L. ˆ Then there is a quaternion fusion packet (G, Ω) 3+6 choosing a maximal parabolic P of Ω+ , the preimage 8 (2) of the form L3 (2)/2 3+6 ˆ ˆ ˆ ˇ P of P in L is of type L3 (2)/2 . Similarly we denote by L3 (2)/23+6 a 2-fusion ˇ ⊆ Sˇ such that (Fˇ , Ω) ˇ is a quaternion fusion system Fˇ such that there exists Ω ◦ ˇ ˇ ˇ ˇ ˇ ∼ packet such that F = F , Z(F) is of order 2, and F /Z(F) = L3 (2)/23+6 . For ˆ example F has such factor systems. Observe that by 5.10.7.3, if (F, Ω) is a quaternion fusion packet with F = F ◦ , Z = Z(F) = 1, and F/Z ∼ = L3 (2)/23+6 , then F is of type Fˆ or Fˇ . In this last lemma of the section we do not assume Hypothesis 5.10.1. Lemma 5.10.13. Assume τ = (F, Ω) is a quaternion fusion packet with F ◦ ˆ 6, L ˆ 3 (2)/E64 , M12 , or L3 (2)/E8 , and F ◦ transitive on Ω. Let T isomorphic to AE ◦ be Sylow in F . Then (1) In the first two cases Ω = {K1 , K2 } is of order 2. (2) In the latter two cases Ω = {K1 } is of order 1 and K2 = CT (K1 ) ∼ = Q8 . (3) There exists no α ∈ AutF (K1 K2 ) of order 3 with [K1 , α] = 1 and K2 = [K2 , α].

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Proof. Parts (1) and (2) follow by construction of F ◦ . Set Q = K1 K2 and suppose α is as in (3). Observe F ◦ , and hence also P = O2 (F ◦ ), is normal in F, so α extends to P Q. But if F ◦ is not M12 then P Q = P Ki for i = 1, 2, so as [K1 , α] = 1 we have α centralizing P Q/P . This is a contradiction as K2 = [K2 , α], so P Q/P = [P Q/P, α]. Thus we may assume F ◦ ∼ = M12 . Let z = z(K1 ); as F ◦  F we have Y = CF ◦ (z)  CF (z). Now a model Y for Y has Q = O2 (Y ) and Y /Q ∼ = S3 faithful on K1 /z for i = 1, 2. But then  α ∈ [α, Y ] ≤ AutY (Q), a contradiction. 5.11. Summary of basics We began this manuscript with statements of our major theorems, followed by a survey of basic notions, notation, and results on fusion systems. The remainder of the first five chapters is devoted to the construction of the foundation of the theory of quaternion fusion packets. In this final section of Chapter 5, we pause to collect in one place a list of the most frequently used concepts, results, and notation from that foundation. Hopefully this list will provide a convenient resource when, in later sections, the reader needs to refresh his or her memory of this theory. Throughout this section, F is a saturated fusion system on a 2-group S. Recall F f denotes the set of fully normalized subgroups (or involutions) in S. Given X ≤ S (or an involution X in S), A(X) denotes the set of α ∈ homF (NS (X), S) such that Xα ∈ F f . For x an involution in S, Fx = CF (x). For Σ a set of subgroups of S, [Σ]F denotes the normal closure of Σ in F: the smallest normal subsystem of F containing Σ. The subnormal closure of T ≤ S in F is sub(F, T ): the limit of the sequence subi (F, T ), where sub0 (F, T ) = F and subi+1 (F, T ) = [T ]subi (F ,T ) . The notion of a set D of 3-transpositions of a finite group G is defined in section 4.1. Let D be a set of 3-transpositions of G. For d ∈ D, d⊥ = CD (d), Dd = d⊥ − {d}, and Ad = D − d⊥ . D(D) denotes the commuting graph on D. Set Vd = {a ∈ D : a⊥ = d⊥ }. By 4.1.7.1, if G is transitive on D then Vd = dO2 (G) . Let G be a graph. For v a vertex of G, G(v) denotes the set of vertices adjacent to v in G. Also G c denotes the complementary graph to G: the graph in which x and y are adjacent if and only if x = y and y ∈ / G(x). From the Introduction, a quaternion fusion packet is a pair τ = (F, Ω), where F is a saturated fusion system on a 2-group S, Ω is an F-invariant set of subgroups of S, and (QFP1) There exists an integer m such that for all K ∈ Ω, K has a unique involution z(K) and K is nonabelian of order m. (QFP2) For each pair of distinct K, J ∈ Ω, |K ∩ J| ≤ 2. (QFP3) If K, J ∈ Ω and v ∈ J − Z(J), then v F ∩ CS (z(K)) ⊆ NS (K). (QFP4) If K, J ∈ Ω with z(J) = z(K), v ∈ K, and φ ∈ homCF (z(K)) (v, S), then either vφ ∈ J or vφ centralizes J. In the remainder of the section we assume τ = (F, Ω) is a quaternion fusion packet. Set F ◦ = [Ω]F and τ ◦ = (F ◦ , Ω). By 2.5.2, τ ◦ is a quaternion fusion packet.

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Set Z = K∈Ω z(K)F and ZS = {z(K) : K ∈ Ω}. Set WS = ZS . For z ∈ ZS , Ω(z) = {K ∈ Ω : z(K) = z}. By 2.5.2, if E is a saturated subsystem of F on T , and Γ ⊆ Ω is contained in T and is E-invariant, then (E, Γ) is a quaternion fusion packet. As a special case, we find in 3.1.1: For z ∈ Z ∩ F f , τz = (Fz , Ω) is a quaternion fusion packet. For z ∈ ZS we set O(z) = Ω(z) , while O(τ ) = Ω . By 3.1.5.1, O(τ ) is a central product of the members of Ω. For K ∈ Ω we define a subsystem OK on K. If m = 8 set OK = FK (K). If m > 8, set OK = O 2 (AutF (Q)) : Q ∈ QK K , where QK is the set of Q8 -subgroups of K. For z ∈ ZS ∩ F f , O(z) is the central product of the OK , for K ∈ Ω(z). By 2.6.11, O(z)  Fz . Write O(τ ) for the normal closure of O(τ ) in NF (WS ). By 3.3.7, O(τ ) is a central product of the OK , for K ∈ Ω. For z ∈ Z, Δ(z) = {(K ∩ CS (z)α)α−1 : K ∈ Ω(zα) and |K ∩ CS (z)α|

> 2}, for α ∈ A(z). By 3.1.3, Δ(z) is independent of the choice of α. Set Δ(τ ) = z∈Z Δ(z) and ZΔ = {z ∈ Z : Δ(z) = ∅}. By 3.1.11.6, each member of Δ(τ ) − Ω is cyclic of order m/2. By 3.1.3 if P ≤ D ∈ Δ(τ ) with |P | > 2, then each conjugate of P is contained in a member of Δ(τ ). Define η(τ ) to consist of the sets Σ of cyclic subgroups of S such that (i) Σ is Σ -invariant. (ii) Each V ∈ Σ is contained in some D(V ) ∈ Δ(τ ) such that either V = D(V ) or D(V ) ∈ Ω and |D(V ) : V | = 2. (iii) The map V → D(V ) is a bijection of Σ with Δ(τ ). Write ηS (τ ) for the set of S-invariant members of η(τ ); by 3.1.10, ηS (τ ) = ∅. Set W (τ ) = {η : η ∈ ηS (τ ) . By 3.1.11 each member of W (τ ) is abelian, and for each η ∈ η(τ ), all members of η are of order m/2. By 3.3.14, if F is transitive on Ω then either |η(τ )| = 1 or m = 8 and Δ(τ ) = Ω. Observe that if η(τ ) = {η} then W (τ ) = {W } is of order 1 and W is weakly closed in S. Let η ∈ η(τ ) and W = η . For V ∈ η, define dV ∈ AutF (W ) by dV = ck α−∗ , where α ∈ homF (W, S) with V α ≤ K ∈ Ω and k ∈ K − V α. By 3.1.20, dV is independent of the choice of α and k. Set Dη = {dV : V ∈ η} and μη = Dη ≤ AutF (W ). By 3.1.10, μη  AutF (W ). By 3.1.26, μη is independent of the choice of η ∈ η(τ ), so we write μ = μ(τ ) for μη . We call μ the Thompson group of τ . By 3.1.25, μ is a direct product of a 3-transposition group and copies of D8 . Let D = Dη ≤ μη = μ. By 4.2.8.6, |Ω| = |D ∩ AutS (W )| is the width of μ. If μ is transitive on D then F ◦ is transitive on Ω by 4.2.8.7. Let D, D∗ be the set of pairs (z, t) in Z×Z such that for some φ ∈ homF (z, t , S), (z, t)φ is in ZS ×ZS , ZΔ ×ZΔ , respectively. Regard D and D∗ as graphs. There are necessary and sufficient conditions for (z, t) to be an edge in the graphs in 3.1.14 and 3.1.15. The subgraph A of D∗ is defined in 3.2.1. Let η ∈ ηS (τ ), W = η , and μ = μη . Then W  S, so W ∈ F f . By 3.3.11, τW = (NF (W ), Ω) is a quaternion fusion packet with μ(τW ) = μ. Further

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NF (W )◦ = [O(τ )]NF (W ) is constrained, and hence has a model M (W ). Set M (τ ) = {M (W ) : W ∈ W (τ )}. Let Φ be a Coxeter diagram of type An , Dn , or En with n ≥ 2 and n ≥ 4 if ¯ = ω ¯ (Φ, m) via Φ is of type Dn . In 4.3.1 we defined a certain universal group G generators and relations. By 4.3.8, if Dη is a class of 3-transposition of μ, then μ ¯ → M (W ) is the Weyl group of type Φ, for some Φ, and M (W ) is an image π : G ¯=ω ¯ of G ¯ (Φ, m) with ker(π) ≤ Z(G). For θ ⊆ ZS define O(θ) = O(z) : z ∈ θ . Define E(τ ) = SO(τ ), NF (O(θ)), NF (W ) : W ∈ W (τ ), ∅ = θ ⊆ ZS with O(θ) ∈ F f S . Often E(τ ) = F. Finally the reader is directed to the list of examples appearing as conclusions to our major theorems; recall this list appears at the end of section 1.1.

Theorems 2 through 5

CHAPTER 6

Theorems 2 and 4 In Chapter 6 we prove Theorem 2 and reduce Theorem 4 to Theorems 1 and 3. Let τ = (F, Ω) be a quaternion fusion packet and S Sylow in F. We begin by showing in Theorem 6.1.5 that if F = F ◦ is transitive on Ω and τ satisfies one of the conclusions of Theorem 1, then, with the exception of the two classes of examples in case (4) of Theorem 4, τ is coconnected: that is D(τ )c is connected. Then in section 6.2 we begin the proof of Theorem 2. We operate under Hypothesis 6.2.1, which, in addition to the hypotheses of Theorem 2, assumes that for K ∈ Ω we have K ∩ O2 (F) = Z(K), and for each nontrivial subgroup Q of S normal in F, we have Q ∩ Z(τ ) = ∅. These hypotheses isolate the interesting minimal cases from Theorem 2. Set E = Z(τ ) . Immediately in 6.2.2 we show that E is the unique minimal normal subgroup of F, and then show in 6.2.4 that F is constrained. Hence F has a model G, and for the most part we work with G rather than with F. There are two cases: E acts on K and E does not act on K. We treat the first case in section 6.2, where we show that G is L3 (2)/E8 , L3 (2)/E64 , or L3 (2)/23+6 . The second case is treated in section 6.3, where we show G is AEn . In section 6.4 we introduce the subnormal closure τ (K) = (s(K), K s(K) ) of K in τ ; here s(K) is the subnormal closure of K in F, defined in 1.2.9. For the most part section 6.4 is devoted to a description of s(K) in the case where z = z(K) ∈ O2 (F), building on the earlier work in sections 6.2 and 6.3. Then in Theorem 6.5.4 we prove that if z ∈ / O2 (F) then z is contained in a unique component C of F, and if OK is not solvable then O 2 (OK ) ≤ C. Building on these results, we show in 6.6.6 that if Theorem 1 holds in all proper subpackets of the form τ (K) for K ∈ Ω, then a weak version of Theorem 4 holds. In particular, as discussed in Remark 6.6.10, Theorems 1 and 3 imply Theorem 4. Lemma 6.6.11 gives information about the embedding of subpackets of τ in τ . Finally Theorem 2 is established in the short section 6.7, using the earlier sections in Chapter 6.

6.1. D(τ )c In this section τ = (F, Ω) is a quaternion fusion packet and S is Sylow in F. Let D = D(τ ) be the graph of τ , and for z ∈ Z, write Σz for the connected component of Dc containing z. Lemma 6.1.1. If all members of ZS are in the same connected component of Dc then D c is connected. 137

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Proof. Suppose t ∈ Z − ZS . Then by 3.1.5.3, there is K ∈ Ω such that t does not centralize K. Set z = z(K); then by 3.3.4, t ∈ / z ⊥ , so Σt = Σz . The lemma follows.  Lemma 6.1.2. If Ω = Ω(z) for some z ∈ ZS then D c is connected. Proof. This is immediate from 6.1.1.



Lemma 6.1.3. Assume μ(τ ) ∼ = Weyl(Φ), where Φ is An for some n ≥ 2, Dn for some n ≥ 3, or En for some n ∈ {6, 7, 8}. Then Dc is connected. Proof. Let η ∈ η(τ ), W = η , M ∈ M (τ ), and D = Dη . Then D is the set of reflections in the Weyl group μ = μ(τ ) = M/W . Let G be the full subgraph of D on ZΔ . By 6.1.1 it suffices to show that G c is connected. By 4.2.8.4, ζ : D(D) → G is a bimorphism of graphs, where ζ : dV → z(V ) for V ∈ η. Thus it suffices to show that D(D)c is connected. But by the hypothesis on μ, μ is transitive on D,  so D(D)c is connected by 4.1.4. Lemma 6.1.4. Assume F = F ◦ and Z ∩ Z(F) = ∅. Set F + = F/Z(F). Then τ = (F + , Ω+ ) is a quaternion fusion packet, F + = (F + )◦ , and ξ : D → D(τ + ) is a morphism of graphs, where ξ : z → z + . Therefore if D(τ + )c is connected then so is Dc . +

Proof. Let Θ : F → F + be the natural map Θ : s → s+ ; then ξ = Θ|Z . By 3.3.2.2, τ + is a quaternion fusion packet, and by 3.3.2.4, (F + )◦ = F + . Recall from 3.1.13 that (z, t) is an edge in D iff z = t and (z, t)φ ∈ ZS × ZS for some φ ∈ homF (t, z , S). Therefore if (z, t) is an edge in D then (z, t)φ = (z(K), z(J) for some K, J ∈ Ω. Then K + , J + ∈ Ω+ , with (z + , t+ )φ+ = (z(K + ), z(J + )), so either (z + , t+ ) is an edge in D(τ + ), or z + = t+ . That is ξ : D → D(τ + ) is a morphism of graphs. Suppose D(τ + )c is connected and let (u, v) ∈ Z × Z, a = u+ , and b = v + . Then there exists a path a = a0 , . . . , an = b in D(τ + )c . As ξ : Z → Z(τ + ) = Z + is a surjection, there is a sequence u = u0 , . . . , un = v in Z such that ui ξ = ai for each i. As ξ is a morphism and (ai , ai+1 ) is not an edge in D(τ + ), also (ui , ui+1 ) is  not an edge in D. Thus Σu = Σv , completing the proof. Theorem 6.1.5. Assume F = F ◦ is transitive on Ω and τ satisfies one of the conclusions of Theorem 1. Then one of the following holds: (1) Dc is connected. (2) F/Z(F) ∼ = P Spn [m] for some n ≥ 6. (3) F/Z(F) ∼ = P SL− n [m] for some n ≥ 6. Proof. Assume τ is a counter example. Then by 6.1.4 and induction on the order of τ , replacing τ by τ + we have: (a) Z(F) = 1. By 6.1.1: (b) There exist z, t ∈ ZS such that Σz = Σt . Similarly by 6.1.2: (c) Ω = Ω(z). Let K ∈ Ω(z); we next show: (d) z ∈ / O2 (F).

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Assume otherwise. The proof of Theorem 2 in section 6.7 does not involve Theorem 6.1.5, so we may appeal to Theorem 2. We conclude that F is constrained with model G, K ≤ O2 (Fz ), and G satisfies one of the conclusions of Theorem 2. By (c), G satisfies neither conclusion (1) nor (2) of Theorem 2 (cf. 6.2.9.5) and if G is AEn then n ≥ 9. Therefore if conclusion (3) of Theorem 2 holds then μ = μ(τ ) ∼ = Weyl(Dk ) for some k ≥ 4, contrary to 6.1.3. Therefore conclusion (4) of Theorem 2 is satisfied by G. Thus G ∈ M (τ ) and μ is generated by a set D of 3-transpositions. As F = F ◦ is transitive on Ω, μ is transitive on D by 4.1.4, so 4.3.8.1 and 5.4.3 supply a contradiction, completing the proof of (d). (e) τ is a Lie fusion packet. By (d) and the hypothesis of our theorem, τ satisfies one of conclusions (2), (3), or (4) of Theorem 1. But by (c), neither conclusion (3) nor (4) holds, establishing (e). (f) F is not the 2-fusion system of an exceptional group. Assume otherwise, By (c), F is not G2 [m]. Therefore by (e), F is F4 [m], E6 [m], E7 [m], or E8 [m], contrary to 6.1.3. (g) F is P Ωn [m] for some n ≥ 9 or P SL+ n [m] for some n ≥ 5. Assume otherwise. By (f), F is the 2-fusion system of a classical group L. If F = P Spn [m] is symplectic, then n ≥ 6 by (c), so F appears in conclusion (2) of our theorem, contrary to the choice of τ as a counter example. Similarly if F is P SL− n [m] then n ≥ 5 by (c), so n = 5 as τ is a counter example. But then by 5.4.6, Ω = {K1 , K2 } is of order 2, and by 5.4.10.4 there exists s ∈ Z nontrivial on K1 and K2 , contrary to (b). We’ve shown that F is P Ωn [m] or P SL+ n [m]. Then by (c), n ≥ 9 in the first case, while n ≥ 5 in the second. Therefore (g) holds. We are now in a position to obtain a contradiction, establishing our theorem. For if F is P Ωn [m] with n ≥ 9 then μ ∼ = Weyl(Dk ) for some k ≥ 4, contrary to [m] for some n ≥ 5. Hence μ is Weyl(An−1 ), 6.1.3. Therefore by (g), F is P SL+ n again contrary to 6.1.3.  Lemma 6.1.6. Let η ∈ η(τ ), W = η , μ = μ(τ ), and G a model for NF (CS (W )). (1) AutF (O(τ ))Ω = AutF (O(τ )W )Ω . (2) If F is transitive on Ω and Dη is a set of 3-transpositions of μ, then either (i) m = 8 and Δ = Ω, or (ii) η(τ ) = {η}, NG (O(τ )) is transitive on Ω, and G is transitive on the connected components of D(Dη ). Proof. Let O = O(τ ), V = OCS (O), Y a model for NF (V ), and X the kernel of the action of Y on Ω. Set R = S ∩ X. Then W ≤ R and by a Frattini argument, Y = XNY (R) = XNY (Δ). Therefore Y Ω = NY (Δ)Ω , so (1) holds. Assume F is transitive on Ω. Then by 3.3.14, either (2i) holds or η(τ ) = {η}, and we may assume the latter. Hence W is weakly closed in S. As F is transitive on Ω, so is AutF (O), and hence also AutF (OW ) by (1). Then as W is weakly closed, NN (O) is transitive on Ω, where N = NF (W ). Therefore NG (O) is transitive on Ω. Now G induces a group of automorphisms on μ via (xCM (W ))g = xg CM (W ) for x ∈ M and g ∈ G. Further D = Dη = {dU : U ∈ η}, so as G permutes η, it also permutes D, and hence the set Σ of connected components of D(D)c . Then as NG (O) is transitive on Ω, also G is transitive on Σ, completing the proof of (2). 

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6.2. Beginning the case z ∈ O2 (F) In this section we assume: Hypothesis 6.2.1. τ = (F, Ω) is a quaternion fusion packet and, setting S0 = O2 (F): (1) F is transitive on Ω. (2) F = F ◦ . (3) For some K ∈ Ω, K ∩ S0 = Z(K). (4) For each nontrivial normal subgroup Q of F, Q ∩ Z = ∅. Set z = z(K) and E = Z . Lemma 6.2.2. (1) F is transitive on Z. (2) E is the unique minimal normal subgroup of F. (3) E ≤ Z(S0 ). (4) AutF (E) is irreducible on E. (5) K does not centralize E. Proof. Part (1) follows from 6.2.1.1. By 6.2.1.3, z ∈ S0 , so E ≤ S0 by (1). Then as Z is F invariant, E  F. If M is a minimal normal subgroup of F then M ∩ Z = ∅ by 6.2.1.4, so E ≤ M by (1). Hence (2) holds, and (2) implies (3) and (4). Suppose K centralizes E. Then by (1), Ω ⊆ CF (E)  F, so E ≤ Z(F) by 6.2.1.2, contrary to 6.2.1.4.  Lemma 6.2.3. (1) [K, NS0 (K)] ≤ z . (2) If t ∈ NZ (K) then either [K, t] = 1 or CK (t) is cyclic of index 2 in K. Proof. First [K, NS0 (K)] ≤ K ∩ S0 = z by 6.2.1.3, establishing (1). Thus  if t ∈ NZ (K) then |K : CK (t)| ≤ 2 by 6.2.2.3, so (2) follows from 3.3.6. Lemma 6.2.4. (1) OK = FK (K). (2) CK (E) = z . (3) F is constrained. Proof. First E ≤ S0 ≤ CF (E) ≤ Fz , so (*) S0 ≤ O2 (Fz ) and CO2 (Fz ) (E) = O2 (CF (E)) = S0 . Suppose (1) fails. Then C = O 2 (OK ) is a component or solvable component of Fz by 2.6.13, so either C ≤ CF (O2 (Fz )) ≤ CF (S0 ) by (*), or C ∼ = SL2 [8]. In the latter case as K ∩ S0 = z , again C centralizes S0 . But the Sylow subgroup Q of C is quaternion and E ≤ CS (Q) ≤ NS0 (K), so E centralizes K by 6.2.3.2, contrary to 6.2.2.5. Thus (1) holds. By (1), K ≤ O2 (Fz ), so CK (E) ≤ K ∩ S0 = z by (*), proving (2). Next as z ∈ S0 , E(F) = E(Fz ) by 10.3 in [Asc11], so E(F) ≤ CFz (K) as K ≤ O2 (Fz ). But now F = F ◦ ≤ CF (E(F) by 6.2.1.2, so E(F) = 1. Therefore (3) follows from 14.2 in [Asc11].  Notation 6.2.5. By 6.2.4.3, F is constrained, so by 1.2.8.1 there exists a model G for F.

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Lemma 6.2.6. (1) E is the unique minimal normal subgroup of G. (2) G = K G . Proof. Part (1) follows from 6.2.2.2. By 1.2.8.4, there is a bijection between the normal subsystems of F and the normal subgroups of G, so (2) follows from 6.2.1.2.  Lemma 6.2.7. (1) m = 8. (2) O(z) is extraspecial of width |Ω(z)|. (3) |E : NE (K)| ≤ 2, and in case of equality, E8 ∼ = E ∩ KK e = [K, e] for e ∈ E − NE (K). (4) For t ∈ Z, one of the following holds: (i) t ∈ z ⊥ and t centralizes O(z). (ii) t ∈ D∗ (z) and for each J ∈ Ω(z), |CJ (t)| = 4. (ii) t is fixed-point-free on Ω(z). Proof. Suppose that E ≤ NS (K). Then by 6.2.3.2, for each t ∈ Z, CK (t) is K or cyclic of index 2 in K. Thus if m > 8 then Z centralizes the unique cyclic subgroup V of K of index 2, contrary to 6.2.4.2. On the other hand suppose e ∈ E − NE (K). Then [K, e] ≤ KK e ∩ E, so Φ([K, e]) = 1. Hence m = 8 in this case too, completing the proof of (1). Moreover as m = 8, KK e ∩ E = [K, e] ∼ = E8 . Then as E is abelian and O(z) is the central product of the members of Ω(z), which are permuted by E, it follows that E acts on KK e , so |E : NE (K)| = 2, completing the proof of (3). Observe (1) implies (2). Next if t centralizes K then t ∈ z ⊥ and [O(z), t] = 1 by 3.3.4, while if |CK (t)| = 4 then t ∈ D∗ (z) and |CJ (t)| = 4 for each J ∈ Ω(z) by 3.3.4 and 3.3.5. Thus (4) holds.  Theorem 6.2.8. Assume E ≤ NG (K) and let μ = μ(τ ). Then (1) m(E) = 3, S0 = CG (E), and AutG (E) = GL(E). (2) One of the following holds: (a) μ ∼ = L3 (2)/E8 . = S3 and G ∼ ∼ (b) μ = S4 and G ∼ = L3 (2)/E64 . (c) μ ∼ = L3 (2)/23+6 . = Weyl(D4 ) and G ∼ Until the proof of Theorem 6.2.8 is complete, assume E ≤ NG (K). Set O0 (z) = O(z) ∩ S0 and for g ∈ G, set O0 (z g ) = O0 (z)g . Lemma 6.2.9. (1) E ∼ = E8 . (2) Let E = z, t, s with t, s ∈ Z. Then G = O(z), O(t), O(s) and CG (E) = S0 = O0 (z)O0 (t)O0 (s). (3) AutG (E) = GL(E). (4) AutO(z) (E) = AutK (E) ∼ = E4 is the group of transvections with center z. (5) Ω = Ω(z). Proof. Let R be the group of transvections on E with center z. By 6.2.3.1, AutO(z) (E) ≤ R, and by 6.2.4.2, AutK (E) ∼ = E4 . Then as G is irreducible on E, it follows from [McL69] that AutG (E) = GL(E), proving (3). In particular G is 2-transitive on E # , so if (5) fails then Z ⊆ z ⊥ . But then E centralizes K, contrary to 6.2.2.5. Thus (5) holds and more generally CE (K) = z .

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By (5) and 6.2.7.4, E induces inner automorphisms on K, so |E : CE (K)| ≤ 4. Therefore as CE (K) = z , we have m(E) ≤ 3, while m(E) > 2 by 6.2.4.2, so (1) holds. Then as E4 ∼ = AutK (E) and AutO(z) (E) ≤ R, (4) follows. Let E = z, t, s with t = z g and s = z h , and set X = CG (E) and G∗ = G/X = AutG (E). By (4), G∗ = H ∗ , where H = O(z), O(z)g , O(z)h . Further X acts on O(z) so [X, O(z)] ≤ CO(z) (E) ≤ O2 (X) = S0 , so [X, O(z)] ≤ O0 (z). Hence H and X, and then also G, acts on T = O0 (z)O0 (t)O0 (s) and H + centralizes X + , where G+ = G/T . Hence G = K G = O(z)H by 6.2.6.2, so H = G. Further G+ = H + is a central extension of X + = Z(G+ ) by L3 (2) with E4 ∼ = K + normal in CG+ (z), + +G+ ∼ so we conclude that G = K = L3 (2). This establishes (2) and completes the proof of the lemma.  Notice 6.2.9 says that part (1) of Theorem 6.2.8 holds, so it remains to establish part (2) of the theorem. Choose notation so that E0 = z, t  S. Recall the definition of the essential subgroups of S from section 1.2. Lemma 6.2.10. (1) z ⊥ = {z}. (2) E − z = D∗ (z) − z ⊥ . (3) ZΔ = E0# . (4) μ ∼ = S3 , D12 , S4 , or Weyl(D4 ) with |Ω(z)| = 2n for n = 0, 1, 1, 2, respectively. n+1 (5) O(z) ∼ = 21+2 . (6) O2 (CG (z)) and O2 (NG (E0 )) are the essential subgroups of S. Proof. Part (1) follows from 6.2.9.5. Then (2) follows from (1) and 6.2.7.4. By 3.1.19, μ = D , where D = Dη = {dV : V ∈ η} for η ∈ η(τ ). By (1) and 4.2.10, either A(z) = ∅ and μ ∼ = D12 , or D is a conjugacy class of 3-transpositions of μ. In the latter case by (1), 4.2.13, and 4.3.5.1, we have μ ∼ = S3 , S4 , or Weyl(D4 ). Next | for V ∈ η by (1) and 4.2.8, so (4) holds. Then (4) and 6.2.7.2 imply |Ω(z)| = |d⊥ V (5). Let M ∈ M (τ ). By (4), M/O2 (M ) ∼ = S3 , so as AutG (E) = GL(E) and S acts on M , we conclude that M S = NG (E0 ). Therefore (3) holds. Let R be an essential subgroup of S. By 1.2.11, S0 ≤ R, so as the radical subgroups of G/S0 are S/S0 and the minimal parabolics CG (z)/S0 and NG (E0 )/S0 , (6) follows.  Lemma 6.2.11. If μ ∼ = S3 or D12 then μ ∼ = S3 and G ∼ = L3 (2)/E8 . ∼ D12 . Then t ∈ A(z) and t ∈ O(z) by 3.2.2.2. Proof. Assume first that μ = Next as X ∈ Syl3 (CG (z)) acts nontrivially on K, E = z [E, X] ≤ O(z), so as X is irreducible on O(z)/E, O0 (z) = E. Then as G is transitive on E # , O0 (t) = E = O0 (s), so E = S0 = CG (E) by 6.2.9.2, and then by 6.2.9.3, G is an extension of E8 by L3 (2). Further for U ∈ Δ(t), S = U KE, so the two members of Ω(z) are normal in S, and hence not conjugate in G, contradicting 6.2.1.1. Therefore μ ∼ = S3 . By 6.2.10.4, O(z) = K, so O0 (z) = z . Thus S0 = E = CG (E) by 6.2.9.2, so, as in the previous paragraph, G is an extension of E8 by L3 (2). Let η ∈ η(τ ), W = η , and M = K NG (W ) . By 4.2.5.4, M is determined up to isomorphism and SM = S ∩M is wreathed of order 32. Therefore |S : SM | = 2

6.2. BEGINNING THE CASE z ∈ O2 (F )

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and m(SM ) = 2, so m(S) ≤ 3. Hence G does not split over E, so by an unpublished result of Alperin, G is determined up to isomorphism. Hence the lemma holds.  Because of 6.2.10.4 and 6.2.11, for the remainder of the proof of Theorem 6.2.8 we assume: ∼ S4 or Weyl(D4 ). Let η ∈ η(τ ) and Hypothesis 6.2.12. E ≤ NG (K) and μ = set W = η and M = K NG (W ) . Thus W = CM (W ) by 4.3.8.4, so M/W ∼ =μ and M ≤ NG (E0 ) by 6.2.10.3. Lemma 6.2.13. If E ≤ F ≤ NS0 (K) with F  G, then F = E. Proof. By 6.2.3.1, [F, K] ≤ z , so as [E, K] = z , we have F = ECF (K). Thus by 6.2.6.2, G = K G centralizes F/E. Suppose f ∈ F − E; without loss F = Ef . Then F ∼ = E16 and |CF (K)| = 4 but CE (K) = z , so K centralizes two members of f E. By 6.2.6.1, CF (G) = 1, so by 6.2.9, either AutS (F ) is regular on f E and AutG (F ) = AutS0 (F )CAutG (F ) (f ), or S0 centralizes F and CAutG (F ) (f ) ∼ = Frob(21). In either case, G is 2-transitive on f E and CG (f ) is transitive on E # . In particular as K centralizes two members of f E, we may choose those members to be f and f z. Therefore CAutG (F ) (f ) is of even order, so AutS0 (F ) is regular on f E. For x ∈ f E, set J (x) = {J ∈ K G : [J, x] = 1} and r = |J (f )|. As CG (f ) is transitive on E # , r = 7j where j = |Ω(z) ∩ J (f )|. Then counting the set Γ of pairs (x, J) with x ∈ f E and J ∈ J (x) in two ways, we conclude that 2 · 7 · |Ω(z)| = 2|K G | = |Γ| = 8r = 56 · j. Then as |Ω(z)| = 2 or 4 by 6.2.12, we conclude |Ω(z)| = 4 and j = 1. Let S1 = CS0 (F ). As j = 1, S1 acts on K, so as [NS0 (K), K] ≤ z , we conclude that S1 = ECS1 (K). Thus Φ(S1 ) = Φ(CS1 (K)) centralizes K, and hence G = K G centralizes Φ(S1 ), so Φ(S1 ) = 1 by 6.2.6.1. Next as AutS0 (F ) is regular on f E, the map s0 S1 → [s0 , f ] is a K-isomorphism of S0 /S1 with E. Therefore S2 = [S0 , K]S1 = s S1 with f s = f z, so s acts on K. Thus S2 = ECS2 (K), so KS2 = P S3 , where P = KE ∼ = Q28 and S3 = CS2 (P ). Now O(z) = K S0 ≤ KS2 as [S0 , K] ≤ S2 . But by 6.2.10.5, O(z) ∼ = 21+8 , so S3 has a Q28 -subgroup, impossible as |S3 : S3 ∩ S1 | = 2 and Φ(S1 ) = 1.  Lemma 6.2.14. (1) G = O 2 (G). (2) G/[G, S0 ] ∼ = L3 (2). (3) [G, S0 ] ≤ E. Proof. By 6.2.12 and 6.2.10.4, |Ω(z)| = 2 or 4, so some X ∈ Syl3 (CG (z)) acts on K. Then from 6.2.9, K = [K, X], so K ≤ O 2 (G). Then (1) follows from 6.2.6.2. Let G∗ = G/[G, S0 ]. From 6.2.9, G∗ /Z(G∗ ) ∼ = L3 (2), so by (1), G∗ ∼ = L3 (2) or ∗ ∼ SL2 (7). However by 6.2.1.3, K = E4 , so (2) follows. If (3) fails then E = S0 by (2), a contradiction since by 6.2.12, μ = M/W ∼ = S4 or Weyl(D4 ).  Lemma 6.2.15. Set T =

 J∈Ω(z)

and let z g = t, z h = s. Then

NS0 (J)

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(1) [O(z), T ] ≤ z ≤ E and (2) E = T ∩ T g ∩ T h . Proof. Set F = T ∩ T g ∩ T h . First [F, O(z)] ≤ [T, O(z)] ≤ z ≤ E by 6.2.3.1, establishing (1). Similarly O(z)g and O(z)h centralizes F/E, so, using 6.2.9.2,  G = O(z), O(z)g , O(z)h centralizes F/E. Therefore F = E by 6.2.13. Lemma 6.2.16. If μ ∼ = S4 then G ∼ = L3 (2)/E64 . Proof. Assume μ ∼ = S4 . Then n = |Ω(z)| = 2 by 6.2.10.4, so T = NS0 (K) is of index at most 2 in S0 . Set F = T ∩ T g ∩ T h . As |S0 : T | ≤ 2, |S0 : F | ≤ 8. But F = E by 6.2.15, so |S0 : E| ≤ 8. Then by 6.2.14.3, S0 /E is a natural module for G/S0 . Suppose e ∈ E ∩ O(z) − z . Then as X ∈ Syl3 (CG (z)) acts on O(z), E = [e, X]z ≤ O(z), so E = O0 (z) as X is irreducible on O(z)/E. But then O0 (t) = O0 (z)g = E = O0 (s), so S0 = E by 6.2.9.2, a contradiction. Therefore E ∩ O(z) = z . Hence by 5.7.4.3, Z(M ) is generated by an involution i, and by 5.7.4.5, M is determined up to isomorphism. By 6.2.6.1, i ∈ / Z(G), and as M  M S0 , S0 centralizes i, so H = M S0 = CG (i) is of index 7 in G. Therefore S0 = iG is the core of the permutation module for G on G/H, using 6.2.6.1 and the fact that E and S0 /E are natural modules for G/S0 . In particular S0 = J(S) is determined in S. Now H = M S0 and M splits over S0 ∩ M by 5.7.4.6, so H splits over S0 . Then as S ≤ H, G splits over S0 by Gaschutz’ Theorem (cf. 10.4 in [Asc86]). So as S0 is the core of the permutation module for G on H, G is determined up to isomorphism, so G ∼  = L3 (2)/E64 . Because of 6.2.16, during the remainder of the proof of Theorem 6.2.8 we assume that μ ∼ = Weyl(D4 ). Lemma 6.2.17. (1) There is a surjective homomorphism π of ω ¯ (D4 , 8) onto M with kernel Z(¯ ω (D4 , 8)). (2) E ≤ O(z). (3) S ≤ M and |S| = 212 . ∼ Weyl(D4 ), there is a surjective homomorphism π : M ¯ = Proof. As μ = ¯ ) is of order 4 by 5.8.7.3, ω ¯ (D4 , 8) onto M by 4.3.8. As Ω = Ω(z), ker(π) = Z(M establishing (1). Then as m = 8, |W | = 26 by 4.3.7.2. Hence as |Weyl(D4 )|2 = 26 , SM = S ∩ M is of order 212 . By 5.8.7.6, t ∈ O(z), so E = [t, Y ]z ≤ O(z), where Y ∈ Syl3 (CG (z)). Thus (2) holds. Define T , g, and h as in 6.2.15, and set R = T ∩ T g . Let T1 be the kernel of the action of S on Ω and R1 = T1 ∩ T1g . Then S/T1 acts faithfully on Ω of order 4, so |S : T1 | ≤ 8. Set H = M S and H + = H/W . Observe that R1 = CS (O(z)+ ) as CS (M + ) acts on the preimage KW of K + in S, and hence also acts on K = KW ∩ Ω. Then R1 = CS (M + ) as M + = O(z)+ , O(z)g+ . Hence as H/S0 ∼ = S4 , we have R1 ≤ S0 , so R1 = R as R is the kernel of the action of S0 on Ω(z) ∪ Ω(z)g . Similarly T1 = CS (O(z)+ ) = O(z)CS (M + ) = O(z)R, so |T1 : R1 | = |O(z) : CO(z) (M + )| = 8. Hence |T1 : R| ≤ 8, so |S : R| ≤ 26 .

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By 6.2.15, E = R ∩ T h , while |T : T ∩ T h | = |T : R| ≤ 8, so |R : E| = |RT h : T | ≤ |T T h : T h | = |T : T ∩ T h | ≤ 8. Thus |S : E| ≤ 29 , so we have |S| ≤ 212 . Therefore (3) holds as |SM | = 212 .  h

Lemma 6.2.18. (1) S0 /E is the direct sum of two natural modules for G/S0 dual to E. (2) S0 is special with center E. Proof. Let X ∈ Syl3 (M ), M ∗ = M/E0 , and G+ = G/E. By 6.2.17.3, S ≤ M . Then from 6.2.17.1 and 5.8.7.5, CS ∗ (X ∗ ) = Z(M ∗ ) ∼ = E8 . So as CE (X) ∼ = Z2 it + + + ∼ follows that CS + (X ) = CS + (M ) = E4 . Thus for u an involution in CS + (M + ), U + = u+G is an image of the permutation module for G/S0 on G/M in which CU + (X + ) = CU + (M + ). Using 6.2.14 and 6.2.17.3, it follows that U + is a natural module for G/S0 , and as M is the stabilizer in G of E0 , U + is dual to E. Then (1) follows. By 6.2.10.5, O(z) ∼ = E4 , so O0 (z) is non= 21+8 and by 6.2.9.4, AutO(z) (E) ∼ abelian. Thus S0 is nonabelian. But by (1), Φ(S0 ) ≤ E, so Φ(S0 ) = [S0 , S0 ] = E by 6.2.6.1, establishing (2).  Lemma 6.2.19. If μ ∼ = L3 (2)/23+6 . = Weyl(D4 ) then G ∼ Proof. Let G∗ = G/S0 , G+ = G/E, and U ∈ Δ(t). Then U + K + is a complement to S0+ in S + , so by Gaschutz’ Theorem (cf. 10.4 in [Asc86]) there exists a complement H0+ to S0+ in G+ . By 6.2.18.1, S0+ = S1+ ⊕ S2+ with S1+ ∼ = S2+ + + ! dual to E as H0 -modules. Let Si be the preimage of Si in G and set G = G/S1 . Claim each involution in G+ − S0+ is in a complement to S0+ in G+ . First for i+ an involution in H0+ , i! S0! contains two S0! -classes of involutions. Moreover H 1 (G∗ , S0! ) ∼ = Z2 with S ∗ regular on the involutions in V −S0! for V a 4-dimensional indecomposable F2 G∗ -module with [V, G∗ ] = S0! . Thus V fuses the two classes and each involution in i! S0! is contained in a complement. Then a similar argument in H0+ S1+ completes the proof of the claim. Let k ∈ K be of order 4; by the claim we may take k ∈ H0 . Then as K = [k, X] for suitable X ∈ Syl3 (CH0 (z)), K ≤ H0 . Further as H0 is transitive on the seven Sylow 2-subgroups of G, S ∩ H0 ∈ Syl2 (H0 ). For i = 1, 2, set Gi = H0 Si and Ωi = K Gi ∩ S. Then Ti = S ∩ Gi ∈ Syl2 (Gi ) and we set Fi = FTi (Gi ). By 2.5.2, τi = (Fi , Ωi ) is a quaternion fusion packet. Claim τi satisfies Hypothesis 6.2.1. By construction, τi satisfies 6.2.1.1. As Gi = K Gi , 6.2.1.2 is satisfied. Condition 6.2.1.3 is inherited from τ . By 6.2.18.2, either 6.2.1.4 holds or Si = E ⊕ F where F is dual to E as an H0∗ -module, and we may assume the latter. But then by 3.3.2.4, F ≤ Z(Gi ), a contradiction. This establishes the second claim. By the second claim and our treatment of earlier cases, Gi ∼ = L2 (3)/E64 . In particular there is a complement L to S0 in G. Also Si = Γi , where Γi = viL is of order 7 and Mi = M ∩ Gi centralizes vi . Then as L is 2-transitive on Γi , the isomorphism type of G is determined by e = [v1 , v2 ] and f = [v1 , w] for w ∈ Γ2 −{v2 } with [w+ , S + ] = v2+ . Now M centralizes vi+ for i = 1, 2, so M centralizes e. Then as CE (M ) = 1, e = 1. Now as CL (v1 ) is irreducible on S2+ /v2+ , it follows that S1 v2 = CS0 (v1 ), so f = 1. Next for g ∈ S, wg = wv2 a and v1g = v1 b for some a, b ∈ E. Therefore f g = [v1 , w]g = [v1g , wg ] = [v1 b, wv2 a] = [v1 , w] = f

146

6. THEOREMS 2 AND 4

as v1 centralizes v2 and E ≤ Z(S0 ). Then as f = 1 and CE (S) = z , we have f = z. Therefore G is determined up to isomorphism by an earlier remark, completing the proof of the lemma.  We are now in a position to complete the proof of Theorem 6.2.8. By 6.2.10.4 and 6.2.11, μ ∼ = L3 (2)/E8 . In = S3 , S4 , or Weyl(D4 ), and in the first case, G ∼ the remaining two cases, G is L3 (2)/E64 or L3 (2)/23+6 by 6.2.16 and 6.2.19. This completes the proof of Theorem 6.2.8. 6.3. The case E ≤ NG (K) In this section we assume the following hypothesis: Hypothesis 6.3.1. Hypothesis 6.2.1 holds, and setting E = Z , E does not normalize K. In addition we adopt the notation of section 6.2. In particular set z = z(K) and, appealing to 6.2.4.3, let G be a model for F. Set Ez = NE (K) and let η ∈ ηS (τ ). Set W = η and M = O(z)NG (W ) . Then μ = μη = M/CM (W ). Set G∗ = G/S0 .

Lemma 6.3.2. (1) |E : Ez | = 2, Ez = NE (J) for each J ∈ Ω(z), and E/Ez is semiregular on Ω(z). (2) [K, E] ∼ = Q28 . = E8 and K[K, E] ∼ E ˜ is the set of quaternion subgroups of K[K, E]. (3) K = {K, K} (4) Z ∩ Ez = {z} ∪ D ∗ (z) and for each t ∈ Z ∩ Ez , |K : CK (t)| ≤ 2. (5) If t ∈ Z ∩ Ez then z ∈ Et . Proof. By Hypothesis 6.3.1, E = Ez , so |E : Ez | = 2 by 6.2.7.3. Part (4) follows from 6.2.7.4. As the relation D∗ is symmetric, (4) implies (5). By 6.2.6.1, G is irreducible on E, so there is t ∈ Z − Ez . By 6.2.7.3, [K, t] = E ∩ KK t ∼ = E8 . As [K, Ez ] ≤ z and E = Ez t since |E : Ez | = 2, it follows that [K, E] = E ∩ KK t ∼ = Q28 , and hence = E8 , and K E = {K, K t }. Then K[K, E] ∼ contains exactly two Q8 -subgroups. Thus (2) and (3) hold. By 6.2.7.4, t is semiregular on Ω(z). Let e ∈ E. Then e = t1 · · · tn with ti ∈ Z, / {z} ∪ D∗ (z)} is of even order. From and from (4), e ∈ Ez iff I(e, K) = {i : ti ∈ 6.2.7.4, I(e, K) = I(e, J) for each J ∈ Ω(z), so Ez = NE (J), completing the proof of (1).  Lemma 6.3.3. (1) A(z) = ∅. (2) Dη is a set of 3-transposition of μ. Proof. Part (1) and 4.2.10.3 imply (2), so it remains to prove (1). Suppose t ∈ ZΔ with t ∈ A(z) and let U ∈ Δ(t) and V = CK (t). By 3.2.2.1, |Ω(z)| = 2, ˜ by 6.3.2.3. By 4.2.6, t = z g for some g ∈ G and X = K, K ˜ g so Ω(z) = {K, K} 2 ∼ ∼ is the split extension of U × V = Z4 by Y = D12 with the involution f ∈ Z(Y ) ˜ ≤ X. But then by 6.3.2.2, E8 ∼ ˜ ≤ X, and inverting U V , and K = [K, E] ≤ K K [K, E] is contained in the elementary abelian normal subgroup E ∩ X of X. We

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147

conclude from the description of X above that E ∩ X = [K, E] = E0 f , where E0 = z, t . Let s ∈ Z −Et ; then 1 = Es ∩E0 , so without loss z ∈ Es . Now s acts on O(z), O(t) = X by 4.2.10.4; set M0 = Xs . Then F = [E, K]s = E ∩ M0  M0 and F = E0 × F0 , where F0 = CF (O(Y )) = f, e . As s and [E, K] act on K, so does F . Thus as F0 centralizes O(Y ) which is transitive on E0# , it follows that F0 acts on K g . But now F = E0 F0 acts on K g , so s ∈ F ≤ NG (K g ), contrary to the choice of s.  Lemma 6.3.4. (1) [G, S0 ] ≤ E. (2) S0 = CG (E). (3) Let {K1 , . . . , Kn } be a set of representatives for the equivalence classes ˜ J ∈ Ω. Then O(τ )∗ = K1∗ × · · · × Kn∗ with K ∗ ∼ {J, J}, i = E4 . (4) If t ∈ ZΔ − z ⊥ and U ∈ Δ(t), then K ∗ U ∗ ∼ = D8 . ∗ ∼ ∗ ˜ ∗ ≤ CG (z)∗ . ∗ (5) K = E4 is a TI-subgroup of G with NG (K ∗ ) ≤ NG (K K) ∗ (6) K acts faithfully as a group of transvections on Ez with center z . (7) For k ∈ K − z , [E, k] ∼ = E4 . (8) Suppose G∗ ∼ = L3 (2) or U3 (4), or G∗ ∼ = An for some n ≥ 5 and K ∗ is a ∗ 2 root 4-subgroup of G . Then G = O (G) and E = S0 . Proof. By 6.2.2.3, E ≤ Z(S0 ), so S0 acts on [K, E]. By parts (2) and (3) of ˜ Therefore each member of Ω−{K, K} ˜ centralizes [K, E], so S0 6.3.2, [K, E] ≤ K K. ˜ On the other hand by 6.2.1.3, K ∩ S0 = z , so K K ˜ ∩ S0 = [K, E]. acts on {K, K}. ˜ Therefore [K, S0 ] ≤ K K ∩ S0 = [K, E] ≤ E. Then (1) follows from 6.2.6.2. As G is a model for F, F ∗ (G) = S0 . Thus (2) follows from (1). By 6.2.1.3, K ∗ ∼ = E4 . Adopt the notation of (3) and let K = K1 and X = ˜ and hence also [E, K]. Thus X ∩ KS0 ≤ Ki : i > 1 . Then X centralizes K K ∗ ∗ CKS0 ([E, K]) = S0 , so X ∩ K = 1, completing the proof of (3). Assume the hypothesis of (4). Then by 4.2.5.2, KU/z, t ∼ = D8 , so as z, t ≤ E and K ∩ S0 = z , (4) follows. Suppose g ∈ G − NG (K ∗ ) with 1 = K ∗ ∩ K g∗ . By (3), K ∗ ∼ = E4 , so K ∗ ∩ K g∗ = g∗ g for some V of index 2 in K. Then, using (3), V ∈ Δ(z g ), contrary to (4). V ˜ ∗ , so NG∗ (K ∗ ) ≤ NG (K K) ˜ ∗, Thus K ∗ is a TI-subgroup of G∗ . By 6.3.2.2, K ∗ = K so (5) holds. By 6.2.3.1, [Ez , K] ≤ z , and by 6.3.2.2, K ∗ is faithful on Ez , so (6) and (7) hold. Assume the hypothesis of (8) and set G+ = G/E and L = O 2 (G). By (1), + S0 ≤ Z(G+ ), so as G∗ is simple in (8) we conclude that G+ is the central product of L+ with S0+ and L+ is quasisimple with Z(L+ ) a 2-group and L+ /Z(L+ ) ∼ = G∗ . ∗ ∗ ∗ ∗ ∼ 2 Moreover K ≤ X ≤ G with X = A4 , so it follows from (5) that K ≤ O (G) = L. Thus G = L by 6.2.6.2. As K + ∼ = E4 , it follows from the structure of the covering group of G∗ that Z(L+ ) = 1, so E = S0 , completing the proof of (8).  Lemma 6.3.5. If μ ∼ = AE5 . = E4 then G ∼ ˜ Proof. Assume μ ∼ Hence K ∗ is strongly closed = E4 ; then Δ(τ ) = {K, K}. ∗ ∗ ∗ ∼ in S with respect to G and by 6.3.4.5, K = E4 is a TI-subgroup of G∗ . As G = K G by 6.2.6.2, it follows from Theorem 1 in [Asc73] that G∗ ∼ = A5 or U3 (4). Thus S0 = E by 6.3.4.8. Suppose G∗ ∼ = Z15 . Set Y = O5 (X) = U3 (4). Then NG (S) = SX where X ∼ ˜ ∼ and T = [S, Y ]. As O = K K = Q28 is Y -invariant, Y centralizes O, so T centralizes

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O. But K ∗ = Z(T ∗ ), so for k ∈ K of order 4, k = te for some t ∈ T and e ∈ E. ˜ = CE (K) as KE = KE. ˜ But then k = te centralizes K, Then e = t−1 k ∈ CE (K) a contradiction. Therefore G∗ ∼ = A5 . Further NG (K ∗ ) = CG (z) is of index 5 in G, so |Z| = 5 by 6.2.2.1 and indeed E is an image of the 5-dimensional permutation module for G∗ . Then E ∼ = E16 is the core of that module by 6.2.6.1. As O splits over [K, E], G splits over E, so G is determined up to isomorphism, completing the proof.  Lemma 6.3.6. Assume g ∈ G with z g ∈ ZΔ − z ⊥ , and let U ∈ Δ(z g ) and ˜ X = Q, J and let V = K ∩ W , V˜ = K ˜ ∩ W, J ∈ K G with U ≤ J. Set Q = K K, EX = E ∩ X, and WX = W ∩ X. Then (1) WX = V V˜ U = CX (WX ) and X/WX ∼ = S4 . (2) O2 (X) = EX WX and S ∩ X = KEX WX . (3) S0 ∩ X = EX and either ˜ WX = V × U , and EX ∼ (i) z g ∈ Q, Ω(z) = {K, K}, = E16 , or (ii) z g ∈ E / Q, WX = V V˜ × U , and EX ∼ with E vzg ∼ = 32 = Z2 X ≥ Z(X) = v˜ ˜ for suitable generators v, v˜ of V and V . (4) For Y ∈ Syl3 (X), CWX (Y ) = Z(X). (5) In each of the two cases in (3), X/EX ∼ = S4 , X is determined up to isomorphism, and X splits over EX . Proof. Set M + = M/CM (W ). By 6.3.3.2 and 4.1.5, X + ≤ M + = μ is ˜ and ηX = V X . Then τX is a quaternion isomorphic to S4 . Let τX = (X, {K, K}) fusion packet with ηX ∈ η(τX ), W0 = ηX  X, and μ(τX ) ∼ = S4 . By 4.3.8.4, W0 = CX (W0 ) so as WX centralizes W0 we conclude that W0 = WX . Then by parts (1) and (2) of 4.3.7, WX = V V˜ U and X/WX ∼ = S4 , so (1) holds. ˜ By 6.3.2.3, K ∈ Ω(z), so by 5.7.4.3, q = |Z(X)| = 1 or 2, with q = 1 iff z g ∈ Q. ˜ by 3.1.6, and as q = 1, |WX | = 16, so Further if z g ∈ Q then Ω(z) = {K, K} WX = V × U . In particular CWX (Y ) = 1, so (4) holds in this case. On the other hand if q = 2 then z g ∈ / Q, so V V˜ ∩ U = 1 and |WX | = 32, so WX = V V˜ × U . Here CWX (Y ) = Z(X) is of order 2 in this case, so again (4) holds. + Next Q+ ∼  X + , we = E4 with Q ∩ W = V V˜ , so [E, K]+ = 1 and hence as EX + + have EX = O2 (X ), so that (2) holds. Moreover EX ≤ S0 ∩X ≤ O2 (X) = EX WX , so S0 ∩ X = EX (S0 ∩ WX ), and then |(S0 ∩ X) : (S0 ∩ WX )| = |EX : E ∩ WX | = 4 and E0 = z, z g ≤ E ∩ WX . As S0 ∩ X = EX (S0 ∩ WX ) it follows from (4) that CS0 ∩X (Y ) ≤ EX Z(X). By 6.3.4.1, [X, S0 ∩ X] ≤ EX , so we conclude that S0 ∩ X ≤ EX Z(X). Suppose q = 1. Then Z(X) = 1 so S0 ∩ X = EX . Also Φ(E) = 1 and E0 = Ω1 (WX ), so EX ∼ = E16 and hence (3i) holds. Observe also that X/EX ∼ = S4 . v ∈ [E, K] ≤ E and z g are in E. So assume q = 2. Then u = v˜ v z g ∈ E as v˜ Thus E ∩ WX contains E0 u ∼ = E32 as u = Z(X) = E8 , and hence S0 ∩ X = EX ∼ by 5.7.4.3. This completes the proof of (3). Observe that again X/EX ∼ = S4 . By 5.7.4.5, in each of the two cases in (3), X is determined up to isomorphism with m2 (X) = m2 (EX ) and X splits over each member of A(O2 (X)) by 5.7.4.6, so (5) holds.  Lemma 6.3.7. If μ ∼ = S4 then G ∼ = AE6 or AE7 . Proof. Assume μ ∼ = S4 . Then M is the group X of 6.3.6, with Q = O(τ ) = ˜ and T = Δ = QW . By 6.3.6.2, SM = S ∩ M = KW EM ; by 6.3.6.3, KK

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∼ M/(M ∩ S0 ) = M/EM ∼ S0 ∩ M = EM ; and then by 6.3.6.5, M ∗ = M S0 /S0 = = S4 , ∗ = (KW EM )∗ = K ∗ U ∗ = T ∗ for z g ∈ ZΔ − {z} and U ∈ Δ(z g ). with D8 ∼ = SM Then as M ∗ ∼ = S4 , each subgroup of T ∗ of order 2 is of the form V1∗ for some V1 of index 2 in some member of K M , so as T = OW , T ∗ is the weak closure of its subgroups of order 2 in S ∗ . Claim T ∗ is strongly closed in S ∗ with respect to G∗ . As T ∗ is the weak closure of its involutions, if the claim is false then the Z4 -subgroup B ∗ of T ∗ is not weakly closed in S ∗ , so by Alperin’s Fusion Theorem there is R∗ ∈ FS ∗ (G∗ )f rc containing B ∗ with B ∗ not normal in N ∗ = NG∗ (R∗ ). As T ∗ is weakly closed in S ∗ and B ∗ char T ∗ , T ∗ ≤ R∗ , so B ∗ = T ∗ ∩ R∗ . Thus as T ∗ is the weak closure of its involutions, Z(T ∗ ) = Z ∗ is strongly closed in R∗ , so Z ∗  N ∗ . But [R∗ , K ∗ ] ≤ Z ∗ so K ∗ ≤ R∗ , a contradiction. This completes the proof of the claim. By 6.2.6.2, G = K G , so by Corollary B3 in [Gol75], S ∗ = T ∗ ∼ = D8 . As S0 = O2 (G), we have O2 (G∗ ) = 1. Therefore, appealing to the classification of groups with D8 -Sylow groups, G∗ is A7 or L2 (q) for some odd q. In the latter case as K ∗ is a TI-subgroup of G∗ , q = 7 or 9. However if q = 7 then CG (z)∗ = NG∗ (K ∗ ) is of index 7 in G∗ , so G is 2-transitive on Z of order 7. But then Z = {z}∪z gCG (z) ⊆ Ez , a contradiction. Therefore G∗ ∼ = A6 or A7 , and up to conjugacy in Aut(G∗ ), K ∗ is ∗ a root 4-subgroup of G . Then by 6.3.4.8, S0 = E. Next let X ∗ be the set of A5 -subgroups of G∗ containing K ∗ and X consist of the subgroups X = Q, Qx for some x ∈ X with X ∗ ∈ X . Then X is 2-transitive on ZX = z X of order 5, so setting EX = ZX , EX is an image of the permutation module for X ∗ on ZX . Thus m(EX ) = 4 or 5. As X = QX and Q ≤ O 2 (NX (Q)), X = O 2 (X), so m(EX ) = 4. Let SX = S ∩ X, FX = FSX (X), and ΩX = K X ∩ S. Then τX = (FX , ΩX ) is a quaternion fusion packet by 2.5.2. As O2 (X ∗ ) = 1, O2 (X) = X ∩ E and as EX is a projective X ∗ -module, E ∩ X = EX ⊕ F for some X-submodule F of E ∩ X. By 3.3.2.4, X centralizes F , so as X = O 2 (X) and K∗ ∼ = E4 , F = 1. Thus τX satisfies Hypothesis 6.2.1. Observe [E, K] = EX ∩ Ez , so τX satisfies Hypothesis 6.3.1. Therefore X ∼ = AE5 by 6.3.5. Next let X1 ∈ X − {X} and H = X, X1 . Then H ∗ ∼ = A6 . Set SH = S ∩ H, ZH = z H , and EH = ZH . Then |ZH | = |H : CH (z)| = 15, and by G.5.3 in [AS04] either m(EH ) = 4 or EH is the core of the permutation module for H ∗ on H ∗ /X ∗ . As in the previous paragraph, O2 (H) = E ∩ H. As [E, K] ≤ EH , [E ∩ H, H] = EH . Then as H = O 2 (H) and K ∗ ∼ = E4 , we conclude that EH = O2 (H). Next M ≤ H and by 6.3.6.5, M is determined up to isomorphism with |M |2 = |H|2 , so SH ∈ Syl2 (M ). Then by 6.3.6.5, SH splits over EH , so H splits ˇ 6. over EH . Therefore H is determined up to isomorphism, so H ∼ = AE6 or AE ∗ ∼ In particular if G = A6 then G = H and the lemma holds, so we may assume G∗ ∼ = A7 . Next as K ∗ is a root 4-subgroup of G∗ , |Z| = |G : CG (z)| = |G∗ : NG∗ (K ∗ )| = 35. Therefore m(E) ≥ 6. But G = H, X2 for X2 ∈ X with X2∗ ≤ H ∗ , so E = [E, G] = [E, H][E, X2 ] with m([E, X2 ]) = 4 and [E, K] ≤ [E, X2 ] ∩ EH . Thus m(E) ≤ m(EH )+1, so m(EH ) = 5 and m(E) = 6. Let e = Z(H). Then E = eG is an image of the permutation module for G on G/H, so as m(E) = 6, E is the core of that module. As H splits over EH and H ∗ contains a Sylow 2-subgroup of G∗ , S splits over E, so G splits over E. Therefore G is determined up to isomorphism,  so G ∼ = AE7 , completing the proof of the lemma. Lemma 6.3.8. Assume Ω = Ω(z) is of order 4. Then μ is nonabelian.

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∼ Proof. Assume μ is abelian and set O = O(z). By 6.3.4.3, O ∗ = K1∗ × K2∗ = ˜ E16 where K = K1 and K2 ∈ Ω − {K, K}. As μ is abelian, Δ(τ ) = Ω(z), so Σ = K ∗ ∪ K2∗ is strongly closed in S ∗ with respect to G∗ . Claim O ∗ is strongly closed in S ∗ with respect to G∗ . Assume otherwise, so that there is x ∈ O and / Σ. g ∈ G with y ∗ = xg∗ ∈ S ∗ − O ∗ . Thus x∗ ∈ Suppose first that y ∗ acts on K ∗ . Then y ∗ centralizes k∗ for some k ∈ K of −1 order 4, so x∗ centralizes kg ∗ . Then as O ∗ is abelian and weakly closed in S ∗ , −1 −1 there is h ∈ CG (x∗ ) with j ∗ = kg h∗ ∈ S ∗ , so j ∗ ∈ Σ and hence kg h ∈ J ∈ Ω(z) −1 as Δ(τ ) = Ω(z). Hence g −1 h ∈ CG (z), so as O  CG (z), xg∗ = x∗h g ∈ O ∗ , ∗ contrary to the choice of y . Thus K ∗y = K2∗ , so B0∗ = CO∗ (y ∗ ) ∼ = E4 . Set B ∗ = B0∗ y ∗ ; then [B ∗ , O ∗ ] = ∗ ∗ B0 , so AutO (B ) is the group of transvections on B ∗ with axis B0∗ . Next, arguing −1 as above, there exists h ∈ CG (x∗ ) with D∗ = B g h∗ ≤ S ∗ . If D∗ ≤ O ∗ then D∗ ∩ K ∗ = 1, contradicting Σ strongly closed. Further by the previous paragraph, −1 D0∗ = ND∗ (K ∗ ) ≤ O ∗ , so there exists d ∈ D with K d∗ = K2∗ . Let θ = O g h and −1 Y = θ, O . Then Autθ (D∗ ) is the group of transvections with axis B0g h∗ and −1 AutO (D∗ ) is the group of transvections with axis D0∗ , so as x∗ ∈ D0∗ − B g h∗ , it −1 follows that AutY (D∗ ) is the stabilizer in GL(D∗ ) of the point D0∗ ∩ B g h∗ . In −1 particular there is k ∈ K of order 4 with kg h ∈ NY (O), contradicting Δ(τ ) = Ω(z). This completes the proof of the claim. By 6.2.6.2, G = K G , so by the claim and Goldschmidt’s Theorem [Gol74],  O ∗ ∈ Syl2 (G∗ ). This is a contradiction as K ∗ and K2∗ are conjugate in G∗ . Theorem 6.3.9. G ∼ = AEn for some 5 ≤ n. During the remainder of the section assume G is a minimal counter example to Theorem 6.3.9. Lemma 6.3.10. Assume Ki ∈ K G for i = 1, 2 with Y = K1Y , K2Y < G. Let SY ∈ Syl2 (Y ) and set ΩY = (K1Y ∪ K2Y ) ∩ SY , FY = FSY (Y ), and τY = (FY , ΩY ). / Ez . Then Take K = K1 ≤ SY , set zi = z(Ki ), and assume z2 ∈ (1) τY is a quaternion fusion packet. ˇ n for some n ≥ 5. (2) K2 ∈ K Y and Y ∼ = AEn or AE Proof. Conjugating in G, we may assume SY ≤ S. Then (1) follows from 2.5.2. Claim Ki ∩ O2 (Y ) = zi . For if |K2 ∩ O2 (Y )| > 2 then K2 ∩ O2 (Y ) ≤ NG (K), contrary to the choice of z2 . This establishes the claim. Set Y1 = K Y and E1 = z Y1 . Assume K Y = K Y1 and let T be maximal subject to z ∈ / T ≤ S and T  Y1 . Set Y1+ = Y1 /T , S1 = S ∩ Y1 and let + + F1 = FS + (Y1 ), Ω1 = K Y1 ∩ S, and τ1+ = (F1+ , Ω+ 1 ). Then by 3.3.2, T ≤ Z(Y1 ) 1 + and τ1 is a quaternion fusion packet. By construction and the claim, τ1+ satisfies Hypothesis 6.2.1, so as Y = G and G is a minimal counter example to Theorem 6.3.9, it follows that either E1 ≤ NG (K) and Y1+ ∼ = AEn for some n ≥ 5, or E1 ≤ NG (K) and from Theorem 6.2.8, Y1+ ∼ = L3 (2)/E8 , L3 (2)/E64 , or L3 (2)/23+6 . ˇ n for some n ≥ 5 Then as T ≤ Z(Y1 ), it follows from 5.1.7 that Y1 ∼ = AEn or AE when E1 ≤ NG (K).

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In particular if K2 ∈ K Y then Y1 = Y and K Y1 = K Y , and as K2 ∈ K Y , / KY . E1 ≤ NG (K), so that (2) holds. Thus we may assume K2 ∈ Suppose z2 ∈ / E1 . Then by 3.3.2.1, K2 centralizes E1 , so Y0 = K2Y centralizes E1 . But then Y0 acts on O(z) and Ω2 = K2Y ∩ S centralizes O(z), so Y0 = ΩY2 0 / Ez . centralizes O(z), contradicting z2 ∈ Thus z2 ∈ E1 . Then as z2 does not act on K, E1 ≤ NG (K), so Y1 ∼ = AEn or ˇ n . Next Ω2 = Ω ∩ Y − Ω1 centralizes Σ = K M ∩Y by 6.3.3 and 4.2.10.2, so for AE J ∈ Ω2 , j = z(J) ∈ CE1 (Σ). But if n = 5 then z = CE1 (K), so as j = z, n > 5. Then by 5.1.9.4 the graph D∗ in that lemma is connected, so j ∈ CE1 (Y1 ), and we get a contradiction as above. This completes the proof in the case K Y = K Y1 . Finally we treat the case where K Y = K Y1 . Consider K2 Y1 . There is y ∈ Y1 with J = K2y ≤ S, so J ∈ Ω acts on K. Now K2 Y1 = JY1 and K K2 Y1 = K JY1 = K Y1 . If Y = K2 Y1 then K Y = K Y1 , contrary to assumption. Therefore K2 Y1 < Y . Set Y2 = K Y1 . If Y2 = Y1 then by induction on |Y |, K2 ∈ K Y1 , a contradiction. Proceeding in this manner, we arrive at Yr subnormal in Y with Yr K2 -invariant and Yr = K Yr . But now by induction, K2 ∈ K Yr , a contradiction. 

Lemma 6.3.11. m(E) > 6. Proof. Assume m(E) ≤ 6. Suppose first that |Ω| = 2. By 6.3.2.3, Ω = ˜ = Ω(z), and by 6.3.5, μ is not E4 , so D(Dη ) is disconnected. Then by {K, K} 4.2.13 and 6.3.3, μ ∼ = S4 . But then 6.3.7 contradict the choice of G as a counter example. ˜ set z3 = z(K3 ) and E0 = [E, K][E, K3 ]. Thus there exists K3 ∈ Ω − {K, K}; ˜ 3 . Then as z = Then E0 ≤ Ez and K centralizes [E, K3 ] as [E, K3 ] ≤ K3 K C[E,K] (K), E1 = [E, K] ∩ [E, K3 ] ≤ z . Therefore m(E0 ) ≥ 5 with z = E1 in case of equality. As m(E) ≤ 6 and |E : Ez | = 2 by 6.3.2.1, we conclude that E0 = Ez is of rank 5, m(E) = 6, and E1 = z . By symmetry, E1 = z3 , so z3 = z and Ω = Ω(z). ˜ and K4 = K ˜ 3 . If K5 ∈ Ω − {K1 , . . . , K4 } then K5 Set K1 = K, K2 = K, centralizes E0 = Ez , a contradiction. Thus Ω = {K1 , . . . , K4 }. By 6.3.8, μ is nonabelian, so μ ∼ = Weyl(D4 ) by 4.2.13 and 4.3.5.1. ˜ and K s = K4 , Let s = z h ∈ Z − Ez and set O = O(z). Then O ∼ = Q48 , K s = K, 3 O ∼ so [O, s] = E32 . Hence as m(E) = 6, E = [O, s]s and |s | = |O : CO (s)| = 16 with zs ∈ / sO . Therefore E − Ez = sO ∪ (sz)O . By 6.2.6, G is irreducible on E, so each member of Ez is fused into E − Ez , and hence G has at most two orbits on E # with representatives s and sz. ˜ Y = K CG (e) , Se = S ∩ Y , Ee = z Y , Let e ∈ [E, K3 ] − z , Ωe = {K, K}, and Fe = FSe (Y ). We may assume Se is Sylow in Y . By 2.5.2, τe = (Fe , Ωe ) is a quaternion fusion packet. By 5.8.7.4, μ(τe ) ∼ = S4 , and setting Me = K CM (e) , e ∈ Me . Choose e ∈ Es and set J = K h . Then |CJ (e)| > 2 and by Sylow there is y ∈ CG (e) with CJ (e)y ≤ S. Then CJ (e)y is contained in a member of Δ, while as Ω = Ω(z), F = z(V ) : V ∈ Δ ∼ = E4 and Me is transitive on F # from 5.8.7.3.ii, so y CG (e) ˇ n for some s ∈z and hence we may take h ∈ Y . Therefore Y ∼ = AEn or AE n ≥ 5 by 6.3.10.2. Then as μ(τe ) ∼ = S4 , n = 6 or 7, and as e ∈ Me , also e ∈ Z(Y ), ˇ 6. so it follows that Y ∼ = AE

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If Y centralizes f ∈ E − Ee then f and e are fused in G, since by an earlier observation, G has two orbits on E # and neither e nor f is in Z since the centralizer of neither involution has a normal subgroup isomorphic to O(z). This is a contradiction as e ∈ Y but f ∈ / Y. Therefore Y is indecomposable on E, so by 5.1.2.3, E is the 6-dimensional permutation module for Y /Ee ∼ = A6 . In particular Y has orbits Γi of length 6, 15, 20, 15, 6, 1 on E # , where Γi is the set of vectors in E of weight i, 1 ≤ i ≤ 6. We’ve seen that G has two orbits on E # with representatives e and z, and that E − Ez = E1 ∪ E2 , where E1 = Z ∩ (E − Ez ) and E2 = eG ∩ (E − Ez ) are of order ˜ = [K K, ˜ X]. As X acts on K3 K4 and 16. Next there is X ∈ Syl3 (CY (z)) with K K ˜ with centralizes e, [X, K3 K4 ] = 1. Similarly there is X1 of order 3 centralizing K K # [K3 K4 , X1 ] = K3 K4 . Then XX1 O has orbits of length 1, 6, 6, 18 on Ez with the orbits [E, K] − z and [E, K3 ] − z of length 6 in eG . We conclude that |Z| = 1 + 18 + 16 = 35 and |eG | = 6 + 6 + 16 = 28. Then it follows that Z = Γ3 ∪ Γ4 and eG = Γ1 ∪ Γ2 ∪ Γ5 ∪ Γ6 . In particular, writing E additively, for f ∈ eG − {e}, f ∈ Ee iff f + e ∈ Z. Define a form ϕ : E × E → F2 by ϕ(x, y) = 0 if y ∈ Ex and ϕ(x, y) = 1 otherwise. The form is symmetric on Z × Z by 6.3.2.5. Similarly z ∈ Ee and e ∈ Ez , so the form is symmetric on (Z × eG ) ∪ (eG × Z). If f ∈ eG − {e}, we saw that f ∈ Ee iff f + e ∈ Z, so ϕ is symmetric on eG × eG . Therefore ϕ is symmetric. Then as |E : Ex | = 2 for each x ∈ E # , ϕ is bilinear. As G is irreducible on E, ϕ is nondegenerate, so ϕ is a symplectic form. Define q : E # → F2 by q(x) = 0 if x ∈ Z and q(x) = 1 otherwise. We check that: (*) for x, y ∈ E, q(x + y) = q(x) + q(y) + ϕ(x, y), so that q is a quadratic form on E. For example x + z ∈ xO for x ∈ Ez − z , so q(x + z) = q(x). Also z + s ∈ eG so q(z + s) = 1. Finally f ∈ Ee iff f + e ∈ Z, so q(f + e) = 0 iff f ∈ Ee , completing the verification of (*). We’ve shown that G∗ ≤ O(E, q) ∼ = O6 (2). As |eG | = 28, 7 divides the order of ∗ ∗ ∗ G , so = +1 and |G | ≥ 28|Y | = |O6+ (2)|/4. Then as G = K G , G∗ ∼ = A8 ∼ = L4 (2). By 6.3.4.8, S0 = E. Let H be the preimage in G of a parabolic H ∗ of G∗ containing S ∗ and with ∗ H /O2 (H ∗ ) ∼ = L3 (2). Set FH = FS (H) and τH = (FH , Ω). By 2.5.2, τH is a quaternion fusion packet; further μ(τH ) ∼ = S4 and EH = z H ∼ = E8 . Indeed τH satisfies Hypothesis 6.2.1 with EH ≤ NG (K), so H ∼ = L3 (2)/E64 by 6.2.16. Therefore S is determined up to isomorphism as is the embedding of E = [O, s]s in S. Hence S splits over E, as this is the case in the overgroup AE8 of H. Thus G splits over E, and hence is determined up to isomorphism. But then G ∼ = AE8 , contrary to the choice of G as a counter example to Theorem 6.3.9.  Lemma 6.3.12. Let J ∈ K G such that z(J) ∈ / Ez and set Y = K, J . Then J ∈ K Y and Y ∼ = AE5 or AE7 . Proof. Set t = z(J), E0 = [E, K][E, J], and EY = z Y . Then m(E0 ) ≤ 2m([E, K]) = 6 and E0 = [E, Y ]  Y with z ∈ [E, K] ≤ E0 . Thus EY ≤ E0 , so m(EY ) ≤ 6. Therefore Y = G by 6.3.11, so by 6.3.10.2, J ∈ K Y and Y ∼ = AEn ˇ n for some n ≥ 5. Let Y + = Y /EY , so that Y + ∼ or AE = An and K + is a root / Ez , it follows from 5.1.8 that 4-subgroup of Y + . Then Y + = K + , J + and as t ∈  Y+ ∼ = A5 or A7 , so the lemma holds.

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∼ AE5 }. Thus for J ∈ θ(K), Notation 6.3.13. Set θ(K) = {J ∈ K G : K, J = z(J) ∈ / Ez and m([E, K][E, J]) = 4 with [E, K] ∩ [E, J] a complement to z(X) in [E, X] for X ∈ {K, J}. Lemma 6.3.14. (1) For t ∈ Z, t ∈ D(z) iff [O(z), O(t)] = 1. (2) For t ∈ Z, t ∈ D∗ (z) iff t ∈ Ez − z . (3) D∗ is connected. (4) G = CG (z), NG (O), M . Proof. By 3.3.4, t ∈ D(z) iff [t, O(z)] = 1, and then as O(t)  CG (t), (1) follows. Similarly by 3.3.5, t ∈ D∗ (z) iff |CK (t)| > 2 iff t ∈ Ez − z , so (2) holds. Assume (3) fails and let Σ and Σ be distinct connected components of D∗ with z ∈ Σ. Let t = z g ∈ Σ and J = K g . By (2), t ∈ / Ez , so by 6.3.12, Y = K, J ∼ = AE5 or AE7 , and as Σ = Σ , the former holds by 5.1.9.4. Thus J ∈ θ(K) for each t ∈ Σ and J ∈ K G with t = z(J). ˜ set Y  = Y, J  . As J  ∈ θ(K), Suppose |Ω| > 2. Then there is J  ∈ Ωg −{J, J};   ˇ 6 (E ∩ Y )[E, J ] is of rank 5 and normal in Y , so Y  = G by 6.3.11. Then Y  ∼ = AE ∗ by 6.3.10, so 5.1.9.4 contradicts D disconnected. Thus |Ω| = 2. Then by 6.3.3 and 4.2.13, μ ∼ = E4 or S4 , contrary to 6.3.5 and 6.3.7. This completes the proof of (3). As G is transitive on Z and D∗ is connected, G = CG (z), Γ , where {z g : g ∈ Γ} is a set of representatives for the orbits of CG (z) on D∗ (z). Next for t ∈ D∗ (z), t is conjugate under CG (z) to a member of ZΔ . Now we appeal to 6.1.6.2. If Δ = Ω then (4) follows as NG (O) is transitive on ZS . On the other hand if Δ = Ω then M (NG (W ) ∩ NG (O)) is transitive on ZΔ by 6.1.6.2, so again (4) follows.  Lemma 6.3.15. Suppose D is disconnected. Then (1) μ ∼ = Weyl(D4 ) and (2) Ω = Ω(z). Proof. By 4.2.13 and 6.3.3, μ is S4 or Weyl(D4 ), so (1) follows from 6.3.7. Suppose (2) fails and let K3 ∈ Ω − Ω(z) and set z3 = z(K3 ). By 5.8.7.2, e = zz3 ∈ Z(M ). Thus if O = O(τ )  CG (z) then e is centralized by CG (z), NG (O), M , contrary to 6.3.14.4. Therefore O is not normal in CG (z), so X = O(z3 )CG (z) = O(z3 ). But X ≤ CG (O(z)), so τX = (FX , Ω(z3 )) is a quaternion fusion packet by 2.5.2, with μ(τX ) ∼ = E4 , where FX = FS∩X (X). As μ(τX ) ∼ = E4 it follows from 3.3.12.3 that K3 ∩ O2 (X) = z3 . Let T ≤ S be maximal subject to T  X + + and z3 ∈ / T , and set X + = X/T and τX = (FX /T, Ω+ ). By 3.3.2, τX satisfies + Hypothesis 6.2.1. Then by 6.2.8, τX satisfies Hypothesis 6.3.1, so it follow from 6.3.5 that X ∼ = AE5 . Let Σ be the connected component of D containing z, H = NG (Σ), and FΣ = FS (H). By 2.5.2, τΣ = (FΣ , Ω) is a quaternion fusion packet. Observe that μ(τΣ ) ∼ = E16 by 6.3.14.4. Let E = [K3 ]FΣ , Γ = K3E , ρ = (E, Γ), T0 maximal subject to / T0 , and E + = E/T0 . By 3.3.2, ρ+ = (E + , Γ+ ) is a quaternion T0  E with z3 ∈ fusion packet satisfying Hypothesis 6.3.1, so by minimality of G, X  K H . It follows that K H = X × X g for g ∈ H with z3g = z. Let f ∈ [E, K3 ] − z3 and Y = K CG (f ) . Set SY = S ∩ Y , FY = FSY (Y ), and τY = (FY , Ω(z)). By 6.3.10, τY is a quaternion fusion packet, and by 5.8.7.4, ˇ 6 . Let b ∈ CX (f ) be μ(τY ) ∼ = S4 , so by 6.3.10, Y ∼ = AEn for n = 6 or 7, or AE of order 3. Then b acts on Y and centralizes X g , so we conclude from 5.1.10 that

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b centralizes Y . Therefore F = f [E ∩ X, b] centralizes Y , a contradiction as F contains a member of Z. This completes the proof of (2).  Lemma 6.3.16. D is connected. ∼ Weyl(D4 ) and O = O(z). Let Proof. Assume otherwise. By 6.3.15, μ = J ∈ θ(K) and Y = O, J . As in the proof of 6.3.12, [E, Y ] = [E, O][E, J] is of rank 6, so by 6.3.10 and 6.3.11, Y ∼ = AE8 . From 5.1.11.2, ZΔ ⊆ O. Further if t = z g ∈ D∗ (z) then |CK g (z)| > 2, so for h ∈ CG (z) with CK g (z)h ≤ S, th ∈ ZΔ . Hence as O  CG (z), D∗ (z) ⊆ O. Indeed from the proof of 6.3.11, CY (z) is transitive on Z ∩ O − {z}, so CY (z) is transitive on D∗ (z). But then as D∗ is connected by 6.3.14, z Y = Z. Thus E = E ∩ Y is of rank 6, contrary to 6.3.11.  Lemma 6.3.17. Either ˜ or (1) Ω(z) = {K, K}, (2) |Ω(z)| = 4 and for J ∈ θ(K), O(z), J ∼ = AE8 . ˜ and let J ∈ θ(K) and Y = K, K3 , J Proof. Assume K3 ∈ Ω(z) − {K, K}, and EY = [E, K][E, K3 ][E, J]. Then m(EY ) = 6, so by 6.3.10 and 6.3.11, Y ∼ = AE8 . ˜ 3 , so by 3.1.6, Ω(z) = {K, K, ˜ 3K ˜ K3 , K ˜ 3 }, By 5.1.11.2, there is t ∈ ZΔ ∩ K KK completing the proof.  Lemma 6.3.18. Let z g ∈ Z − z ⊥ and set Y = O(z), O(z g ) . Then (1) O(z) and O(z)g are conjugate in Y . (2) K ≤ O2 (Y ). (3) CY (z) is transitive on Ω(z). Proof. Suppose first that t = z g ∈ D∗ (z). Then t is fused under CG (z) into ZΔ , so without loss, t ∈ ZΔ . Then by 6.3.3 and 4.2.5, (2) and (3) hold and Y ≤ M is transitive on z, t # , so that (1) holds. Thus we may assume that t ∈ / Ez , so by 6.3.12, Y0 = K, K g ∼ = AE5 or AE7 . In particular K ≤ O2 (Y0 ) and t ∈ z Y0 . Thus (1) and (2) hold in this case too. ˜ so if |Ω(z)| = 2 then (3) holds. On the other hand if |Ω(z)| > 2 Further K t = K, then (3) follows from 6.3.17.  Notation 6.3.19. Set D = O(z)G , and for P ∈ D let z(P ) be the involution generating Z(P ) and DP = CD (P ). For α a set of imprimitivity for G on D define Dα to be the set of conjugates of α centralizing α and Dα∗ = {z g : O(z g ) ∈ Dα }. ˜ y] = K K ˜ and Define Y(K) to consist of those elements y of order 3 with [K K, m([E, y]) = 2. Observe for each J ∈ θ(K) that K, J ∩ Y(K) = ∅. Define D(D) to be the commuting graph on D. Lemma 6.3.20. (1) The map t → O(t) is an isomorphism of D(τ ) with D(D). (2) For P, Q ∈ D, either [P, Q] = 1 or Q ∈ P P,Q . (3) D(D) is connected. (4) G is transitive on D. (5) There exists a unique maximal set of imprimitivity α for G on D containing O(z). (6) Dα = ∅ and Dα is transitive on Dα∗ by conjugation. (7) G = Dα , Dβ for each β ∈ Dα .

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Proof. Part (1) follows from 6.3.14.1, (2) follows from (1) and 6.3.18.1, and (3) follows from (1) and 6.3.16. By 6.2.1.1, F is transitive on Ω, so G is transitive on Z, and hence (4) holds. Finally (5)-(7) follow from (2)-(4) and 7.8 in [Asc97].  Notation 6.3.21. Choose α as in 6.3.20.5 and set X = Dα . Replacing S by a suitable conjugate, we may assume SX = S ∩ X ∈ Syl2 (X). Set ΩX = Ω ∩ X, FX = FSX (X), and τX = (FX , ΩX ). Set EX = Dα∗ . Let z g ∈ Dα∗ . Lemma 6.3.22. (1) τX is a quaternion fusion packet. ˇ n for some n ≥ 5. (2) X ∼ = AEn or AE Proof. Part (1) follows from 2.5.2. If Dα∗ = {z g } then α = {O(z)} and {O(z), O(z g )} is a set of imprimitivity for G on D, contrary to 6.3.20.5. Hence |Dα∗ | > 1. Then as X is transitive on Dα∗ by 6.3.20.6, it follows that there exists / z g⊥ . In particular K g ≤ O2 (X) and CX (z g ) is transitive on Ω(z g ) x ∈ X with z x ∈ by 6.3.18. Therefore FX is transitive on ΩX . Suppose K g ∩ O2 (X) = z g . Then by 3.3.12, WX = ηX  X for some ηX ∈ η(τX ). Now ηX ⊆ η ∈ η(τ ), so for J ∈ K gX , J ≤ NG (W ). Thus X = K gX ≤ NG (W ). As W ≤ CG (z g ) ≤ NG (X g ), by symmetry, X g ≤ NG (W ), so by 6.3.20.7, G = X, X g ≤ NG (W ), contrary to 6.2.1.3. Therefore K g ∩ O2 (X) = z . / T and T  X. Set X + = X/T , Let T ≤ SX be maximal subject to z g ∈ + + + + + FX = FS + (X ), and τX = (FX , ΩX ). By 3.3.2, τX is a quaternion fusion packet X

+ satisfies conditions (1) and (3) of Hypothesis yy.1. and T ≤ Z(X). We saw that τX By the choice of T , 6.2.1.4 is satisfied, and as X is transitive on Dα∗ , 6.2.1.5 holds. + As X = K gX , 6.2.1.2 holds. Thus τX satisfies Hypothesis 6.2.1. g Suppose EX ≤ NG (K ). Then by Theorem 6.2.8, X + ∼ = L3 (2)/E8 , L3 (2)/E64 , ˜ = [K K, ˜ y]. Now y ∈ or L3 (2)/23+6 . Hence there is y of order 3 in CX g (z) with K K g CG (O(z )) acts on X, so by 5.10.8, y centralizes X. Then P = [O2 (CX g (z)), y] ≤ CG (X). But by 5.10.9.2, X g = K h : z h ∈ P, h ∈ X g , and so as X ≤ CG (z h ) ≤ NG (O(z h )), X centralizes K h , and hence X centralizes X g . Thus X  X, X g = G by 6.3.20.7, a contradiction.  Therefore EX ≤ NG (K g ), so (2) follows from 6.3.10.

Lemma 6.3.23. (1) α = {O(z)}. ˜ (2) Ω(z) = {K, K}. ˜ (3) For y ∈ Y(K) ∩ X g , K Ky  CX g (z) and y centralizes X. Proof. From a remark in 6.3.19, there exists y ∈ Y(K)∩X g , and from 5.1.12.1, ˜ either K Ky  CX g (z) or X ∼ = AE8 . In any event y ∈ CG (z) ≤ NG (X). As [E, y] = [E, K, y], y centralizes EX , so [X, y] ≤ CX (EX ) = EX , and then (3) holds unless X ∼ = AE8 . ˇ 8, From 6.3.22 and 5.1.12.2, CX g (z) is a maximal subgroup of X g unless X ∼ = AE where NG (ΩX g ) is maximal. In the first case O(z) is a maximal set of imprimitivity for X g on D ∩ X g , so that (1) holds in this case as α ⊆ Dαg . In the second case ˜ K3 K ˜ 3 } for K3 ∈ ΩX g − {K, K}. ˜ Similarly, working in (1) holds unless α = {K K, X g , either |Ω(z)| = 2 or X ∼ = AE8 and |Ω(z)| = 4, and in the first case (2) holds. Thus the lemma holds unless n = 8, which we now assume.  ˜ 3 for K3 ∈ ΩX g − {K, K}. ˜ Set Now L = O 2 (CX g (y)) ∼ = AE5 contains K3 K B = L, X . Then B ≤ CG (y) and L does not centralize X by 7.8.5 in [Asc97], as ˇ k for some k > 8. α ≤ ΩX g and α is maximal. Thus by 6.3.10, B ∼ = AEk or AE

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˜ g } = Ω(z g ) ∩ B, so it follows that As k > 8, CB (z g ) is maximal in B with {K g , K ˇ 8 , so (3) also holds. (1) and (2) hold. Also X ∼  = AE Lemma 6.3.24. For y ∈ Y(K), O(z)y = Y(K)  CG (z). Proof. Choose y as in 6.3.23.3 and set B = O(z)y . By 6.3.23.2, X is not AE8 . Then by 5.1.12.1, B char CX g (z)  CG (z, z g ). Further X  CG (z) and X is transitive on D(z), so by a Frattini argument, CG (z) = XCG (z, z g ). Therefore B  CG (z). Now let x ∈ Y(K). Conjugating in O(z), we may take x ∈ yCG (B); in particular x centralizes y so F = x, y is a 3-group. As x, y ∈ Y(K), [E, x] = [E, y] ≤ [E, K], so xy −1 centralizes [E, y] and E/[E, y], so as F is a 3-group,  xy −1 ∈ CG (E) = S0 by 6.3.4.2. Hence x = y ∈ B, completing the proof. Lemma 6.3.25. Let e ∈ [E, K g ] − z g and Ye = K CG (e) . Then ˇ n+2 . (1) Ye ∼ = AEn+2 or AE (2) Let x ∈ CK g (e) be of order 4. Then x acts on Ye and induces the transposition on Ye∗ centralizing X g∗ . (3) Let y ∈ Y(K g ) with e ∈ [E, y], and set Yy = K CG (y) . Then X g ≤ Yy ≤ Ye ˇ n+1 . with Yy ∼ = AEn+1 or AE Proof. Set B = CX (e) and Y = X gB . Observe that X g centralizes O(z g ), g and hence also e, so Y ≤ CG (e). By 6.3.22.2, X g = K X . As X centralizes O(z), g gb Y for each b ∈ B, O(z) ≤ X ∩ X , so Y = K . Hence by 6.3.10, Y ∼ = AEp or ˇ p for some p ≥ 5. AE Suppose n > 5 and let Γ consist of those O(t) ∈ Dα centralizing e and set ˇ n−2 when n > 6. L = Γ . Then L is isomorphic to Q28 when n = 6 and to AE In particular L is transitive on Γ. By construction, L acts on Y , so from 6.3.20.2, either Γ ⊆ O(z)Y or L centralizes Y . But in the latter case as X g ≤ Y it follows from maximality of α that Y = X g and Γ = {O(z g )}, a contradiction as e does not centralize O(z g ). Therefore Γ ⊆ O(z)Y . We conclude that Γ consists of those O(t) ∈ O(z)Y centralizing O(z), and then that p = n + 2. So take n = 5. Then Ω ∩ Y = Ω(z), so 5 ≤ p ≤ 7. For b ∈ B of order 3, z gb = z g so X gb = X g and hence b acts nontrivially on Y and centralizes O(z), so ˇ n+2 . p = 7. Thus in any event Y ∼ = AEn+2 or AE Next if J ∈ K CG (e) − K Y then z(J) ∈ / z ⊥ , so J is conjugate to K in Ye by ˇ k for some k ≥ 5. As 6.3.18. Thus Ye = K Ye . Then by 6.3.10, Ye ∼ = AEk or AE Y ≤ Ye with Γ ⊆ Y we conclude that Y = Ye , completing the proof of (1). Next x ∈ K g so x centralizes X g , and as x centralizes e, x acts on Ye . Thus g either (2) holds or x centralizes Ye∗ . But in the latter case x acts on J ∈ K Ye −K X , so z(J) ∈ D∗ (z g ) − (z g )⊥ . Thus [x, J ∗ ] = 1, contradicting [x, Ye∗ ] = 1. Hence (2) holds. Choose y as in (3). By 6.3.23.3, y centralizes X g . By 6.3.24, y ∈ X so y ∈ Yf for f ∈ [E, K] − z . Then by (1), Y0 = K gYf ∩ CG (y) is isomorphic to AEn−1 ˇ n−1 if n > 5 and isomorphic to Q28 if n = 5. Further Y0 centralizes [E, y], or AE / Ezg . Then as Y0 centralizes Y0 = K gYf ∩ CG (e) , and Y0 contains O(z h ) with z h ∈ [E, y] and e ∈ [E, y], we have z h ∈ z Ye by 6.3.18.1, so Y1 = K Ye ∩ CG (y) = X g ˇ n+1 . As Yy acts on [E, y] and is perfect, Yy or Ye , and hence Y1 ∼ = AEn+1 or AE centralizes [E, y], so Yy = Y1 , completing the proof of (3). 

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Lemma 6.3.26. The group presented by generators yi , 1 ≤ i ≤ n + 2, and the following set of relations, is the alternation group An+4 : (1) y13 = yi2 = 1 for 1 < i ≤ n + 2. (2) (yi yi+1 )3 = 1 for 1 ≤ i < n + 2. (3) (yi yj )2 = 1 for i ≥ 1 and j > i + 1. Proof. See for example Theorem 265 in [Dic58].



Define e and Ye as in 6.3.25, and let x = CK g (e). By 6.3.25 we can represent B = Ye x S0 on I = {1, . . . , n + 4} so that S0 is the kernel of the representation, Ye∗ = Alt(I)1,2 is the stabilizer in the alternation group on I of 1, 2, X g∗ = Alt(I)1,2,3,4 , and x∗ = (1, 2)(3, 4). Next choose y and Yy as in 6.3.25.3. Then X g ≤ Yy ≤ Ye with Yy∗ ∼ = An+1 generated by root 4-subgroups of Ye∗ , so we may choose Yy∗ = Alt(I)1,2,3 . As K ≤ X g , we can choose notation so that K ∗ is the root 4-subgroup moving {n + 1, . . . , n + 4}. Choose f ∈ [K, E] − z ; thus Yf∗ ∼ = An+2 by 6.3.25.1. Let x0 ∈ xYe with x∗0 = (1, 2)(n + 3, n + 4).. Then x∗0 acts on K ∗ , so x0 acts on K, and we choose / X g , x∗0 is nontrivial on K g∗ , so, using 6.3.24, f so that x0 centralizes f . As x0 ∈ x∗0 inverts two subgroups of K g∗ y ∗ of order 3, and we choose y so that x∗0 inverts Y y ∗ . For 3 ≤ i ≤ n + 2, let xi ∈ x0 y with x∗i = (1, 2)(i + 1, i + 2), and set yi = x∗i and ∗ ∗ ∗ y1 = y . As x0 inverts y , yi inverts y1 . Set y2 = x∗ . Then by direct computation: Lemma 6.3.27. yi , 1 < i ≤ n + 2, satisfy the relations in 6.3.26. Next we verify that the remaining relations in 6.3.26, involving y1 , are also satisfied. As |y| = 3 and y1 = y ∗ , y13 = 1. As y2 = x∗ ∈ K g∗ and K g∗ y ∗ ∼ = A4 , (y1 y2 )2 = 1. Recall that for i ≥ 3, yi inverts y1 , so (y1 yi )2 = 1. This completes the verification. Thus it follows from 6.3.26 that G∗0 = yi : 1 ≤ i ≤ n + 2 ∼ = An+4 . Next each yi is in G∗ , so G∗0 ≤ G. Further X g∗ ≤ Ye∗ ≤ G∗0 and X ∗ = K g∗ , (Ye ∩ X)∗ ≤ x∗ , y ∗ , (Ye ∩ X)∗ ≤ G∗0 , so by 6.3.20.7, G∗0 = G∗ . Thus we have shown: Lemma 6.3.28. G∗ ∼ = An+4 with K ∗ a root 4-subgroup of G∗ . Then by 6.3.28 and 6.3.4.8: Lemma 6.3.29. E = S0 and G = O 2 (G). Remark 6.3.30. By 6.3.28 and 6.3.29 we can represent G on I = {1, . . . , n + 4} with kernel E as the alternating group on I; that is G∗ = Alt(I). Moreover we can choose notation so that K ∗ , K g∗ are the root 4-subgroups moving {n+1, . . . , n+4} and {1, . . . , 4}, respectively, and X ∗ , X g∗ are G∗n+1,...,n+4 , G∗1,...,4 , respectively. Choose e, x, and Ye as in 6.3.25. From that lemma, we may choose Ye∗ = G∗1,2 to be the stabilizer of 1 and 2, and x∗ = (1, 2)(3, 4). Then Ye∗ x∗ is the global stabilizer of {1, 2} and centralizes e ∈ E. Choose y ∈ Y(K g ) as in 6.3.25.3; from that lemma we may choose y ∗ = (1, 3, 2) and ey ∼ = E4 . Lemma 6.3.31. E is the natural module for G∗ .

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Proof. By 6.2.6.1, G is irreducible on E. From 6.3.30, the global stabilizer in G∗ = Alt(I) of {1, 2} centralizes e ∈ E # and ey ∼ = E4 . Hence the lemma follows from 5.1.14.  ˇ n+3 . Lemma 6.3.32. Let Hn+4 = K gGn+4 . Then Hn+4 ∼ = AE ˇ n+3 , and the latter holds by 6.3.31. Proof. By 6.3.10, Hn+4 ∼ = AEn+3 or AE  Lemma 6.3.33. n is even. Proof. Assume n is odd and set H = Hn+4 . By 6.3.32, HE splits over E. But as n is odd, H ∗ contains a Sylow 2-subgroup S ∗ of G∗ , so S splits over E. Therefore by Gaschutz’ Theorem (cf. 10.4 in [Asc86]) G splits over E. Hence  G∼ = AEn+4 , contrary to the choice of G as a counter example. By 6.3.31, E is the natural module for G∗ , so E = V¯ = V /vI , where V is the core of the permutation module for G∗ on I, and for J ⊆ I of even order, vJ is the vector in V with support J. Then e = v¯1,2 and f = v¯n+3,n+4 ∈ [E, K] − z , so Yf is described in 6.3.25.1. Next from 6.3.32, there is a complement H to E in Hn+4 . Then CH (f ) is a complement to E in EYf . By 6.3.25.1, and as E is the natural module for G∗ , ˇ n+2 and E is the permutation module for Y ∗ on n + 2 points. Therefore Yf ∼ = AE f by 5.1.2.2, H 1 (CH (f ), E) = 0, so E is transitive on the complements to E in Yf E. Let t∗ = (1, 2)(n + 3, n + 4). Then CH (f )∗ = Yf∗ t∗ ∼ = Sn+2 , and as E is transitive on complements to E in Yf , we may choose t to be an involution acting on CH (f ), so that Hf = CH (f )t is a complement to E in CG (f ). Define y as in 6.3.30 and set Gy = NG (y ). Then, using 6.3.30 and 6.3.25.3, Gy = (y ×Yy )t , with t inverting y and t∗ inducing a transposition on Yy∗ ∼ = An+1 , and Yy ∼ = AEn+1 . Let H1 = NH (y ), so that H1 is a complement to E in the stabilizer Gy,n+4 of n + 4 in Gy . Next O2 (Yy ) is the permutation module for CH (y) ∼ = An on n points, so by 5.1.2.2, O2 (Yy ) is transitive on its complements in CH (y)O2 (Yy ). Thus H1 is contained in a complement H2 to O2 (Yy ) in Gy . By 5.1.2.2, O2 (Yy ) is regular on such complements, so H1 is contained in exactly |CO2 (Yy ) (H1 )| = 2 such complements. Similarly F = CO2 (Yy ) (CH (f )) ∼ = E4 is regular on the four complements to O2 (Yy ) in Gy containing CH (f ), and F t ∼ = D8 , so each such complement contains an F -conjugate of Hf = CH (f )t . Thus replacing Hf by a suitable F -conjugate, we may assume Hf ≤ H2 . Set y1 = y and for 1 < i < n + 2, let yi be the element of H with yi∗ = (1, 2)(i+1, i+2). Observe that by construction, {y1 , . . . , yn+1 } satisfies the relations in 6.3.26. Next set yn+2 = t. As yi ∈ CH (f ) for 1 ≤ i ≤ n and CH (f ) ≤ Hf , also yi ∈ Hf . But also t ∈ Hf , so the relations in 6.3.26 involving t and yi , 1 ≤ i ≤ n are satisfied. Finally t and yn+1 are contained in H2 , so (yn+1 t)3 = 1, verifying the last of the relations in 6.3.26. Thus we conclude from 6.3.26 that: Lemma 6.3.34. B = yi : 1 ≤ i ≤ n + 2 ∼ = An+4 , so B is a complement to E in G.

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It follows from 6.3.31 and 6.3.34 that G ∼ = AEn+4 , contrary to the choice of G. This final contradiction completes the proof of Theorem 6.3.9. 6.4. Subnormal closure In this section we assume τ = (F, Ω) is a quaternion fusion packet and adopt Notation 2.5.1 and 3.1.1. Moreover pick K ∈ Ω and set z = z(K). Definition 6.4.1. Recall Definition 1.2.9 where the subnormal series subi (F, K), 0 ≤ i ≤ n(F, K), and the subnormal closure sub(F, K) of K in F are defined. Set s(K) = sub(F, K), τi (K) = (subi (F, K), K subi (F ,K) ), and τ (K) = (s(K), K s(K) ). Write S(K) for the Sylow group of s(K). Lemma 6.4.2. (1) τ (K) and τi (K) are quaternion fusion packets and subi (F, K) and s(K) are subnormal subsystems of F. (2) O2 (s(K)) = S(K) ∩ O2 (F). (3) z ∈ O2 (F) iff z ∈ O2 (s(K)). (4) z < K ∩ O2 (F) iff z < K ∩ O2 (s(K)). (5) K ∩ O2 (F) is nonabelian iff K ∩ O2 (s(K)) is nonabelian. (6) OK ≤ s(K). Proof. By 2.7.3.1 in [Asc19], subi (F, K) is subnormal in F; then 2.5.2 completes the proof of (1). Part (2) is 2.7.3.6 in [Asc19]. Then (2) implies (3)-(5). Finally set si = subi (F, K). If (6) fails then OK is not contained in si for some i, and we choose i minimal subject to that constraint. Then i > 0 and OK ≤ si−1 , so replacing τ by τi−1 (K), we may assume i = 1, so si = s1  F. As FK (K) ≤ s1 , OK = FK (K), so C = O 2 (OK ) is a component or solvable component of Fz by 2.6.13. Next Cs1 (z)  Fz by 8.23.2 in [Asc11], so as K ≤ s1 , we conclude from 7.2.2 in [Asc11] that OK ∧ Cs1 (z) is a normal subsystem of OK on K, and hence  that OK ≤ Cs1 (z). But then OK ≤ s1 , contrary to the choice of i. Lemma 6.4.3. Assume z ∈ O2 (F) and OK = FK (K). Then (1) [K]F = O(F, K F ) is the central product of the members of {OJ : J ∈ K F }. (2) τ (K) = (OK , {K}). (3) [K]F centralizes OJ for J ∈ Ω − K F . Proof. By 3.3.13, C = O 2 (OK ) is a component or solvable component of F. Let Φ = {O 2 (OJ ) : J ∈ K F } and T ∈ Syl(E+ (F)). Then E+ (F)  F and AutF (T ) permutes Φ transitively as F is transitive on K F , and  E= D D∈Φ

is the central product of the members of Φ by 2.2.2. Therefore E  F by 7.4 in [Asc11]. Hence Z = z(J) : J ∈ Φ  F, so by 3.3.7.2, O(F, K F )  F. Thus, by definition of [K]F , (1) holds. Moreover OK  O(F, K F ), so s2 (K) = OK . Hence (2) also holds. Part (3) follows from (1) and 3.3.7.1.  Lemma 6.4.4. Set S0 = O2 (F) and assume K0 = K ∩ S0 is nonabelian. Then [K]F = O(F, K F ) is the central product of the members of {OJ : J ∈ K F }, and τ (K) = (OK , {K}).

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Proof. By 3.3.8.3, O(F, K F ) is the central product of the members of {OJ : J ∈ K F } and is normal in F. Then, as in the proof of the previous lemma, [K]F = O(F, K F ) and τ (K) = (OK , {K}).  Lemma 6.4.5. E = NS (K)s(K) is a saturated fusion system with s(K)  E and O 2 (E) = O 2 (s(K)). Further (E, K s(K) ) is a quaternion fusion packet. Proof. The first few statements follow from 2.7.3.7 in [Asc19], while the last statement follows from the first and 2.5.2.  Lemma 6.4.6. Assume z ∈ O2 (F) and OK = FK (K). Then (1) [K]F ≤ CF (E(F)). (2) [K]F , s(K), NS (K)[K]F , and NS (K)s(K) are constrained. (3) Let G be a model for NS (K)[K]F . Then there is a unique subnormal subgroup H of G such that H is a model for s(K). Moreover NS (K) acts on H, HNS (K) is a model for NS (K)s(K), and H = K H . (4) Suppose J ∈ Ω − K s(K) and J ≤ O2 (Js(K)). Then J centralizes s(K). Proof. As z ∈ O2 (F), E(F) = E(Fz ) by 10.3 in [Asc11]. As OK = FK (K), K ≤ O2 (Fz ) ≤ CFz (E(Fz )), so (1) holds. As s1 = [K]F  F, E(s1 ) ≤ E(F), so K ≤ Cs1 (E(s1 )) by (1). Therefore s1 ≤ CF (E(s1 )), so E(s1 ) = 1. Hence s1 is constrained by 14.2 in [Asc11], establishing the first statement in (2), while the second statement follows by induction on n(F, K). By 6.4.5, E = NS (K)s(K) is a saturated fusion system with s(K)  E and O 2 (E) = O 2 (s(K)). Let C = E(E). By 7.17.3 in [Asc11], C = O 2 (C) ≤ O 2 (E) = O 2 (s(K)). Then by 1.11 in [Asc19], C  s(K), contradicting E(s(K)) = 1. A similar argument shows NS (K)s1 is constrained. This completes the proof of (2). Part (3) follows from (2) and 2.1.7. Assume J ∈ Ω−K s(K) and let T ∈ Syl(Js(K)). Suppose that J ≤ O2 (Js(K)). Then as O2 (Js(K)) = O2 (HJ), Q = J H is a normal 2-subgroup of HJ. As K∈ / J H , K centralizes Q by 3.1.5.1, so H = K H centralizes Q. This establishes (4).  Lemma 6.4.7. Assume K ∩ O2 (F) = z and OK = FK (K). Let H be a model for s(K). Then (1) H/Z(H) ∼ = L3 (2)/E8 , L3 (2)/E64 , L3 (2)/23+6 , or AEn for some n ≥ 5. (2) If H/Z(H) ∼ = L3 (2)/E8 then Z(H) = 1. Proof. By parts (1), (4), and (6) of 6.4.2, we may assume that F = s(K). Thus F = [K]F , and replacing Ω by K F , we may assume that F is transitive on Ω. As K ∩ O2 (F) = z and OK = FK (K), z ∈ / Z(F). Let T be maximal subject to T ≤ S, z ∈ / T , and T  F. Set F + = F/T . By 3.3.2, T ≤ Z(F), τ + = (F + , Ω+ ) is a quaternion fusion packet, F + is transitive on Ω+ , and F + = [K + ]F + . By maximality of T , z + ∈ Q+ for each 1 = Q+ ≤ S + with Q+  F + . Therefore τ + satisfies Hypothesis 6.2.1, so a model H + for F + satisfies (1) by Theorems 6.2.8 and 6.3.9. As T ≤ Z(F), T ≤ Z(H) and H + = H/T . Thus (1) holds. Part (2) follows from 5.10.6.  Lemma 6.4.8. Assume K ∩ O2 (F) = z and OK = FK (K). Assume J ∈ Ω − K s(K) and let H be a model for s(K). Then either (1) J centralizes s(K), or

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(2) H ∼ = L3 (2)/E8 and τ0 = (s(K), {K, J}) is a quaternion fusion packet with z = z(J) and z H − {z} ⊆ A(z). Proof. As in 6.4.6.3, let G be a model for NS (K)[K]F and choose H subnormal in G. By 6.4.6.3, J ≤ NG (H) and H0 = HJ is a model for Js(K). Let T ∈ Syl(s(K)), T0 = T J, Ω0 = K H ∪ J H , and τ0 = (Js(K), Ω0 ). By 2.5.2, τ0 is a quaternion fusion packet. Set t = z(J) and assume J does not centralize s(K); equivalently, J does not centralize H. Set E = z H . As J acts on H and centralizes K, E  H0 . Set Z = Z(H), Q = ˜ 0 = H0 /Z. Then H ˜ is described in 6.4.7.1. In particular CT (E) ˜ = Q. O2 (H), and H As J does not centralize H, it follows from 6.4.6.4 that J ≤ O2 (H0 ) = Q0 . ˜ ∼ ˜ = GL(E). ˜ Then by 6.2.8 there is Q8 ∼ ˜0 Suppose E = E8 and H/CH (E) =K 2 H ˜ =K ˜K ˜0 ∼ ˜ ∈ Ω(˜ subnormal in CH˜ (˜ z ) with X z ), or = Q8 , and either K0 ∈ K and K ∼ ˜ ˜ ˜ ˜ ˜ H = L3 (2)/E8 . Further E ≤ X, so if J centralizes K K0 then J ≤ CH0 (E) ≤ Q0 , ˜ 0 , so K ˜0 ∈ a contradiction. Therefore J does not centralizes K / Ω(˜ z ) by 3.3.4, and ∼ hence H = L3 (2)/E8 . If J = K0 then (2) holds, so we may assume otherwise. Now J acts on but does not centralize K0 , so J ∩ K0 = 1 by 3.1.4. Thus z ∈ J so J ∈ Ω(z). Also JK0 = CH0 (K)  CH0 (z), so as CH (z) is irreducible on K0 /z we have [J, K0 ] ≤ z . Hence JK0 ∼ = Q8 does not centralize = Q28 or D8 Q8 and as J ∼ K0 it is the latter. Then Ω0 (z) = {K, J} by 3.1.5.1. But K0 = [JK0 , Y ] for Y of order 3 in CH (z) and CJK0 (Y ) ∼ = D8 , so Y does not act on J. This contradicts Ω0 (z) = {K, J}. ˜ ∼ Hence we may assume H = AEn for some n ≥ 5. Now J centralizes R = O(τ (K)), while by 5.1.15, |JQ0 : Q0 | ≤ 2, so J0 = J ∩ Q0 is of order at least 4. ˜ 0 ), so [H, ˜ t˜] = 1 and hence J ≤ Q0 , a Therefore t˜ is the unique involution in Φ(Q contradiction.  Lemma 6.4.9. Assume K ∩ O2 (F) = z and OK = FK (K). Set n = n(F, K), si = subi (F, K), and let Si ∈ Syl(si ). Then (1) Either s(K) is transitive on Ω ∩ S(K) or H ∼ = L3 (2)/E8 and Ω ∩ S(K) = {K, K0 } with z H − {z} ⊆ A(z), where H is a model for s(K). (2) If n > 1 then sn−1 is the central product of the members of Φ = {s(K)α : α ∈ AutF (Sn−1 )}. (3) n ≤ 2 and if F = F ◦ then n ≤ 1. Proof. Choose G and H as in 6.4.6.3. Then H = K H by 6.4.6.3. Set Ω1 = Ω ∩ S(K). If K1 ∈ Ω1 − K H then K1 ∈ Ω1 ⊆ H. Therefore as K1 is nonabelian, K1 does not centralize H, so (1) follows from 6.4.8. If n ≤ 1 then (2) and (3) hold, so we may assume n > 1. By 2.1.7, for i ≥ 1, there is a model Xi for si that is subnormal in G. Let X = Xn−1 ; as s(K)  sn−1 , we have H  X and for each α ∈ AutF (Sn−1 ), s(K)α  sn−1 by 7.3.4 in [Asc11]. Observe that s(K) = [K]sn−1 = [Ω1 ]sn−1 as Ω1 ⊆ S(K). By 2.8.3, α extends to α ˇ ∈ Aut(X). If Ω1 α = Ω1 then using (1), s(K) = [Ω1 ]sn−1 = [Ω1 α]sn−1 = s(K)α. On the other hand if Ω1 α = Ω1 then Ω1 ∩ Ω1 α = ∅ by (1), so Ω1 α centralizes s(K) by 6.4.8, and hence as H  X, H α ˇ = KαH αˇ centralizes H. Thus the 2-fusion system Y = FS∩Y (Y ) of the central product Y of the conjugates of H is normal in sn−2 . Therefore Y = sn−1 , so (2) holds.

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It remains to prove (3). For each J ∈ Ω − Sn−1 , J centralizes s(K)α for α ∈ AutF (Sn−1 ) by 6.4.8. Thus J centralizes sn−1 by (2), so if n > 2 then L = X Autsn−3 (Sn−2 ) is the central product of the Autsn−3 (Sn−2 )-conjugates of X, so L = Xn−2 . But then H  Xn−2 , so s(K)  sn−2 , and hence s(K) = sn−1 , contrary to the definition of n. Thus n ≤ 2. Similarly if F = F ◦ then n ≤ 1, completing the proof.  Lemma 6.4.10. Assume F = F ◦ and set Ω0 = {J ∈ Ω : z(J) ∈ / O2 (F)}. Then τ = τ0 ∗ τ1 ∗ · · · ∗ τk , where Fi = [Ωi ]F , τi = (Fi , Ωi ) is a quaternion fusion packet, and for each i > 0 and K ∈ Ωi , either (1) Fi = s(K) and Ωi = K F = K Fi , or ˜ = Ω(z), Fi is constrained with model H, z H − {z} ⊆ A(z), (2) Ωi = {K, K} and either H ∼ = μ(τi ) ∼ = D12 . = L3 (2)/E8 or W = η  H for η ∈ η(τi ) and H/W ∼ Proof. Set Q = O2 (F). If K ∩ Q is nonabelian then by 3.3.8, [K]F is the central product of the OJ for each J ∈ K F , and centralizes [Ω − K F ]F , so as F = F ◦ we conclude that {K} is an orbit of F on Ω and OK = [K]F = s(K). Thus we may assume K ∩ Q is cyclic for each K ∈ Ω. Suppose next that |K ∩ Q| > 2 for some K ∈ Ω. Then by 3.3.12, there is Γ ⊆ Ω containing K such that FΓ = [Γ]F centralizes [Ω − Γ]F , τΓ = (FΓ , Γ) is a quaternion fusion packet, and η ∈ η(τΓ ) is normal in FΓ . Moreover either FΓ is ˜ is of order 2 and A(z) = ∅. Then by 3.3.9, FΓ is transitive on Γ or Ω(z) = {K, K} constrained with model H and for η ∈ η(τΓ ), W = η  H. Finally if A(z) = ∅ then H/W ∼ = μ(τΓ ) ∼ = D12 by 4.2.10. Thus we may assume |K ∩ Q| ≤ 2 for each K ∈ Ω, and there exists some K with z = z(K) ∈ Q. Suppose OK = FK (K). Then by 6.4.3.3, s1 = [K]F centralizes [Ω − K F ]F and then by 6.4.3.1, K F = {K}. Therefore we may assume OK = FK (K) for each K ∈ Ω with z(K) ∈ Q. As F = F ◦ , we conclude from 6.4.9.3 that n(F, K) ≤ 1, so s(K) = s1 . Now s1  F, so O2 (s1 ) ≤ Q. If J ∈ Ω − S(K) then by 6.4.8, J centralizes s1 , so that [J]F ≤ CF (s1 ). In particular [Ω0 ]F centralizes [L]F for each L ∈ Ω − Ω0 , so we may assume Ω0 = ∅. Then as [J]F centralizes s1 for each J ∈ Ω − S1 , and as F = F ◦ , it follows that F = F1 ∗ · · · ∗ Fk is a central product, where Fi = s(Ki ) for some  Ki ∈ Ω. Now the lemma follows from 6.4.9.1. 6.5. F ∗ (F) In this section we assume τ = (F, Ω) is a quaternion fusion packet and adopt Notation 2.5.1 and 3.1.1. Moreover pick K ∈ Ω with z = z(K) ∈ F f . Lemma 6.5.1. (1) O(z)  Fz . (2) For each K ∈ Ω(z), OK is subnormal in Fz . (3) O(z) is the central product of the systems OK , K ∈ Ω(z). (4) For each K ∈ Ω(z), OK = FK (K) or OK ∼ = SL2 [m] or SL12 [m]. Proof. Parts (1) and (2) are a restatement of 2.6.11, part (3) is a restatement of 2.6.8, and part (4) was established in 2.5.7. 

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Lemma 6.5.2. Suppose C is a component of F centralized by z. Then (1) C is a component of Fz . (2) One of the following holds: (i) OK centralizes C. (ii) OK = C. (iii) OK ∼ = SL12 [m], C = O 2 (OK )), and OK centralizes each component of F distinct from C. Proof. Part (1) follows from 2.2.3, so it remains to prove (2). Appealing to (1) and 2.5.3.1, we may assume F = Fz . Thus OK is subnormal in F by 6.5.1.2. Therefore if OK = FK (K), then by 9.10 and Theorem 6 in [Asc11], K ≤ O2 (F) ≤ CS (C), so that (2i) holds. Thus we may assume that m > 8. Similarly if OK ∼ = SL2 [m], then as m > 8, OK is quasisimple, so OK is a component of F, and hence (2i) or (2ii) holds as E(F) is the central product of its components. Therefore by 6.5.1.4, we may assume OK ∼ = SL12 [m]. Set B = O 2 (OK ), so that ∼ B = SL2 [m/2], and let B be Sylow in B. Then either B is a component of F, or m = 16 and B ∼ = SL2 [8] is a solvable component. In either case by 2.2.2, either B = C or B centralizes C. In the first case, for E ∈ Com(F) − {B} and E Sylow in E, we have E ≤ CS (B) ≤ NS (K) by 2.6.6. Thus [K, E] ≤ CK (B) = z , so K acts on E by 9.7.2 in [Asc11], so [K, E] ≤ z ∩ E ≤ Z(E). Hence K centralizes E by 9.5 in [Asc11], so OK = KB centralizes E, and therefore (2iii) holds. Finally assume B centralizes C. Then the argument in the previous paragraph applied to C in the role of E, shows that OK centralizes C, so that (2i) holds, completing the proof.  Lemma 6.5.3. If z ∈ / O2 (F) then (1) OK ≤ CF (O2 (F)) and (2) z ∈ E(F) − Z(E(F)). Proof. As z ∈ / Q = O2 (F), K centralizes Q by 3.1.4. Then 2.6.6 implies (1). Let E = E(F) and E ∈ Syl(E). Then E  S, so K acts on E. Hence by 3.1.4, either z ∈ E or [K, E] = 1, and we may assume the latter. By 2.3.1 in [Asc19], z centralizes E. Then by 6.5.2.2, OK ≤ CF (E(F)) = B. By 9.12.1 in [Asc11], Q = F ∗ (B). But then K ≤ CB (Q) = Z(Q) by Theorem 6 in [Asc11], contradicting K nonabelian.  Theorem 6.5.4. Assume z ∈ / O2 (F). Then (1) z is contained in a unique component C = C(z) of F. (2) OK centralizes each component of F distinct from C. (3) If OK is not solvable then O 2 (OK ) ≤ C. We prove Theorem 6.5.4 in a series of reductions. Assume τ = (F, Ω) is a minimal counter example. That is F, Ω is a counter example, but Theorem 6.5.4 holds for all quaternion fusion packets τ1 = (F1 , Ω1 ) such that F1 is of smaller order than F, or Ω1 is of smaller order than Ω. Lemma 6.5.5. z is contained in no component of F.

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Proof. Suppose z is contained in a component C of F. As z ∈ / O2 (F), C is the unique component containing z, so K acts on C. Then K also acts on the product E of the remaining components of F. Let E ∈ Syl(E). Then K acts on E and z ∈ / E, so [K, E] = 1 by 3.1.4. Claim OK centralizes E. For by 6.5.3.2 applied to KE, we have z ∈ O2 (KE), so that z centralizes E and the claim follows from 6.5.2. Further if OK is not solvable, then L = O 2 (OK ) is a component of Fz by 6.5.1, so by L-balance (cf. Theorem 7 in [Asc11]), L ≤ E(F). Hence L ≤ CE(F ) (E) = X = Z(E(F))C, so by 7.17.3 in [Asc11], L = O 2 (L) ≤ O 2 (X ) = C. But now all conclusions of Theorem 6.5.4 hold, contrary to the choice of τ as a counter example.  Lemma 6.5.6. OK = FK (K). Proof. Assume otherwise. Then L = O 2 (OK ) ∼ = SL2 [k] with k = m or m/2. In particular L is a component or solvable component of Fz containing z, on some L ≤ K, so by 2.2.6.4, either L ≤ C for some component C of F, or L centralizes E(F). The former is impossible by 6.5.5, and the latter contradicts 6.5.3.  Lemma 6.5.7. F = E(F)K and F is transitive on Ω. Proof. Let E = KE(F). By Theorem 5 in [Asc11], E is a saturated fusion system with O 2 (E) = E(F). Thus E(E) = E(F). Let Σ = K E . By 2.5.2, ρ = (E, Σ) is a quaternion fusion packet. Thus if F = E or Ω = Σ, then by minimality of τ , the theorem holds for ρ. But then z is contained in some component of F, contrary to 6.5.5.  Lemma 6.5.8. O2 (F) = 1. / Q, so K centralizes Q Proof. Suppose Q = O2 (F) = 1. By hypothesis, z ∈ by 3.1.4. By Theorem 6 in [Asc11], E(F) centralizes Q, so Q = Z(F) by 6.5.7. Set F + = F/Q. By 3.3.2.2, τ + = (F + , Ω+ ) is a quaternion fusion packet, so by minimality of τ , z + is contained in some component C + of F + , and K + acts faithfully on C + . By 2.1.11, the preimage C of C + in F is QB, with B = O 2 (C) a component of F. Now K acts on C, and hence also on B. By 6.5.5, z ∈ / B, contrary to 6.5.3.2 applied to KB.  Lemma 6.5.9. K is transitive on the components of F. Proof. Let C be a component of F), E = C K , and E¯ the product of the components not in C K . By 7.4 in [Asc11] and 6.5.7, E and E¯ are normal in F. We may assume E¯ = 1. By Theorem 5 in [Asc11], B = KE and B¯ = K E¯ are saturated ¯ By 2.5.2, (B, K B ) and (B, ¯ K B¯) are subsystems with E = E(B) and E¯ = E(B). quaternion fusion packets, so by minimality of F, z is contained in some component ¯ contrary to 6.5.5. of B or B,  Let {Ci : i ∈ I) be the set of components of F, with I = {1, . . . , s} and Ci a system on Ti . Then E(F) is a system on T = T1 × · · · × Ts by 6.5.8. As z ∈ T , z = z1 · · · zs , where zi is the projection of z on Ti , and by 6.5.9, zi = 1 for each i ∈ I, and K is transitive on U = {zi : i ∈ I}. By 6.5.5, s > 1. Set R = CT (z), Q = NT1 (K), and X = K ∩ T1 T2 . Lemma 6.5.10. If X = 1 then s = 2 and z1 ∈ Q.

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Proof. Assume X = 1. Then as z is the unique involution in K, z ∈ X, so z = z1 z2 , and then as zi = 1 for all i ∈ I and as s > 1, it follows that s = 2. As K is transitive on U of order 2, |CK (z1 )| = m/2 > 2, so z1 ∈ Q.  Lemma 6.5.11. X is cyclic. Proof. Assume X is noncyclic. Then by 2.5.2, (E(F), X E(F ) ) is a quaternion fusion packet, so by minimality of F, z is contained in a component of F, contrary to 6.5.5.  Lemma 6.5.12. Choose notation so that there is k ∈ K of order 4 with C1k = C2 . Then the map ϕ : y → [y, k] is an isomorphism of Q with X. Proof. Let u, v ∈ Q. Then [u, k] ∈ X, so ϕ : Q → X. Further uϕ = vϕ iff uv −1 = uk v −k = (uv −1 )k iff k centralizes uv −1 . However uv −1 ∈ T1 and T1k = T2 with T1 ∩ T2 = 1, so k centralizes uv −1 iff u = v. Thus ϕ is an injection. Hence if |X| ≤ 2 then the lemma follows from 6.5.10, so we may assume X = x is of order greater than 2. Now x = x1 x2 with xi ∈ Ti . Further x1 ∈ CT1 (x) ≤ NT1 (K) = Q as |x| > 2. Therefore |Q| ≥ |X|, so ϕ is a bijection and Q = x1 . Then QX = x1 , x2 is abelian, so from 8.5.4 in [Asc86], ϕ : Q → X is a homomorphism, completing the proof.  Lemma 6.5.13. R1 = R ∩ T1 is noncyclic, and either Φ(R1 ) = 1, or R1 is nonabelian dihedral and |R1 : Q| = 2. Proof. By 9.1.2 in [Asc11], T1 is not cyclic, so R1 = CT1 (z) = CT1 (z1 ) is noncyclic. But Q is cyclic by 6.5.11 and 6.5.12, so there exists r ∈ R1 − Q. Then u = [r, k] = k−r k, and as r does not act on K, [k, k−r ] ∈ [K, K r ] = 1, so [u, k] = 1. But k inverts u, so u2 = 1, and hence r is an involution. We have shown that each element in R1 − Q is an involution, so each such element inverts Q, and hence as Q is cyclic either |Q| ≤ 2 or |R1 : Q| = 2 and R1 is nonabelian dihedral. Thus the lemma holds.  Lemma 6.5.14. |R1 : Q| = 2.

 Proof. Assume otherwise. Then by 6.5.13, Φ(R1 ) = 1, so as R = i CTi (z) and K is transitive on the components of F), R = R1 ×· · ·×Rs with Ri = CTi (z) ∼ = R1 . Thus Φ(R) = 1, so |X| ≤ 2 by 6.5.11, Let W = O(z). If K = W then R1 = Q, whereas R1 is noncyclic by 6.5.13, while Q is cyclic by 6.5.11 and 6.5.12. Therefore W = K. But W = KWR , where WR = W ∩ R, so W/K ∼ = (W ∩ R)/(K ∩ R) is elementary abelian, and hence as W = K we conclude that m = 8. Then W ∼ = 21+2w is extraspecial, as W is the central product of w copies of K ∼ = Q8 . As Φ(WR ) = 1, m(WR ) ≤ w + 1. But |W : WR | = |K : K ∩ R| = 4, so m(WR ) = 2w − 1. Then 2w − 1 ≤ w + 1, so w = 2. As R permutes the two members of Ω(z), |R1 : Q| ≤ 2. By 6.5.13, |R1 | > 2, so Q = 1 and by 6.5.12, X ∼ = Q so X = 1. By the previous paragraph, |X| ≤ 2 so  |Q| = 2 and |R1 : Q| = 2. Thus the lemma holds. We are now in a position to establish Theorem 6.5.4. By 6.5.11 and 6.5.12, Q is cyclic, so by 6.5.13 and 6.5.14, R1 is dihedral and |R1 : Q| = 2. In particular Q = 1, while X ∼ = Q by 6.5.12, so X = 1. Therefore s = 2 by 6.5.10.

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If R1 is nonabelian, then z1 = Z(R1 ), so as R1 = CT1 (z1 ), we have T1 = R1 dihedral. On the other hand if R1 ∼ = E4 , then by a lemma of Suzuki (cf. Exercise 8.6 in [Asc86]), T1 is dihedral or semidihedral. So in any event, T1 is dihedral or semidihedral. Then as C1 is simple, it follows from 2.1.10 that C1 is transitive on its involutions. But as z ∈ F f , z1 ∈ C1f , so R1 = T1 is dihedral and z is in the center of S = T1 T2 k ∼ = R1 wr Z2 . By 9.1.2 in [Asc11], R1 is nonabelian, so |X| = |Q| > 2. Let E = C1 C2 . As |X| > 2, K0 = Xk is quaternion, so replacing τ by τ0 = (k E, K0E , we may assume τ = τ0 , so K = K0 . As R1 = T1 , |T1 : Q| = 2 by 6.5.14, so as Q ∼ = X is cyclic, Q is the cyclic subgroup of T1 of index 2. Let a1 ∈ T1 − Q; then a = a1 ak1 is in z E and centralizes k, so a ∈ D∗ (z). Thus there is z = t ∈ z E such that there is U = u ∈ Δ(t). As z is weakly closed in QX, t ∈ T − QX, so t ∈ aQX. Then as S/QX ∼ = D8 it follows that m = 8 and u ∈ a1 kQX. But then X u = X, where is u acts on K by condition (QFP3) in the definition of quaternion fusion packets, and hence also on X = K ∩ T . This contradiction completes the proof of Theorem 6.5.4.

6.6. z not in O2 (F) In this section we assume τ = (F, Ω) is a quaternion fusion packet and adopt Notation 2.5.1 and 3.1.1. Moreover pick K ∈ Ω with z = z(K) ∈ F f . Lemma 6.6.1. Assume F = F ◦ and Z ∩ O2 (F) = ∅. Set Q = O2 (F) and F + = F/Q. Let I = {1, . . . , r}, {Ci : i ∈ I} the set of components of F, Si ∈ Syl(Ci ), Ωi = {J ∈ Ω : z(J) ∈ Si }, Fi = [Ωi ]F , and τi = (Fi , Ωi ). (1) Q = Z(F). (2) τ + = (F + , Ω+ ) is a quaternion fusion packet with F + = [Ω+ ]F + and O2 (F + ) = 1. (3) For each  i ∈ I, Ci  F and Ωi = ∅. (4) Ω = i∈I Ωi is a partition of Ω. (5) F = F1 · · · Fr is a central product of the subsystems Fi , so τ = τ1 ∗ · · · ∗ τr is a central product of quaternion fusion packets. (6) F ∗ (Fi ) = Ci Z(Fi ). Proof. By hypothesis, Z ∩ Q = ∅, so Ω centralizes Q by 3.1.4. Then (1) follows as F = F ◦ . Now (2) follows from (1) and 3.3.2. Next by Theorem 6.5.4, z ∈ Cj for a unique j ∈ I and OK centralizes Ci for i ∈ I − {j}. In particular K acts on each component of F, so as F = F ◦ , each component of F is normal in F by 2.1.6. Further if Ωi = ∅ then each member of Ω centralizes Ci , so F = [Ω]F centralizes Ci , a contradiction. Hence (3) and (4) hold. For i ∈ I, set  Cj and Di = CF (Ei ), Ei = j∈I−{i}

and let Ti ∈ Syl(Di ). From the previous paragraph, Ωi ⊆ Ti and Di  F, so Fi ≤ Di . Observe F ∗ (Di ) = Ci Q. Also for i = j, CKQTi (Ci ) = KQ and CDi (Ci ) = Q, so KQ = O2 (KDi ) and hence {K} = Ω ∩ O2 (KDi ) so K  KDi . Hence Fi = [Ωi ]Fi centralizes K. Then Fj = [Ωj ]F centralizes Fi , proving (5). As F ∗ (Di ) = Ci Q, (6) holds. 

6.6. z NOT IN O2 (F )

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Lemma 6.6.2. Assume O2 (F) = 1 and F = [K]F . Then (1) F is almost simple. (2) z ∈ F ∗ (F). (3) If OK = FK (K) then O 2 (OK ) ≤ F ∗ (F). Proof. By 6.6.1, there exists a unique component C of F such that z ∈ T ∈ Syl(C), K centralizes all other components of F, and each component of F is normal in F. Hence as F = [K]F , C is the unique component of F, so C = F ∗ (F) since  O2 (F) = 1. In particular (1) and (2) hold, while (3) follows from 6.5.4.3. Recall the definition of s(K) = sub(F, K) and τ (K) = (s(K), K s(K) ) from Definitions 1.2.9 and 6.4.1. Lemma 6.6.3. Assume F ∗ (F) = Z(F)C for some component C of F, and z∈ / O2 (F). Then / O2 (s(K)). (1) F ∗ (s(K)) = Z(s(K))C and z ∈ (2) τ (K) = (s(K), K s(K) ) is a quaternion fusion packet with s(K) = [K]s(K) . Proof. By 6.4.2.1, τ (K) is a quaternion fusion packet, and by definition of s(K), we have s(K) = [K]s(K) . Thus (2) holds. As z ∈ / O2 (F), also z ∈ / O2 (s(K)) by 6.4.2.3. By 6.4.2.1, s(K) is subnormal in F, so as F ∗ (F) = Z(F)C, it follows that either C ≤ s(K) or F ∗ (s(K)) ≤ Z(F). But in the latter case as Cs(K) (F ∗ (s(K)) ≤ F ∗ (s(K)), we have s(K) abelian, contradicting K ≤ s(K). Therefore C ≤ s(K), so F ∗ (s(K)) = Z(s(K))C. Thus (1) holds.  Lemma 6.6.4. Assume A(τ ) = ∅. Then τ ◦ = τ1 ∗ · · · ∗ τr is a central product of quaternion fusion packets τi = (Fi , Ωi ) such that Fi = [Ωi ]Fi and for Ki ∈ Ωi and zi = z(Ki ), (1) if zi ∈ O2 (Fi ) then Fi is transitive on Ωi , D(τi )c is connected, and either Fi is constrained or Fi = OKi , and (2) if zi ∈ / O2 (Fi ) then F ∗ (Fi ) = Z(Fi )Ci for some component Ci of F such that Ci ≤ [Ki ]Fi . Proof. We may assume τ = τ ◦ . Then by 6.4.10, τ = τ0 ∗ τ1 ∗ · · · ∗ τs , where / O2 (F)}, and for τi = (Gi , Γi ) is a quaternion fusion packet, Γ0 = {J ∈ Ω : z(J) ∈ j > 0 and Jj ∈ Γj , we have z(Jj ) ∈ O2 (Gj ) and Gj is transitive on Γj . By Theorem 6.1.5, D(τj )c is connected. Further if OJj = FJj (Jj ) then Gj = OJj by 6.4.3. On the other hand if OJj = FJj (Jj ) then Gj is constrained by 6.4.6.3. Therefore, replacing τ by τ0 , we may assume Ω = Γ0 . Now the lemma follows from 6.6.1 and 6.6.3.  Lemma 6.6.5. Assume (a) A(τ ) = ∅. (b) F ∗ (F) = Z(F)C for some component C of F, and Z ∩ O2 (F) = ∅. (c) If τ (K) = τ ◦ then τ (K) satisfies one of the conclusions of Theorem 1. Then (1) F ◦ is transitive on Ω. (2) F ◦ = [K]F . (3) If τ ◦ = (F ◦ , Ω) satisfies one of the conclusions of Theorem 1 then either D(τ )c is connected or F/Z(F) is P Spn [m] or L− n [m] for some n ≥ 6.

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Proof. Set E = s(K), Γ = K E , and ρ = τ (K). By (b) and 6.6.3, ρ satisfies the hypotheses of Theorem 1, so in particular ρ satisfies (1) and (2). Suppose (1) or (2) fails; then as ρ satisfies (1) and (2), we conclude τ ◦ = ρ. Therefore by (c), ρ satisfies one the conclusions of Theorem 1. By 6.6.3.1, z ∈ / O2 (E), so ρ satisfies one of conclusions (2)-(4) of Theorem 1. Further by 6.6.3.1, F ∗ (E) = Z(E)C, so C is described in Theorem 1. If E is L2 [2m](1) then there is a unique quaternion subgroup Q of order m normal in a Sylow group T of E. Therefore K = Q and Γ = {K}. Indeed if J ∈ Ω − Γ then J ≤ CS (Q), so from the structure of Aut(L2 (q)), J ≤ CS (C), so J ≤ CS (F ∗ (F)) = Z(F), contradicting J nonabelian. Therefore Ω = {K}, contradicting the assumption that (1) or (2) fails. (1) Similarly if E is L+ , then there is a unique quaternion subgroup Q of KT 3 [m] of order m such that KT = QT , so K = Q and Γ = {K}. Then J ≤ CS (K) = CS (Q) = Z(F)P , where P ∼ = K is Sylow in a fundamental subgroup of Lπ3 (q). Also 2 J ≤ O (Z(F)C) = C, so J = P is determined. But now z = z(J) and A(z) = ∅, contrary to (a). Therefore ρ satisfies conclusion (2) or (4) of Theorem 1, or E ∼ = M12 . In particular each member of Γ is in Z(F)C, so F ◦ = Z(F)C. But also Γ ≤ O 2 (F ◦ ), so F ◦ = C. Suppose C is M12 or G2 [m]. Then Γ = {K}, and if J ∈ Ω − Γ then z(J) = z and A(z) = ∅, contrary to (a). Thus ρ satisfies conclusion (2) or (4) of Theorem 1, but C is not G2 [m]. Therefore Γ is uniquely determined in E = C, so Ω = Γ, contrary to the assumption that (1) or (2) fails. This completes the proof of (1) and (2). Further (3) follows from 6.1.5.  Lemma 6.6.6. Assume A(τ ) = ∅ and Theorem 1 holds in all proper subpackets of τ of the form τ (K) for some K ∈ Ω with z(K) ∈ F f . Then (1) τ ◦ = τ1 ∗ · · · ∗ τr is a central product of quaternion fusion packets τi = (Fi , Ωi ), where Fi = [Ωi ]Fi is transitive on Ωi . (2) If either τ = τi or τi satisfies one of the conclusions of Theorem 1, then either D(τi )c is connected or Fi /Z(Fi ) is P Spn [m] or L− n [m] for some n ≥ 6. Proof. We may assume τ = τ ◦ . Then by 6.6.4, τ = τ1 ∗ · · · ∗ τr is a central product of quaternion fusion packets τi = (Fi , Ωi ), where Fi = [Ωi ]Fi satisfies (1) or (2) of 6.6.4. In 6.6.4.1, Fi is transitive on Ωi and D(τi )c is connected, so we may assume τi appears in 6.6.4.2. Let K ∈ Ωi . If τi = τ (K), then we have transitivity by 6.6.3.2, so assume otherwise. Then τ (K) is a proper subpacket of τ , so by hypothesis, τ (K) satisfies one of the conclusions of Theorem 1. But now we have transitivity by 6.6.5.1. This completes the proof of (1). / O2 (Fi ). Observe that It remains to prove (2). By 6.6.4 we may assume zi ∈ τi = τ (K) for K ∈ Ωi with z(K) ∈ F f . Thus if τ = τi , then τi is proper in τ , so τi satisfies one of the conclusions of Theorem 1 by hypothesis. On the other hand if τ = τi this also holds by the hypotheses of (2). Now 6.6.5.3 completes the proof of (2).  Definition 6.6.7. We often assume the following Inductive Hypothesis : Theorem 1 holds in each proper subpacket of τ satisfying the hypothesis of Theorem 1 and of the form (Y, Γ) with Y ≤ F and Γ ⊆ Ω.

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Sometimes we instead assume the stronger Extended Inductive Hypothesis : For each Q ≤ S centralizing F ◦ with Q∩Z(τ ) = ∅, the Inductive Hypothesis is satisfied + by τQ = (F ◦+ , Ω+ ), where (QF ◦ )+ = QF ◦ /Q. Observe that if τ is a counter example to Theorem 1 of minimal order, then τ satisfies the Extended Inductive Hypothesis. Rather than work in such a minimal counter example, we often assume instead the Inductive Hypothesis or the Extended Inductive Hypothesis is satisfied. This approach has the virtue, first, of keeping track of which properties of a minimal counter example are actually needed in a given proof, and, second, once Theorem 1 is established, results established under either of the two hypotheses become real theorems. Lemma 6.6.8. Assume A(τ ) = ∅ and the Inductive Hypothesis holds. Then Theorem 4 holds in each proper subpacket of τ of the form (Y, Γ) with Y ≤ F and Γ ⊆ Ω. Proof. For K ∈ Ω with z(K) ∈ F f , τ (K) is a subpacket of τ satisfying the hypothesis of Theorem 1. Then if τ (K) is proper Theorem 1 holds for τ (K) by the Inductive Hypothesis. Hence the lemma follows from 6.6.6.  Lemma 6.6.9. Assume τ satisfies the Extended Inductive Hypothesis. (1) If F ◦ ≤ Y ≤ F with Y saturated then (Y, Ω) is a quaternion fusion packet satisfying the Extended Inductive Hypothesis. (2) Suppose Q ∈ F f with F ◦ ≤ N = NF (Q) and Q∩Z(τ ) = ∅. Set N + = N /Q and ρ+ = (N + , Ω+ ). Then Q centralizes F ◦ and ρ+ is a quaternion fusion packet satisfying the Extended Inductive Hypothesis. Proof. Part (1) is straightforward, so it remains to prove (2). Assume the setup of (2). Replacing F by N and appealing to (1), we may assume F = N . By 3.3.2, Q centralizes F ◦ and τ + is a quaternion fusion packet. Suppose P + ≤ S + centralizes F ◦+ with P + ∩ Z(τ + ) = ∅. We may assume Q ≤ P ≤ S. Then as P + ∩ Z(τ + ) = ∅ = Q ∩ Z(τ ), also P ∩ Z(τ ) = ∅. Then P centralizes F ◦ by 3.3.2. Set (P F ◦ )! = P F ◦ /P . As F satisfies the Extended Inductive Hypothesis, we have τP! = (F ◦! , Ω! ) satisfies the Inductive Hypothesis, completing the proof of (2).  Remark 6.6.10. Observe that 6.6.6 says that Theorems 1 and 3 imply Theorem 4. Theorem 3 treats the case where A(τ ) = ∅, and will be proved later in section 7.2. Similarly, with a bit of work, Theorem 1 (together with Theorem 2) implies Theorems 3 through 8, although in practice Theorem 1 is established by partitioning the class of quaternion fusion packets τ = (F, Ω) with F = F ◦ transitive on Ω into a number of subclasses corresponding to the hypotheses of Theorems 3 through 8, and treating each subclass by proving a version of the theorem corresponding to that subclass. From time to time we appeal to the following observation, often without comment: Lemma 6.6.11. Assume ρ = (Y, Γ) is a subpacket of τ with Y = Y ◦ and Γ ⊆ Ω. Assume A(ρ) = ∅ and E  F. Then (1) If ρ satisfies one of the conclusions to Theorem 1 and E contains a member of Γ then Y ≤ E. (2) If ρ satisfies one of the conclusions of Theorem 1 then Y ≤ F ◦ .

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(3) If τ satisfies the Inductive Hypothesis and E contains a member of each orbit of Y on Γ then Y ≤ E. (4) If τ satisfies the Inductive Hypothesis then Y ≤ F ◦ . (5) If τ satisfies the Inductive Hypothesis and ρ is coconnected then Y ≤ X for some coconnected component (X , Σ) of τ . Proof. We first prove (1). Let T be Sylow in Y and E Sylow in E. Then some K ∈ Γ is contained in T ∩ E and as E  F we have T ∩ E strongly closed in T with respect to Y. Hence by condition (a)of Remark 5.6.19, we have T ≤ E. Now by  Remark 5.6.19 it suffices to show B∗ (Y) ≤ E. Let R ∈ Y f c and N = O 2 (NY (R));  it suffices to show N ≤ E. Let G be a model for NF (R); then O 2 (G) = H is a H model for N . Now Q = NT (R) is Sylow in G and H = Q . As Q ≤ T ≤ E and Λ = AutE (R)  AutF (R), we have AutH (R) ≤ Λ. Then as CG (R) ≤ R it follows that N = FQ (H) ≤ E, completing the proof of (1). Part (2) follows from (1) with E = F ◦ . It remains to prove (3)-(5), so we may assume the Inductive Hypothesis. Suppose ρ = τ . Then (4) and (5) are trivial. Further in (3), F = Y = Y ◦ , so as E  F we have Y ≤ E and hence (3) holds. Therefore we may assume τ = ρ. Next (3) implies (4). Assume the setup of (5). Then Y ≤ F ◦ by (3). By 6.6.6, ◦ τ = τ1 ∗ · · · ∗ τr is the central product of its coconnected components τi = (Xi , Σi ). Now K ∈ Σi for some i and Xi  F ◦ , so Y ≤ Xi by (1), completing the proof of (5) and the lemma.  Lemma 6.6.12. Assume τ satisfies the Inductive Hypothesis and ρ = (Y, Γ) is a proper subpacket of τ with Γ ⊆ Ω. Then ρ satisfies the Extended Inductive Hypothesis. Proof. Let T be Sylow in Y and assume Q ≤ T centralizes Y ◦ with Q∩Z(ρ) = ∅. Replacing Y by QY ◦ we may assume Y = QY ◦ . Set Y + = Y/Q. By 3.3.2, ρ+ is a quaternion fusion packet, Y + = (Y + )◦ , and the map K → K + is a bijection of Γ with Γ+ . We must show ρ+ satisfies the Inductive Hypothesis, so assume ξ + = (X + , Σ+ ) is a quaternion fusion packet with Σ+ ⊆ Γ+ and satisfying the hypothesis of Theorem 1. Let X be the preimage of X + and Σ the inverse image of Σ+ in Γ. Then X + = (X + )◦ so X + = (X ◦ )+ by 8.10.2 in [Asc08a]. Set ζ = (X ◦ , Σ). As K → K + is a bijection of Σ with Σ+ , and X + = (X ◦ )+ is transitive on Σ+ , we have X ◦ transitive on Σ, so ζ satisfies the hypohtesis of Theorem 1. Hence as ρ satisfies the Inductive HYpothesis, ζ satisfies one of the conclusions of Theorem 1. But then ζ + also satisfies one of the conclusions, so ρ+ satisfies the Inductive HYpothesis, completing the proof. 

6.7. The proof of Theorem 2 In this section we assume the hypothesis of Theorem 2 from the introduction. Thus τ = (F, Ω) is a quaternion fusion packet, K ∈ Ω, and z = z(K) ∈ O2 (F) but z∈ / Z(F). In addition F = [K]F and F is transitive on Ω. Lemma 6.7.1. OK = FK (K).

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Proof. Assume otherwise. By hypothesis, F = [K]F , so by 6.4.3.1, F is the central product of the members of {OJ : J ∈ K F }. As F = [K]F , it follows that F = OK , contradicting z ∈ / Z(F).  Lemma 6.7.2. K ≤ O2 (Fz ). Proof. By 2.6.11, OK is subnormal in Fz , and by 6.7.1, OK = K, so K ≤ O2 (Fz ).  Lemma 6.7.3. F is constrained. Proof. This is a consequence of 6.7.1 and 6.4.6.2, given the hypothesis that F = [K]F .  By 6.7.3 and 2.8.1, F has a model G. Set S0 = O2 (G). As F = [K]F we have: Lemma 6.7.4. G = K G . Lemma 6.7.5. K ∩ S0 is cyclic. Proof. If not by 6.4.4, G is the central product of the members of K G , so G = K by 6.7.4, contradicting z ∈ / Z(G).  Lemma 6.7.6. Assume K ∩ S0 = z . Then for η ∈ η(τ ), η  G and μ(τ ) is generated by 3-transpositions. Proof. By 6.7.5, V = K ∩ S0 is cyclic, and by assumption, |V | > 2. Then by 3.3.12.4, |K : V | = 2 and, as F is transitive on Ω, for η ∈ η(τ ), we have η  G. Therefore Hypothesis 4.2.1 is satisfied. As F is transitive on Ω, G is transitive on η, so by 4.2.10.2, G = G(V ), and by 4.2.10.4, A(z) = ∅. Then by 4.2.10.3, μ(τ ) is generated by 3-transpositions.  Lemma 6.7.7. Assume K ∩ S0 = z . Then one of the following holds: (1) G ∼ = L3 (2)/E8 . (2) G/Z(G) ∼ = L3 (2)/E64 or L3 (2)/23+6 . ∼ (3) G/Z(G) = AEn for some n ≥ 5. Proof. As F = [K]F is transitive on Ω it follows from 6.4.10 that F = s(K).  By 6.7.1, OK = FK (K). Now 6.4.7 completes the proof. Observe that by 6.7.2 and 6.7.3, K ≤ O2 (Fz ) and F is constrained, so the initial statements in Theorem 2 hold. Then G satisfies one of the four conclusions of Theorem 2 by 6.7.6 and 6.7.7.

CHAPTER 7

Theorems 3 and 5 In Chapter 7 we prove Theorems 3 and 5. Let τ = (F, Ω) be a quaternion fusion packet. Recall that Theorem 3 deals with the case where A(τ ) = ∅, while Theorem 5 determines those τ with Ω = {K} of order 1. If z ∈ ZS (τ ) with A(z) = ∅ and F = [Ω(z)]F ◦ , then either a member of W (τ ) is normal in F or F is one of the systems appearing as a conclusion to Theorem 5. The difference is that in the first case, F is intransitive on Ω = Ω(z) = {K1 , K2 }, while in the second Ω = {Ki } is of order 1 for i = 1 or 2. Thus the two results are related. We prove Theorem 5 first in section 7.1, and then reduce Theorem 3 to Theorem 5 in lemma 7.2.18. Indeed at other points in the proof of Theorem 3 we essentially repeat corresponding arguments from the proof of Theorem 5. As we observed in Remark 6.6.10, given Theorem 3 it can be shown that Theorem 4 follows from Theorem 1; this is Corollary 7.2.27. 7.1. Packets of width 1 In this section we assume the following hypothesis: Hypothesis 7.1.1. (1) τ = (F, Ω) is a quaternion fusion packet with Ω = {K} of order 1. (2) z = z(K) ∈ / Z(F). Notation 7.1.2. Let Zin , Zout consist of those s ∈ Z inducing a nontrivial inner, outer automorphism on K, respectively. Set Fz = CF (z). Let η ∈ η(τ ) be Sinvariant and set W = η . Let GW be a model for NF (CS (W )) and M = K GW . Let O = OK , μ = μ(τ ), V = K ∩ W , and let v be an element of order 4 in V . Lemma 7.1.3. O  Fz , so K is strongly closed in S with respect to Fz . Proof. This follows from 7.1.1.1 and 2.6.11.



Lemma 7.1.4. Either (1) μ ∼ = Z2 and W = V is of index 2 in K, or (2) μ ∼ = S3 and W is weakly closed in S with respect to F. Proof. If Δ = {K} then (1) holds, so assume otherwise. By 7.1.1.1 the graph D is disconnected, so μ ∼  = S3 by 4.2.13. Thus (2) holds by 3.3.14. Recall the subsystem E(τ ) from Definition 3.4.2. Lemma 7.1.5. E(τ ) = SO, NF (K), FS (SM ) . 173

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Proof. By Definition 3.4.2 and 7.1.1.1, E = E(τ ) = SO, NF (K), NF (W ) . If μ ∼ = Z2 then W = V is of index 2 in K by 7.1.4, so NF (W ) ≤ NF (K). If μ ∼ = S3 then by 3.3.9, FS∩M (M )  NF (W ), so as KW = S ∩ M and K is weakly closed in KW , we conclude that NF (W ) = FS (SM ), NNF (W ) (K)) from 1.3.2 in [Asc19]. This completes the proof.  Theorem 7.1.6. Assume z ∈ O2 (F). Then F ◦ is constrained, μ ∼ = S3 , and for G a model of F ◦ , either (1) G ∼ = L3 (2)/E8 , or (2) G = M is the split extension of W ∼ = Z2m/2 by S3 . Proof. By 7.1.1 and the assumption that z ∈ O2 (F), the hypotheses of Theorem 2 are satisfied by τ ◦ . Then the lemma follows from that theorem.  Theorem 7.1.7. Assume z ∈ / O2 (F) and μ ∼ = Z2 . Then F ◦ ∼ = L2 [2m](1) or

L− 3 [m].

∼ Z2 , SM = S by 7.1.4, so E = E(τ ) = SO, NF (K) by Proof. As μ = 7.1.5. In particular, z ∈ Z(E), so E = F by 7.1.1.2. Therefore by 3.4.8 and 3.4.6, Z = {z} ∪ tK with T = Kt ∼ = B ≤ T we have = SD2m , and for E4 ∼ AutF (B) = Aut(B). Also as μ ∼ = Z2 , Δ = {K}, so xF ⊆ K for each x ∈ K − {z}. Claim T is strongly closed in S with respect to F. As T is semidihedral and Z = {z} ∪ tK , Z is the set of involutions in T . Let x ∈ T with 2 < |x| < m. Then x ∈ K and hence xF ⊆ K by the previous paragraph. Finally let Y = y ≤ T be of order m and φ ∈ homF (Y, S). Then Φ(Y ) ≤ K, so Φ(Y )φ ≤ K and hence zφ = z, so φ is an Fz -map. But T = KZ , so T is weakly closed in S. Then as K is strongly closed in S with respect to Fz , T /K is weakly closed in S/K with respect to Fz /K by 8.9.1 in [Asc08a], so T is strongly closed in S with respect to Fz . Therefore Y φ ≤ T as φ is an Fz -map, completing the proof of the claim. Let Q8 ∼ = Q ≤ T and B = Aut(B), AutF (Q) T . Observe AutF (Q) = Aut(Q) if O = FK (K) or m = 8 and AutF (Q) is not a 2-group, while AutF (Q) = AutT (Q) otherwise. Therefore B is F-invariant by 2.1.4. By Example 2.1.3, B ∼ = L2 [2m](1) − or L3 [m]. In particular, B is saturated. Therefore B is weakly normal in F, so B  F by 1.2.6. Hence B = F ◦ .  Given Theorems 7.1.6 and 7.1.7, in the remainder of the section we assume: Hypothesis 7.1.8. Hypothesis 7.1.1 holds with z ∈ / O2 (F) and μ ∼ = S3 . Lemma 7.1.9. F = E(τ ). Proof. If not then by 3.4.8 and 3.4.6, Zin = ∅, contrary to μ ∼ = S3 in 7.1.8.



Notation 7.1.10. Set E = Ω1 (W ) and let t ∈ E − z . Lemma 7.1.11. (1) KW ∈ Syl2 (M ) and KW is the wreath product of Zm/2 by Z2 . (2) If s = kc ∈ Zin with k ∈ K and c ∈ CS (K) then there exists γ ∈ homFz (s, k , W ).

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(3) Fz is transitive on Zin . (4) M is determined up to isomorphism. (5) For each y ∈ CS (E), there exists an integer n(y) such that cy : W → W is the map w → wn(y) . (6) W  NF (E). Proof. Parts (1) and (4) follow from 4.2.5. Choose s as in (2). Then CK (s) = U = CK (k) is cyclic of order at least 4. Let β ∈ A(s). Then sβ = z and U β ≤ D ∈ Δ(zβ), so there is δ ∈ AutF (W ) with Dδ = V . Then γ = βδ ∈ homFz (U s , W ), so that (2) holds. Part (3) follows from (1) and (2). By 7.1.6 there is a complement X to W in M . Let y ∈ CS (E). Then y acts on V , so y acts as u → un for some integer n. Then if y centralizes X, y also has this action on W as W = V × V x for suitable x ∈ X. Finally CS (E) = W × CS (X) by a Frattini argument, so y = ws for some w ∈ W and s ∈ CS (X), and y and s have the same action on W , completing the proof of (5). Finally V = CK (E)  CFz (E) = CF (E), so W = V M  CF (E). Now (6) follows from 7.1.4.  Lemma 7.1.12. If s ∈ Zout then s inverts W . Proof. As s ∈ Zout , s inverts V by 3.3.6, so if s centralizes E then s inverts W by 7.1.11.5, and hence the lemma holds in this case. So assume [E, s] = 1. Then for k ∈ K − V , y = sk centralizes E and y 2 generates V . Thus by 7.1.11.5, y centralizes W , so W y = CM y (E)  M y , a contradiction as z = (W y )m while M is transitive on E # .  Lemma 7.1.13. Assume s ∈ Z − E centralizes E and set A = Es . Then (1) s inverts W . (2) One of the following holds: (i) s ∈ Zin , AutF (A) = CGL(A) (z), NK (A)A ∼ = Q28 is normal in NF (A), and / V but kFz ∩ V = ∅. writing s = kc with k ∈ K and c ∈ CS (K), k ∈ (ii) s ∈ Zin , AutF (A) = GL(A), and writing s = kc with k ∈ K and c ∈ / V but kFz ∩ V = ∅. CS (K), k ∈ (iii) s ∈ Zout , AutF (A) = CGL(E) (e) for some e ∈ {t, tz}, and for h ∈ M with h e = z, the pair sh , Ah is described in (i). (3) A# ⊆ Z. Proof. Let α ∈ A(s) with sα = z. By 3.3.6, s inverts or centralizes V , so by 7.1.11.5, s inverts or centralizes W . But if s centralizes W then W α = W by 7.1.4.2, contradicting s ∈ / W . Thus (1) holds. Observe (2) implies (3), so it remains to prove (2). By (1), AutW (A) is the group of transvections on A with axis E. Further Aα acts on K with Eα faithful on K by 7.1.1.1 and as z ∈ / Eα, so ζ = cv α−∗ ∈ Aut(A) is a transvection with center s . Therefore B = ζ, AutW (A) is the stabilizer in GL(A) of some e ∈ E # . / V by (1) but kα ∈ D ∈ Δ − {K}, so there Suppose s = kc ∈ Zin . Then k ∈ exists β ∈ A(kα) with Dαβ = V . Then γ = αβ is a Fz -map with kγ ∈ V , so kFz ∩ V = ∅. Further k induces a transvection ξ on A with center z such that ξ / B and (2ii) centralizes s, so either ξ ∈ B and CE (ξ) = z , so that e = z, or ξ ∈ holds. In the first case, (2i) or (2ii) holds.

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Thus we may assume s ∈ Zout . If B centralizes z then s = [E, ζ] ≤ KCS (K), contradicting s ∈ Zout . Therefore we may assume B is the stabilizer of t in GL(A). Now there is h ∈ M with th = z; replacing s, A by sh , Ah we are in the case B = CGL(A) (z). As B is transitive on A − z and t ∈ Zin , also sh ∈ Zin , so (2i) holds after conjugation by h. Thus A satisfies (2iii), completing the proof.  Lemma 7.1.14. If E8 ∼ = A ≤ S with A# ⊆ Z then CZ (A) = A# , so z ∈ A. Proof. Suppose r ∈ CZ (A) − A. Let γ ∈ A(r) and B = Aγ. As B # ⊆ Z and z ∈ / B, B is faithful on K by 7.1.1.1. Now Bv = CB (v) is of index at most 2 in B and as B # ⊆ Z, Bv induces a group of inner automorphism on K. This is a  contradiction as CK (v) = V is cyclic. Lemma 7.1.15. Zin = E # . Proof. Assume otherwise. Then if s ∈ Z − E, we have s ∈ Zout . Hence by 7.1.12, s centralizes E, so by 7.1.13.2, Zin is not contained in E, a contradiction. Therefore Z = E # , so E is a strongly closed abelian subgroup of S, and hence E  F by 14.1 in [Asc11], contrary to 7.1.8.  Lemma 7.1.16. Set WC = CW (K). Then (1) WC is cyclic of order m/2 with z = WC ∩ V . (2) WC ≤ CS (O). (3) CS (K) is strongly closed in S with respect to Fz . (4) Let P ≤ WC and φ ∈ homFz (P, S). Then if |P | > 4 or Zin ⊆ KW , then P φ ≤ WC . Proof. Let k ∈ K − V . By 7.1.11.1, KW is a wreath product so V = [W, k] is inverted by k and V ∼ = CW (k) = WC , so (1) holds. From 2.6.12, C = CF (K) = CF (O), so (2) follows and C  Fz . Thus (3) holds. Assume the setup of (4). By (2), P φ ≤ CS (K). As WC is cyclic and normal in S, so is P , so P ∈ Fzf , and hence there is α ∈ AFz (P φ) with P φα = P . As P φ ≤ CS (K), K ≤ CS (P φ), so as K is weakly closed in S, Kα = K. Further P V = V × P  where P  ≤ D ∈ Δ(t) is of order |P |/2, so if |P | > 4 then P  α−1 ≤ W as W = Δ − {K} . Hence P φ = P α−1 ≤ (P V )α−1 = (P  V )α−1 ≤ KW. Therefore if |P | > 4 then P φ ≤ CKW (K) = WC . Similarly if |P | = 4 and Z ⊆ KW then P φ ≤ (KZ ∩KP )α−1 ≤ KW , so P φ ≤ WC , completing the proof of (4).  Lemma 7.1.17. Assume m = 8 and set S0 = KCS (K) and T = KW . Let G0 be a model for N = NF (S0 ), and H0 = T G0 . (1) If WC  G0 then T ∈ Syl2 (H0 ) and either (i) AutF (K) is a 2-group and T  G0 , or (ii) AutF (K) ∼ = S4 , L = O 2 (H0 ) ∼ = SL2 (3) with K = O2 (L) and H0 = Lx where x ∈ T of order 8 induces an involutory outer automorphism on L in P GL2 (3). (2) Set K0 = WCG0 and assume K0 = WC . Then K0 ∼ = Q8 , T0 = T K0 ∈ Syl2 (H0 ), and T0 = T a , where a is an involution inverting W in the center of the group CS (X0 ) for X0 a complement to W in M . Moreover one of the following holds: (i) AutF (K) is a 2-group and H0 = T1 L0 where L0 = O 2 (H0 ) ∼ = SL2 (3), K0 = O2 (L0 ), and T1 is semidihedral of order 16 with K = CT1 (L0 ) and H0 /K ∼ = P GL2 (3).

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∼ S4 , O 2 (H0 ) = L ∗ L0 where L = ∼ L0 ∼ (ii) AutF (K) = = SL2 (3), K = O2 (L), K0 = O2 (L0 ), and H0 = O 2 (H0 )W with y ∈ W − O 2 (H0 ) inducing an outer automorphism in P GL2 (3) on L and L0 . (iii) AutF (K) ∼ = S4 , O2 (H0 ) = KK0 ∼ = S3 , and A = = Q28 , H0 /O2 (H0 ) ∼ G0 ∼ # E = E8 with A ⊆ Z. Proof. As F is saturated, AutF (K) = AutN (K) = AutG0 (K). Observe that S = S0 W . As K is weakly closed in S, K is normal in G0 and then also CS (K)  G0 . Set X = NG0 (V ) ∩ NG0 (WC ). Then E = Ω1 (V WC )  X, so W  X by 7.1.11.6. Thus T = KW  X. In particular in (1), if AutF (K) is a 2-group then (1i) holds. Next let Y = NG0 (WC ), set HY = T Y , and assume Y = X. Then AutY (K) ∼ = S4 . Set Y ∗ = Y /WC and observe that [W, CS (K)] ≤ CW (K) = WC , so [W , CS (K) ] = 1. Further K ∗ W ∗ ∼ = D8 and as S = S0 W we have [S ∗ , W ∗ ] = ∗ ∗ ∗ ∼ [K , W ] = V . As AutY (K) = S4 and O2 (Y ∗ ) = S0∗ is of index 2 in S ∗ , it follows ∗ that HY∗ = K ∗ W ∗Y ∼ = S4 . Thus HY satisfies the conclusions of (1ii), completing the proof of (1). ¯ ¯ 0 = G0 /z , I0 = CG (K), So assume the hypotheses and notation of (2). Set G 0 and I = CG0 (K). Observe I0 = KI with CS (K) Sylow and normal in I, so S0 = ¯ C ≤ Z(S¯0 ) of KCS (K) is Sylow in I0 . Further [T, CS (K)] ≤ CT (K) = WC with W ¯ 0 = W ¯ G0 ≤ Z(S¯0 ). By hypothesis, K0 = WC , so K ¯ 0 = W ¯ C. order 2, so K C ¯ 0 ), and let t1 be an involution in T − W . Then T ! = Set G!0 = G0 /CG0 (K ¯ 0] ≤ W ¯ C . If t!1 centralizes K ¯ 0 then T = KW ≤ I0 , t!1 is of order 2 and [t!1 , K so G0 = I0 NG0 (T ) as T is weakly closed in S. But now WC  G0 , contrary ¯ 0 with center W ¯ C . Further to assumption. Thus t!1 induces a transvection on K !G0 ! ¯ ¯ ¯ H0 = t1 . As K0 ≤ Z(S0 ), S0 ≤ CG0 (K0 ), so I/S0 , and hence also I ! is of odd order. Next G0 /I = AutG0 (K) = AutH0 (K) is either Aut(K) ∼ = S4 or a Sylow ˆ 0 = G0 /S0 = tˆ1 O(G ˆ 0 ) with |O(G ˆ 0 ) : I| ˆ =1 group T /WC of Aut(K). Therefore G ! ! ! ! ¯ ¯ or 3. Hence G0 = t1 O(G0 ), and as t1 is a transvection on K0 = WC , we conclude ¯ 0 ) = 2. Then as WC ∼ that H0! = G!0 ∼ = S3 and m(K = Z4 , we have K0 ∼ = Q8 and 2 ∼ K1 = KK0 = Q8 . ¯ 1 , S0 = K1 S1 , where S1 = CS (K1 )  G0 . Now [T, S1 ] ≤ As S0 centralizes K CT (K1 ) = z , so T centralizes S¯1 . + Set G+ is of order 2 and as T is weakly closed in S with re0 = G0 /K1 . Then T spect to G0 , it follows from Glauberman’s Z ∗ -Theorem that H0+ = T + O(H0+ ). Further as T centralizes S¯1 , so does H0 , so O 2 (CH0 (K1 )) = O 2 (CH0 (S0 )) ≤ O 2 (S0 ) = 1. Therefore CH0 (K1 ) = z . Hence H0+ ≤ Out(K1 ) ∼ = O4+ (2). In particular one of the three cases listed in (2) hold. For example if AutF (K) is a 2-group take T1 = Kt1 ; as t1 induces an outer automorphism on K, T1 is semidihedral. Finally for w a generator for WC , t = wv, so for j ∈ K0 − WC , [t, j] = [w, j] = z, so for k ∈ K − V , b = kj is an involution inverting V WC . Hence by 7.1.11.5, b inverts W . Now a complement X0 to W in M acts on b W = CT0 M (E) and Cb W (X0 ) = a with a an involution inverting W . Further [a, CS (X)] ≤ CT0 (M ) = 1, completing the proof of (2).  ∗

∗

Theorem 7.1.18. Assume Zin ⊆ KW . Then S = KW is wreathed and F ◦ ∼ =

L+ 3 [m].

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Proof. Let T = KW . By 7.1.16.4, WC is strongly closed in S with respect to Fz , so by 14.1 in [Asc11], WC  Fz . Thus WC O  Fz . By 7.1.15 there is s ∈ Zin − E, but by hypothesis, Zin ⊆ KWC . Then as W is transitive on the involutions in T − W since T is a wreath product, it follows that E # ∪ sW is the set I of involutions in T and I = {z} ∪ Zin . Further Fz is transitive on I − {z} by 7.1.11.3, so V is not normal in Fz since E centralizes V but s inverts V . Therefore O ∼ = SL2 [m] if m > 8, while AutF (K) ∼ = S4 if m = 8. Assume for the moment that m = 8 and adopt the notation of 7.1.17. It follows from 7.1.17.1 that L = O 2 (H0 ) ∼ = SL2 (3) with K = O2 (L) and H0 = Lx where x ∈ T of order 8 induces an involutory outer automorphism on L in P GL2 (3). Set H = FT (H0 ). Observe that the hypotheses of 2.1.5 are satisfied by Fz , T, KWC , W in the role of “F, T, Q, W ”. For example O 2 (Aut(K)) ∼ = A4 ∼ = O 2 (AutH0 (K)) and N = 2 NFz (W ) = CNF (W ) (z), so O (AutN (T )) centralizes E, and hence also W , and centralizes T /W of order 2, so the group is indeed trivial. Therefore H  Fz by 2.1.5. Next assume m > 8. In this case set H = T O. We claim H  Fz in this case too. For as m > 8, T = NH (KWC ). Next as T is weakly closed in S with respect to Fz and T /KWC is of order 2, T is strongly closed in NFz (KWC ), and hence T  NFz (KWC ) by 14.1 in [Asc19]. Hence the claim follows from Theorem 1.5.2 in [Asc19]. Note H ∼ = SL2 [m](1) in both cases, as by 2.1.2: (*) H = O 2 (Aut(QWC )) T , where Q is any Q8 -subgroup of K. This follows since, up to conjugation in H, QWC is the unique member of Hrc in D(H). Next we claim K is the unique quaternion subgroup of its order in T . Namely suppose K1 is a second such subgroup. As T /E is wreathed, it is transitive on it involutions in T /E − W/E, so as K  T , all elements of order 4 in T − W are in K. As K and K1 are generated by such elements, the claim follows. By (*) and the second claim, it follows from 2.1.2 that Aut(H) = Aut(T ). Let E = H, FT (M ) . We next claim that E is determined up to isomorphism. We just saw H is determined, and FT (M ) is determined by 7.1.11.4. Hence as Aut(H) = Aut(T ), the claim follows from 1.6.8 in [Asc19]. In particular as L+ 3 [m] [m], so E is saturated. has such a fusion system, E ∼ = SL+ 3 From 7.1.5 and 7.1.9, F = Fz , FS (SM ) . Then by reductions above, Hypotheses 1.4.7 in [Asc19] is satisfied by F, E, and the families (Fz , FS (SM )) and (H, FT (M )). Therefore by 1.4.8 in [Asc19], E  F. But then F ◦ = E, completing the proof.  Lemma 7.1.19. Let ZV = (Z − E) ∩ V CS (K) and assume s ∈ ZV . Then (1) s inverts WC and W is transitive on sWC . (2) ZV = sWC . (3) Let v ∈ V be of order 4 and set K0 = WC sv and Ω0 = {K0 }. Then τ0 = (F, Ω0 ) is a quaternion fusion packet. (4) If m > 8 then OK0 ∼ = SL2 [m]. (5) If m = 8 then K0 is Sylow in E0  Fz with E0 ∼ = SL2 [8]. Further if AutF (K) is not a 2-group then K is Sylow in E1  Fz with E1 ∼ = SL2 [8]. (6) For η0 ∈ η(τ0 ), η0 = W .

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(7) Set T = KW and T0 = T K0 . Then T0 = T a , where a is an involution inverting W in the center of CS (X0 ), for X0 a complement to W in M . (8) K and K0 are the quaternion subgroups of T0 of order m. Proof. As s ∈ ZV , s = vc with v ∈ V of order 4 and c ∈ CS (K) with c2 = z. If s centralizes E then conclusion (i) or (ii) of 7.1.13.2 holds by that lemma, so s = kc with k ∈ / V , a contradiction. Thus [t, c] = [t, s] = z. Let k ∈ K − V and i = kc. Then i is an involution centralizing E and inverting V , so i inverts W by 7.1.11.5. Therefore as k centralizes WC , c and s invert WC . Therefore K0 = WC c is quaternion of order m and D = WC s ∼ = Dm . Moreover as [t, s] = z, W s is wreathed, so as s inverts WC , WC = [W, s] and W is transitive on sWC . That is (1) holds. Suppose ZV = sWC . As K, V , and E are normal in S, and S acts on Z, S also acts on ZV . Thus there exists r ∈ (NS (D) ∩ ZV ) − D. Hence [s, r] ∈ WC . If [s, r] generates WC then setting f = sr, we have f 2 generating WC , so by 7.1.11.5, f centralizes W . But now z = (W f )m is characteristic in W f , a contradiction as M acts on CM f (W ) = W f and is transitive on E # . Therefore sr = sw for some w ∈ WC , so wr ∈ CS (s). But from (1), wr ∈ r W , so wr ∈ Z. Also r = vc1 for some c1 ∈ CS (K), so swr = vcwvc1 ∈ CS (K). By 7.1.11.3 there is α ∈ AFz (s) with sα = t. Then by 7.1.13.3, (wr, s, z α)# ⊆ Z, so swr ∈ Z, contradicting swr ∈ CS (K). Hence (2) holds. Claim K0 is strongly closed in S with respect to Fz . If not, by 7.1.16.4=AA.16.4 there is X = x of order 4 in K0 and φ ∈ homFz (X, S) with Xφ ≤ K0 . Observe that we can extend φ to KX. For by 1.3.2 in [Asc19], Fz = SO, NF (K) , and as X ≤ CS (O), each θ ∈ homSO (X, S) is induced by conjugation in S, while of course K  NF (K). By 7.1.16.3, Xφ ≤ CS (K). By (1), W is transitive on sWC , so as W centralizes v, also W is transitive on svWC . Then as x ∈ svWC ∪ WC , r = xv ∈ E ∪ sWC = E ∪ ZV by (2). Suppose V φ = V . Then rφ ∈ V (xφ) ≤ V CS (K), so rφ ∈ E ∪ ZV = E ∪ sWC by (2), so xφ = rφ(v −1 φ) ∈ vE ∪vSWC , and hence xφ ∈ WC ∪cWC = K0 , contrary to the choice of φ. Thus V φ = V , so in particular m = 8 and AutF (K) ∼ = S4 . Adopt the notation of 7.1.17 and let WC = w . We showed that NG0 (V ) acts on K0 ∼ = Q8 . Further by (2), (wc)v = ws = r0 ∈ Z, so by 7.1.11.2, there is γ ∈ homFz (r0 , v , W ). Thus γ acts on V and WC is generated by (r0 v −1 )γ = (wc)γ. As WC  S, we have WC ∈ F f , so there is α ∈ ANFz (V ) (wc ) with WC = (wc)α . We can choose α = cg for some g ∈ NG0 (V ) with (wc)g ∈ WC . N (V ) Therefore by 7.1.17.2, W G0 = K0 ∼ = Q8 . Thus conclusion (ii) of 7.1.17.2 C

NG (V ) holds as AutF (K) ∼ = S4 and WC 0 ∼ = Q8 . Hence there is l ∈ L with ϕ = cl φ fixing v and xϕ = xφ, so replacing φ by ϕ, we obtain the contradiction V ϕ = V . This completes the proof of the claim. Further (3) follows from the claim. Again by 7.1.11.2 applied to s = vc, cFz ∩ WC = ∅, so AutF (K0 ) is transitive on its elements of order 4. Then (4) follows from (3) and 2.5.4 and 2.5.7 applied to K0 . So assume m = 8. Then K0  Fz and by another application of 7.1.11.2, there exists γ ∈ AutF (KK0 ) with vγ = v and cγ = w a generator of WC . Again the hypotheses of 7.1.17.2 are satisfied with γ induced by some g in the group G0 of that lemma. If follows that conclusion (i) or (ii) of the lemma is satisfied, so in particular there is SL2 (3) ∼ = L0  H0 with K0 = O2 (L0 ). Further L0 centralizes CS (K0 ), and therefore also CFz (K0 ) by 2.3 in [Asc11]. Thus E0 = FK0 (L0 )  Fz . Similarly

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∼ S4 then conclusion (ii) of 7.1.17.2 holds and E1 = FK (L)  Fz , if AutF (K) = completing the proof of (5). Observe WC is the member of η(τ0 ) containing z, so as W = WCM , (6) follows from 3.4.3.1 applied to the quaternion fusion packet τ0 . The proof of (7) is the same as that of the corresponding statement in 7.1.17, appearing in the last paragraph of the proof of 7.1.17. We next prove (8). Suppose Q is a quaternion subgroup of T0 of order m. As W is abelian, Q ≤ W and Q ∩ W is cyclic. From (7), all members of aW are involutions, so Q ∩ a W = Q ∩ W . From (7), T0 /W ∼ = E4 , so |Q : Q ∩ W | = 2 and QW = KW or K0 W . Thus we may take QW = KW , and now (8) follows from the step in the proof of 7.1.18 showing K is the unique quaternion subgroup of KW of order m.  Notation 7.1.20. Suppose ZV = ∅ and define K0 and Ω0 as in 7.1.19.3. If m > 8 define E0 = OK0 and E1 = OK . If m = 8 define E0 as in 7.1.19.5, set E1 = K if AutF (K) is a 2-group, and if AutF (K) is not a 2-group, define E1 as in 7.1.19.5. Thus in any case K0 is Sylow in E0 ∼ = SL2 [m], K is Sylow in E1 ∼ = SL2 [m] or K, and E0 and E1 are normal in Fz . Also define T0 = KK0 W as in 7.1.19.7 and set M0 = T0 M and H = T0 E0 E1 . Sometimes we partition the proof into two cases: In Case I, E1 ∼ = SL2 [m], while in Case II, E1 = K. Lemma 7.1.21. Assume ZV = ∅ and let L = G2 (q) for some odd prime power q with (q 2 − 1)2 = m. Then (1) T0 is isomorphic to a Sylow 2-subgroup of L, so we may take T0 ∈ Syl2 (L). Set F(L) = FT0 (L). (2) τL = (F(L), Ω) is a quaternion fusion packet with ZV = ∅ appearing in Case I, and F(L) ∼ = G2 [m]. (3) We may choose q to be a power of 3. In that event there is a graph-field automorphism γ of L acting on T0 inducing an automorphism of F(L) interchanging K and K0 . (4) T0 ≤ X ≤ L with X the extension of SL3 (q) by a graph automorphism, where q ≡ mod 4. Further, choosing K0 Sylow in a fundamental subgroup of L, τX = (F(X), Ω) is a quaternion fusion packet with ZV = ∅, K  CF (X) (z), and AutF (X) (K) a 2-group, so the packet appears in Case II. Moreover F(X) ∼ = (1) L+ . 3 [m] Proof. Let TL ∈ Syl2 (L) and zL = Z(TL ). Then O 2 (CL (zL )) = L0 ∗L1 with ∼ Li = SL2 (q) normal in CL (zL ), so by 2.4.1, τL = (FTL (L), {KL }) is a quaternion fusion packet, where KL ∈ Syl2 (L1 ). As L has one class of involutions, ZV = ∅ in this example, and as Li ∼ = SL2 (q), the packet appears in Case I. By 7.1.11.4 and 7.1.19.7, TL is determined up to isomorphism by these conditions, so T0 ∼ = TL . Thus we have proved (1) and (2). Observe for r ≡ 1 mod 4, (r 2 − 1)2 = (r + 1)2 · (r − 1)2 = 2(r − 1)2 . Thus k 2k+1 − 1)2 = (92 − 1) = 2k · 8, proving the first statement in (3). From 2.5.13 (3 in [GLS98], L admits a graph-field automorphism γ which we may choose to be a 2-element, and hence acting on T0 by Sylow’s Theorem. From the discussion in [GLS98], γ interchanges long and short root subgroups of L, and hence K and K0 , completing the proof of (3).

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Next from Example 2.6.3 in [GLS98] (and with more detail in [Asc87]), the subgroup X of (4) exists. As K0 is Sylow in a fundamental subgroup of X, K  CF (X) (z) and AutF (X) (K) is a 2-group. By 2.5.2, τX = (F(X), {K}) is a quaternion fusion packet, and as E(X) has one class of involutions, ZV = ∅ (1) in this example. By definition, F(X) ∼ , completing the proof of the = L+ 3 [m] lemma.  Lemma 7.1.22. Assume ZV = ∅. (1) FT0 (M0 )  NF (E). (2) T0 = CM0 (z)  NFz (E). (3) AutFz (T0 ) is a 2-group. (4) H  Fz . (5) In Case I, H ∼ = CF (L) (z) and Aut(T0 ) = Aut(H). (6) In Case II, H ∼ = CF (X) (z) and Aut(T0 ) = Aut(H)Aut(FT0 (M0 )). (7) Let E = H, FT0 (M0 ) . Then E is isomorphic to G2 [m] in Case I, and to L3 [m](1) in Case II. Proof. Recall from 7.1.11 that T ∈ Syl2 (M ) and FT (M )  NF (E) = M. As z = Z(T ), NM (T ) ≤ CM (z). As W  M and E0 and E1 are normal in Fz , we have KK0  CM (z), so T0 = KK0 W  CM (z), proving (2). Then (1) follows from Theorem 1.5.2 in [Asc19]. Set A = O 2 (AutFz (T0 ). By 7.1.4.2, A acts on W . Then as A centralizes z, A also centralizes E, so A centralizes W (cf. 24.3 in [Asc86]). Also A acts on K and K0 and centralizes K ∩ W and K0 ∩ W , so A centralizes K and K0 . Therefore A centralizes KK0 W = T0 , establishing (3). Assume for the moment that m = 8. Claim the hypotheses of 2.1.5 are satisfied on T0 in Fz with Q = KK0 . For example hypothesis (b) follows from 7.1.17.2 with O 2 (AutH0 (Q)) = O 2 (Aut(Q)) in Case I and isomorphic to A4 in Case II. Hypothesis (c) follows from (3). Then (4) follows when m = 8 from the claim and 2.1.5, which also says that Q is the greatest member of D(T0 , Fz ) and H = O 2 (AutF (Q)) as z  NF (Q). Next suppose that m > 8. Then NFz (Q) ≤ NFz (E), so T0  NFz (Q) by (2). Hence (4) follows from 1.5.2 in [Asc19] when m > 8. This completes the proof of (4). We next prove (5) and (6). Suppose m = 8. We saw Q is the greatest member of D(T0 , Fz ) and D = O 2 (AutF (Q)). In Case I, O 2 (AutF (Q)) = O 2 (Aut(Q)), so H is determined up to isomorphism, and hence is isomorphic to CF (L) (z) by 7.1.21.2. Moreover Aut(T0 ) = Aut(H) by 2.1.2. Similarly in Case II, given 7.1.19.8, 2.1.2 says that Aut(D) = NAut(T0 ) (K) is of index 2 in Aut(T0 ), so by 7.1.21.4, D∼ = CF (X) (z) or CF (X) (z)ρ where ρ is the automorphism of F(L) in 7.1.21.3. As ρ ∈ Aut(FT0 (M0 )), we have Aut(T0 ) = Aut(H)Aut(FT0 (M0 )). This completes the proof of (5) and (6) when m = 8. So assume m > 8. This time in Case I a set U of representatives for the H-orbits on Hrc are {KQ0 , QK0 , QQ0 }, where Q, Q0 are Q8 -subgroups of K, K0 , respectively. By 2.1.2, H = O 2 (Aut(U )) : U ∈ U , so once again H ∼ = CF (L) (z) by 7.1.21.2 and Aut(T0 ) = Aut(H) by 2.1.2 and 7.1.19.8. This completes the proof of (5). Finally in Case II, U = {KQ0 }, and then arguing as in the case m = 8, (6) also holds when m > 8.

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Finally we prove (7) via an appeal to 1.6.8 in [Asc19], applied to E in the role of F, H in the role of F1 , and FT0 (M0 ) in the role of F2 . By definition, E = H, FT0 (M0 ) , so Hypothesis 1.6.1 of [Asc19] is satisfied and E is a member of the set Λ defined there. By 7.1.11.4 and 7.1.17.2, FT0 (M0 ) is determined up to isomorphism, while by (5) and (6), H is also determined up to isomorphism. Further by 7.1.21, F(L), F(X) is in Λ in Case I, Case II, respectively. Thus it suffices to show all members of Λ are isomorphic, which is the conclusion of 1.6.8 in [Asc19] under either of its two hypotheses. Thus it remains to observe that one of those hypotheses does indeed hold by (5) and (6).  Theorem 7.1.23. Assume ZV = ∅. Then either (1) Case I holds and F ◦ ∼ = G2 [m], or (1) (2) Case II holds and F ◦ ∼ . = L+ 3 [m] Proof. We will show that Hypothesis 1.4.7 in [Asc19] is satisfied by F, E, and the families (Fz , M) and (H, FT0 (M0 )). Then by Lemma 1.4.8 in [Asc19] and 1.2.6, E  F. But then F ◦ = E, so that 7.1.22.7 completes the proof. So it remains to verify Hypothesis 1.4.7 in [Asc19]. By 7.1.5 and 7.1.9, F = Fz , M , so Hypothesis 1.4.1 in [Asc19] is satisfied. By 7.1.22.4, H  Fz and by 7.1.21.1, FT0 (M0 )  M. As E and z are characteristic in T0 , AutF (T0 ) = AutM (T0 ) = AutFz (T0 ), so AutF (T0 ) is Aut(H) and Aut(FT0 (M0 )) by parts (1) and (4) of 7.1.22. By construction, E = H, FT0 (M0 ) . By 7.1.22.7, E is saturated. This completes our verification of Hypothesis 1.4.7 in [Asc19] and hence also the proof of the lemma.  Because of 7.1.18, and 7.1.23, in the remainder of the section we assume: Hypothesis 7.1.24. Hypothesis 7.1.8 holds with Zin not contained in KW and ZV = ∅. In addition we adopt the following notation: Notation 7.1.25. Set S0 = KCS (K) and T = KW . Let G0 be a model for NF (S0 ), H0 = T G0 , K0 = WCG0 , Q = KK0 , and T0 = T K0 . Let X0 be a complement to W in M . Set H0 = FT0 (M0 ), H = FT0 (H0 ), and E = H, H0 . Lemma 7.1.26. (1) WC  CF (K). (2) V is not normal in Fz . (3) m = 8. (4) T0 ∈ Syl2 (H0 ) and T0 = T a , where a is an involution inverting W and centralizing X0 . Further K0 ∼ = Q28 , H0 /Q ∼ = S3 , and = Q8 , O2 (H0 ) = Q ∼ G0 ∼ A = E = E8 . (5) K0  Fz . (6) H = FT0 (H0 )  Fz . Proof. Suppose (1) fails and let P ≤ WC and β ∈ homCF (K) (P, CS (K)) with P β ≤ WC . As β is a CF (K)-map, β extends to ρ ∈ homFz (KP, S) centralizing K. Now t = vw with v ∈ V and w ∈ P , so tρ = vρwβ = v(wβ) ∈ V CS (K) − V WC , contradicting the condition in Hypothesis 7.1.24 that ZV = ∅. So (1) holds.

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By Hypothesis 7.1.24, Zin is not contained in KW , so there is s = kc ∈ Zin with k ∈ K and c ∈ CS (K) − WC . By 7.1.11.2 there is γ ∈ homFz (s, k , W ); in particular tγ −1 = s or sz. Suppose V  Fz . Then tγ −1 = vγ −1 wγ −1 ∈ V CS (K), so as tγ −1 ∈ {s, sz} and s = kc, it follows that k ∈ V . Hence s ∈ V CS (K), so as ZV is empty it follows that s ∈ E, contradicting the choice of s. This proves (2). Suppose m > 8. Then by (2) and as W interchanges the two classes of Q8 subgroups of K, O ∼ = SL2 [m], = SL2 [m]. From 2.6.12, CS (K) = CS (O), so as O ∼ there is ψ ∈ AutFz (Kc ) centralizing c with kψ ∈ V . But then sψ = kψcψ = kψc ∈ ZV , a contradiction. This establishes (3). By (3), KCF (K)  Fz , so γ = ϕφ with ϕ ∈ AutFz (S0 ) and φ ∈ homKCF (K) (s, k ϕ, W ). By (1), V WC  KCF (K), so s, k ϕ ≤ V WC . In particular cϕ ∈ CW (K) = WC , so WC ϕ = WC . But ϕ = cg|s,k for some g ∈ G0 , so WC is not normal in G0 ; that is K0 = WC , so the hypotheses of 7.1.17.2 are satisfied. Then by (1) and (2), conclusion (iii) of that lemma holds, establishing (4). N (S ) By (1), WC Fz = WC Fz 0 = WCG0 , so as K0  G0 it follows that K0 = WCFz , so (5) holds. Next from (1)-(5), the hypotheses of 2.1.5 are satisfied with Fz , T0 , KK0 , W in the roles of “F, T, Q, W ”. It follows from 2.1.5 that (6) holds.  Lemma 7.1.27. (1) H and H0 are determined up to isomorphism. (2) H0  NF (E). (3) Aut(H0 ) ≤ Aut(H) and Aut(H) is of index 2 in Aut(T0 ). (4) Either (i) a ∈ A, A# = z E , and E = FT0 (L) ∼ = L3 (2)/E8 , where L is a nonsplit extension of E by L3 (2) with Sylow group T0 , or (ii) a ∈ / A and E ∼ = M12 . Proof. First M is determined up to isomorphism by 7.1.11.4. Then M a is determined as a inverts W and centralizes X0 . Thus H0 is determined. Then the proof of 7.1.22.1 establishes (2). ˜ 0 = H0 /z and H + = H0 /Q. Recall H = FT (H0 ). As there exists an Let H 0 0 involution in aW −Q, H0 splits over Q. Further H0+ ≤ Out(Q) ∼ = O4+ (2), and H0+ is determined up to conjugacy as an S3 -subgroup acting on each of the Q8 -subgroups K and K0 of Q, and with CQ˜ (O(H0+ )) = 1. Therefore H0 , and hence also H, is determined up to isomorphism, completing the proof of (1). ˜ = J(T˜0 ), so Q is characteristic in T0 . Let D = Aut(T0 ) and B = Observe Q Aut(Q). Then [CD (Q), T0 ] ≤ CT0 (Q) = z , so CD (Q) = α where yα = yz for y ∈ T0 − Q. Observe α extends to α ˆ ∈ Aut(H0 ), where α ˆ centralizes O 2 (H0 ) and 2 hα ˆ = hz for h ∈ H0 − O (H0 ). Next the restriction ζ : β → β|Q is a homomorphism of D into B with kernel α . Let c : H0 → B be the conjugation map. Then Inn(T0 )ζ = T0 c is of order 25 with NB (T0 c) = B0 a Sylow group of B. Further B1 = NB0 (H0 c) is of index 2 in B0 . We conclude that |D : Aut(H)| = 2. We now apply results in section 1.6 of [Asc19] to E, H, H0 , T0 in the roles of “F, F1 , F2 , S” in Hypothesis 1.6.1 of [Asc19]. By the previous paragraph, A1 = Aut(H) is of index 2 in D = Aut(T0 ), so either A2 = Aut(H0 ) ≤ A1 and Δ =

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A1 \D/A2 = {A1 , A1 γ} for γ ∈ D −A1 , or A2 ≤ A1 , so that D = A1 A2 and |Δ| = 1. In the second case, E is determined up to isomorphism by 1.6.8 in [Asc19], while in the first case there are at most two possibilities for E by 1.6.5.4 and 1.6.6 in [Asc19]. Thus to complete the proof of (3), it suffices to exhibit two nonisomorphic systems in the set Λ of 1.6.2 of [Asc19]. Hence (4) implies (3). ¯ = G2 (3) (cf. [Asc87]) there is a subgroup Y which is an extension of In G ¯ which we may take to be T0 by E8 by L3 (2) and contains a Sylow 2-group of G, 7.1.21.1. Then by 2.5.2, τY = (FT0 (Y ), {K}) is a quaternion fusion packet satisfying Hypothesis 7.1.24, supplying our first example. ˆ be M12 and Sˆ a Sylow 2-subgroup of G. ˆ Then for K ˆ a Q8 -subgroup Next let G ˆ ˆ {K}) ˆ is a quaternion fusion packet satisfying Hyτˆ = (FSˆ (G), of O2 (CGˆ (Z(S))), pothesis 7.1.24, supplying us with our second example. This completes the proof of (4) and the lemma.  Theorem 7.1.28. Assume Hypothesis 7.1.24. Then F ◦ ∼ = M12 . Proof. We apply 1.4.8 in [Asc19] to F and the families (Fz , M), (H, H0 ) of Hypothesis 1.4.1 in [Asc19] to show that E  F. Then F ◦ = E. Now by 7.1.27.4, either z ∈ O2 (F) or the lemma holds, and by 7.1.8 it is the latter. By 7.1.5 and 7.1.9, F = Fz , M , so Hypothesis 1.4.1 in [Asc19] is indeed satisfied. By 7.1.26.6 and 7.1.27.2, 1.4.7.1 in [Asc10] is satisfied. As z and M are characteristic in T0 , 1.4.7.2 holds. By definition, E = D, H0 , so 1.4.7.3 holds. By 7.1.27.4, 1.4.7.4 holds. Thus we have verified Hypothesis 1.4.7 in [Asc19]. Therefore E  F by 1.2.6 and 1.4.8 in [Asc19], completing the proof.  Theorem 7.1.29. Assume Hypothesis 7.1.1 with z ∈ / O2 (F). Then F ◦ is iso+ morphic to one of L2 [2m](1) , L3 [m], L3 [m](1) , G2 [m], or M12 . Proof. By Theorem 7.1.7, we may assume that μ ∼ = S3 , so Hypothesis 7.1.8 is satisfied. Then by Theorems 7.1.18 and 7.1.23, we may assume Hypothesis 7.1.24 holds. But now Theorem 7.1.28 completes the proof.  Remark 7.1.30. Observe that Theorems 7.1.6 and 7.1.29 imply Theorem 5. For the hypothesis of Theorem 5 is just Hypothesis 7.1.1 together with the condition that F = F ◦ , so we can appeal to the results in this section. If z ∈ O2 (F) then conclusion (1) or (2) of Theorem 5 holds by Theorem 7.1.6. Therefore we may assume z ∈ / O2 (F). Then Theorem 5 follows from Theorem 7.1.29. 7.2. A(z) = ∅ In this section we assume the following hypothesis: Hypothesis 7.2.1. τ = (F, Ω) is a quaternion fusion packet, K ∈ Ω, z = z(K) ∈ F f , and A(z) = ∅. ˜ is of order 2. Lemma 7.2.2. (1) Ω(z) = {K, K} (2) There exists a unique 4-subgroup E of S containing z such that A(z)∩ZΔ = E − z . (3) D∗ (z) − D(z) = A(z) = Z ∩ O(z) − {z}. (4) Fz is transitive on A(z).

7.2. A(z) = ∅

Proof. Part (1) is 3.2.2.1, while parts (2)-(4) appear in 3.2.4.

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The following theorem is the main result of this section: Theorem 7.2.3. Assume Hypothesis 7.2.1 and let F0 = [O(z)]F , Ω0 = K F ∪ K , and z F ∩ ZS = {zi : i ∈ I}. For i ∈ I, set Fi = [O(zi )]F0 and τi = (Fi , Ω(zi )). Then (1) For each i ∈ I, τi is a quaternion fusion packet with A(zi ) = ∅ and μ(τi ) ∼ = D12 .  (2) F0 = i∈I Fi is a direct product. (3) For each i ∈ I, one of the following holds: (i) Wi  Fi for Wi ∈ W (τi ). (ii) Fi ∼ = L3 (2)/E8 . (iii) Fi ∼ = L3 [m](1) . ∼ (iv) Fi = G2 [m]. (v) Fi ∼ = M12 . (4) Suppose Ω = Ω0 and set Ω∗ = Ω−Ω0 and F∗ = [Ω∗ ]F . Then F ◦ = F0 ×F∗ . ˜F

We prove Theorem 7.2.3 via a series of reductions. Assume the theorem is false and choose a counter example τ with τ minimal. Adopt the notation from Theorem 7.2.3. Lemma 7.2.4. Ω = Ω0 . Proof. Assume otherwise. By 2.5.2, τ0 = (F, Ω0 ) is a quaternion fusion packet, and of course τ0 satisfies Hypothesis 7.2.1, so by minimality of |Ω|, τ0 satisfies the conclusions of Theorem 7.2.3. Hence τ also satisfies conclusions (1)-(3) of the theorem. By 2.5.2, τ∗ = (F∗ , Ω∗ ) is a quaternion fusion packet. Let J ∈ Ω∗ and t = z(J). By 3.1.5, J centralizes O(zi ) so J acts on the factor Fi of F0 . Let Ti ∈ Syl(Fi ). If t ∈ Ti then t ∈ CTi (O(zi )) = zi as 7.2.3.3 holds. But this contradicts the choice of t and 7.2.2.1. Therefore t ∈ / Ti , so J centralizes Ti by 3.1.4. Then by 2.3.1 in [Asc19], t centralizes Fi . As Ti centralizes J it centralizes O(t) by 2.6.12. Therefore Fi = [O(zi )]Fi ≤ CFt (O(t)). Thus as F0  F, we have F∗ = [Ω∗ ]F ≤ CF (F0 ), so (4) holds, contradicting the choice of τ as a counter example to Theorem 7.2.3.  Lemma7.2.5. (1) SetO = O(τ ), Oi = O(zi ), O = O(τ ), and Oi = O(zi ). Then O = i∈I Oi , O = I∈I Oi , and AutF (O) is transitive on {Oi : i ∈ I} and {Oi : i ∈ I}. (2) There is a unique η ∈ η(τ ). Let W = η , GW a model for NF (CS (W )), M = O(z)GW , and μ = μ(τ ).   M (3) Set i . Then M = i∈I Mi , W = i∈I Wi ,  Mi = Oi and Wi = W ∩ M and μ = i∈I μi , where μi = AutMi (Wi ) ∼ = D12 . ˜ F , and GW is transitive on {Mi : i ∈ I}. (4) NGW (O) is transitive on K F and K Proof. By 7.2.4 and 7.2.2.1, F is transitive on ZS = {zi : i ∈ I} and hence also on {Oi : i ∈ I}, so (1) follows from 3.3.7. Part (2) follows from the proof of 3.3.14. By (1) and as A(z) = ∅, 6.1.6.1 says that NGW (O) is transitive on K F and ˜ F . Now (3) follows from 4.2.10. Then (3) and the transitivity of GW on O(z)F K says GW is transitive on {Mi : i ∈ I}, completing the proof of the lemma. 

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Notation 7.2.6. Adopt the notation in 7.2.5. Set Ti = S ∩ Mi and Ei = ˜ ∩ W . Set Ω1 (Wi ). Take z = z1 and set T = T1 , Q = O1 , V = K ∩ W , and V˜ = K Fz = CF (z) and H = [T ]Fz . Lemma 7.2.7. F = F0 = [Q]F . Proof. Set Y = F ◦ . If F = Y then F = F0 = [Q]F by 7.2.4, so assume F = Y. By 2.5.2, ξ = (Y, Ω) is a quaternion fusion packet, and ξ satisfies Hypothesis 7.2.1, so by minimality of |τ |, ξ satisfies the conclusions of Theorem 7.2.3. Let  {zj : j ∈ J} = z Y ∩ ZS , Y0 = [Q]Y , and Yj = [Oj ]Y0 . Then Y0 = j∈J Yj with Yj satisfying 7.2.3.3. Now Y0 ≤ Y, and as F0  F, also F0  Y by 1.2.6. If z Y = z F it follows that Y0 = F0 , while if z Y = z F then as ξ satisfies 7.2.3.4, we have F0 = Y0 ×Y∗ . Further by minimality of |τ |, Y∗ satisfies conclusions (1)-(3) of 7.2.3, so τ satisfies those conclusions too. By 7.2.4, 7.2.3.4 holds vacuously. Thus we have a contradiction to the choice of τ as a counter example.  Lemma 7.2.8. (1) T = QW and |T : Q| = 2. (2) FT (M1 )  NF (E1 ) = NF (W1 ), and T = CM1 (z)  NFz (E1 ). (3) OK ∼ = SL2 [m] or OK = K. (4) T is Sylow in H and either (a) H = T O(z), or (b) m = 8, H is constrained with Q = O2 (H) ∼ = Q28 , and, taking H to be a model for H, H/Q ≤ Out(Q) is isomorphic to E3k extended by Z2 , where 3k = |AutF (Q)|3 . Proof. The group M1 is described in 4.2.6, from which (1) follows. Let E = E1 and M = NF (E). By 2.5.2, ξ = (M, Ω) is a quaternion fusion packet satisfying Hypothesis 7.2.1 by 7.2.5, so by minimality of |τ |, ξ satisfies the conclusion of Theorem 7.2.3. As E centralizes Oi for i > 1 but not Q, we conclude M1 = [Q]M◦  M. Next CO(z) (E) ≥ V V˜ is normal in CFz (E) by 8.23.2 in [Asc11]. Then as CF (E)  M, V V˜ ≤ O2 (M), so W1 = (V V˜ )M ≤ O2 (M). Hence as ξ satisfies 7.2.3, we get M1 = FT (M1 ). From the structure of M1 in 4.2.6, T = CM1 (z), completing the proof of (2). If m > 8 then from 4.2.6, W1 interchanges the two K-classes of Q8 -subgroups of K, so (3) holds in this case. Further E is characteristic in Q, so by (2), W and T are normal in NFz (Q). Therefore H = W O(z) = T O(z) by 1.5.2 in [Asc19], establishing (4) when m > 8. Thus we may assume that m = 8. Then OK = K, so once again (3) holds, completing the proof of (3). We now argue as in the proof of 7.1.17 to show that (4) holds. Let S0 = QCS (Q), G a model for NF (S0 ), and H = T G . Set G∗ = G/Q, G+ = G/z , and G! = OutG (Q). Now S acts on W by 7.2.5.2, so CS (z) acts on T and E by (2). Therefore [CS (Q), T ] ≤ CT (Q) = z , so [H, CS (Q)] ≤ z . Thus O 2 (H) ≤ CGz (CS (Q)), so a Sylow 3-subgroup X of H is faithful on Q. If G! is a 2-group then E  Fz , so H = T by (2). Thus we may assume G! is not a 2-group. As X is ˜ + , t! inverts O3 (G!z ), faithful on Q, X ∼ = X ! . As t ∈ T −Q is nontrivial on K + and K ! ! ! ! ∼ ! so X ≤ H , so H is X = E3k extended by the involution t , where k ∈ {1, 2}.

7.2. A(z) = ∅

187

Let Y = CFz (Q). By (2), Y acts on T and centralizes Q and T /Q, so O 2 (Y) centralizes T . Therefore FT (H) ≤ H ≤ CFz (O 2 (Y)) = X . But X is constrained, so H = FT (H), completing the proof of (4) and the lemma.  ˜ Theorem 7.2.9. Ω = {K, K}. We prove Theorem 7.2.9 via a series of reductions. Assume the theorem is false. Then |I| = n > 1. Lemma 7.2.10. (1) Fz◦ = O(z) × F2 × · · · × Fn , where Fi = [Oi ]Fz◦ and τi = (Fi , Ω(zi )) is a quaternion fusion packet. (2) Ti is Sylow in Fi and NFi (Ei ) = FTi (Mi ). (3) Let αi ∈ A(zi ) with zi αi = z. Then F1 = Fi αi∗ is independent of i and αi . (4) F1 = H, FT (M1 ) and τ1 = (F1 , Ω(z)} is a quaternion fusion packet appearing in one of the cases in 7.2.3.3. (5) F1 × · · · × Fn ≤ F. Proof. By 7.2.5.3, FTi (Mi ) ≤ Fz◦ for 1 < i ≤ n. Therefore the quaternion fusion packet τz = (Fz , Ω) satisfies Hypothesis 7.2.1, so by minimality of |F|, τ satisfies the conclusions of Theorem 7.2.3. In particular (1) and (2) hold. As Fz◦ satisfies 7.2.3, we conclude from 7.1.5 and 7.1.9 that Fi = Hi , FTi (Mi ) , where Hi = CFi (zi ). By 7.2.5.2, W αi = W , so by 7.2.5.4, Ti αi = T and FTi (Mi )αi∗ = FT (M1 ). As Hi = [Ti ]Hi we have Hi αi∗ ≤ H. Also by 7.2.5.1, Oi α∗ = O(z), so by 7.2.8.4, either Hi = Ti Oi and Hi α∗ = H or m = 8. In the latter case as AutF (Oi )α∗ = AutF (Q), again Hi α∗ = H. Therefore Fi αi∗ = H, FT (M1 ) , proving (3) and (4). Next T centralizes Ti and is strongly closed in CS (z) with respect to Fz as T is Sylow in H  Fz by 7.2.8.4. Therefore T centralizes Fi by 9.5 in [Asc11]. Let Ti = Tj : j = i . Then Ti αi = T1 and Fi αi∗ = F1 , so as Ti centralizes Fi , also T1 centralizes F1 . Therefore by 2.3 in [Asc11], F1 × (F2 × · · · × Fm ) ≤ F, proving (5).  Lemma 7.2.11. Let 1 = P ∈ F1f , α ∈ A(P ), and L = F2 · · · Fn . Then Lα∗  NF (P α). Proof. Set R = P α, Y = NF (R), and Γ = {J ∈ Ω : J ≤ NS (R)}. By 2.5.2, ξ = (Y, Γ) is a quaternion fusion packet. As Fi α∗ ≤ Y for 1 < i ≤ n, it follows that ξ satisfies Hypothesis 7.2.1, and hence 7.2.3. Set B = [Ti α]Y ◦ ; then Fi α∗ ≤ B. From 7.2.10.4, B = HB , FTi (Mi )α∗ , where HB = CB (zi α). Now zα centralizes FTi (Mi )α∗ , and as zα centralizes Ti α ∈ Syl(B), zα centralizes HB from the structure of HB in 7.2.8. Thus zα centralizes B. Therefore for β ∈ A(zα) with zαβ = z, we have E = Bβ ∗ ≤ Fz and E = [Oi αβ]E , so as Fi α∗ β ∗ ≤ E we conclude from 7.2.10 that E = Fj for some j, and hence E ∼ = Fi by 7.2.10.3. Therefore  Fi α∗ = B, so Lα∗  Y, completing the proof. Lemma 7.2.12. Let T  = Ti : 1 < i ≤ n . Then T  ∈ F f and F1  NF (T  ). Proof. As z ∈ F f also T  ∈ F f . Set Y = NF (T  ); then ξ = (Y, Ω(z)) is a quaternion fusion packet by 2.5.2 and T  centralizes F1 by 7.2.10.5, so ξ satisfies Hypothesis 7.2.1. Now complete the proof as in the previous lemma.  

Lemma 7.2.13. For each P ∈ F1f and α ∈ A(P ), O 2 (NF1 (P ))α∗  NF (P α).

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Proof. Adopt the notation used in 7.2.11 and its proof. By 7.2.11, Lα∗  Y.  Set P = O 2 (NF1 (P )) and Q = Pα∗ . By 7.2.10.5, Q = [NT (P )α]Q ≤ CY (Lα∗ ) = X . By 7.2.12, P  NNF (T  ) (P ). Then as X ≤ NY (T  α), it follows from 1.2.6 that Q  X . Let S0 be Sylow in X Lα∗ . Then NY (S0 ) ≤ NY (T  α), so NY (S0 ) acts on Q and hence Q  Y by 7.4 in [Asc11]. The proof is complete.  Lemma 7.2.14. F1 is tightly embedded in F. Proof. We must verify conditions (T1)-(T3) if Definition 3.1.2 in [Asc19]. Condition (T1) is satisfied by 7.2.13. Condition (T3) is satisfied as H  Fz , as FT (M1 )  NF (E1 ), and as z and E1 are characteristic in T , so that AutF (T ) ≤ Aut(F1 ) using 7.2.10.4. It remains to verify (T2), so we must show that if u ∈ T is an involution and ψ ∈ homF (u, T ), then ψ = ϕφ for some ϕ ∈ AutF (T ) and φ ∈ homF1 (uϕ, T ). As condition (T1) is satisfied, it follows from 3.1.5 in [Asc19] that uF ∩ T ∩ F f = ∅, so we may take uψ ∈ F f . Then ψ extends to ζ ∈ homF (u T  , S) acting on T  . Then (T2) follows from 7.2.12.  Lemma 7.2.15. F1 is subnormal in F and [T ]F = F1 × · · · × Fn . Proof. By 7.2.10, F1 appears in 7.2.3.3. In particular F1 = T , m(T ) > 1, Φ(T ) = 1, and T is not dihedral, so the lemma follows from Theorem 3.0.3 in [Asc19].  We can now complete the proof of Theorem 7.2.9. For parts (1)-(3) of Theorem 7.2.3 hold by 7.2.15 and 7.2.10.4, while part (4) holds by 7.2.4. This contradicts the choice of τ as a counterexample, establishing Theorem 7.2.9. Remark 7.2.16. By Theorem 7.2.9, |I| = 1, so by 7.2.5.3, μ(τ ) ∼ = D12 . By 7.2.7, F = [O(z)]F . Therefore parts (1) and (2) of 7.2.3 hold with F1 = F. By 7.2.4, part (4) of 7.2.3 holds vacuously. Therefore it remains only to prove part (3) of Theorem 7.2.3. Lemma 7.2.17. H is transitive on D∗ (z) = A(z) = Z ∩ O(z) − {z}. Proof. By 7.2.9, D(z) = ∅, so D∗ (z) = A(z) = Z ∩ O(z) − {z} by 7.2.2.3. By 7.2.2.4, A(z) = tFz for t ∈ E − z , so it remains to show tFz = tH . By 7.2.8.4, T is Sylow in H, so as H  Fz , tFz = {tϕφ : ϕ ∈ AutFz (T ) and φ ∈ homH (tϕ, T )}.  Hence as E1 is characteristic in T , tFz = tH , as desired. Lemma 7.2.18. F is transitive on Ω. ˜ Ωi = {Ki }, Mi = K M , Proof. Assume otherwise and let K1 = K, K2 = K, i and Ti = Ki W . By 2.5.2, τi = (F, Ωi ) is a quaternion fusion packet. Observe μi = μ(τi ) ∼ = S3 with ηi = ViMi , where Vi = Ki ∩ W . By 4.2.5, W = ηi . Set Fi = [Ki ]F and let Si be Sylow in Fi . Thus W ≤ Ti ≤ Si ; in particular E = Ω1 (W ) is determined. Suppose first that z ∈ O2 (F). Then Fi appears in Theorem 7.1.6. If 7.1.6.2 holds then W = ηi  Fi = FTi (Mi ), so 7.2.3.3.i holds in this case, contrary to Remark 7.2.16 and the choice of τ as a counter example. So assume 7.1.6.1 holds. Then O2 (Fi ) = E CFi (z) ≤ O by 7.2.2.3, so O = KO2 (Fi ) ≤ Si and hence F = [O]F = Fi . Therefore 7.2.3.3.ii holds in this case, again a contradiction.

7.2. A(z) = ∅

189

Therefore z ∈ / O2 (F), so Fi appears in Theorem 7.1.29. Then as μi ∼ = S3 we (1) conclude that Fi is L+ [m], L [m] , G [m], or M . In the last three cases Ti = 3 2 12 3 CFi (z) Ki E W = T , so Fi = [O]F = F and hence 7.2.3.3 holds, a contradiction. + Therefore Fi is L+ 3 [m]. Here Ti = Si . But now as W ≤ T1 ∩ T2 and L3 [m] is  quasisimple, F1 = [W ]F = F2 , so T = T1 T2 ≤ Si = Ti , a contradiction. Notation 7.2.19. Set E = H, FT (M ) . Lemma 7.2.20. T is Sylow in E, H = CE (z), and FT (M ) = NE (W ). Moreover one of the following holds: (1) H = T , E = FT (M ), and E and T are normal in Fz . (2) O 2 (H) ∼ = SL2 [m] ∗ SL2 [m] and E ∼ = G2 [m]. (3) m = 8, H is constrained with model H ∼ = S3 /Q28 , and E ∼ = L3 (2)/E8 or M12 . ˜ ∈ K Fz . Set M = FT (M ). Proof. Observe that by 7.2.18, K First suppose that m > 8. Then by 7.2.8, T is Sylow in H and either E and ˜ ∈ K Fz ) O 2 (H) ∼ T are normal in Fz or (using the fact that K = SL2 [m] ∗ SL2 [m]. Further in the first case, H = T and E = M, so CE (z) = CM (z) = T , and hence (1) holds. So assume that m = 8. Then by 7.2.8, H is constrained, T is Sylow in H = FT (H), and either H = T , or O 2 (H) ∼ = SL2 [8] ∗ SL2 [8], or H ∼ = S3 /Q28 . Of course in the first case, (1) holds. Suppose that O 2 (H) ∼ = SL2 [m]∗SL2 [m]. Here we argue as in the proof of parts (5) and (7) of 7.1.22 to show that E ∼ = G2 [m]; hence (2) holds in this case. Thus we may assume m = 8 and H is constrained with model H ∼ = S3 /Q28 . Here we argue as in the proof of 7.1.27 to show that (3) holds.  Lemma 7.2.21. AutF (T ) = AutFz (T ) = AutNF (E) (T ) ≤ Aut(E). Proof. This follows as z and E are characteristic in T ; for example z =  Z(T ) and E = Ω1 (Z2 (T )). Lemma 7.2.22. Z ⊆ NS (K). ˜ Let v be a generator of V . Then s Proof. Suppose s ∈ Z with K s = K. s centralizes vv = w ∈ W with E ≤ z, w , so s centralizes E. Set L = M s ≤ GW . Then by 4.2.6, U = s, a, W = CL (E)  L, where a is an involution in M inverting W . Thus U = O2 (L) = W × CU (O(X)), where X is a complement to W in M . Let {u} = sW ∩ CU (O(X)). Then u is an involution centralizing W0 = wO(X) = Φ(W ), so s ∈ uW centralizes W0 . But s inverts v −1 v s ∈ Φ(W ), so Φ(W0 ) = 1 and hence m = 8 and W0 = E is the set of elements of W inverted by s. Therefore s E = Ω1 (s W ) and hence u ∈ sE. But W is transitive on sE, so we may take s = u. Therefore O(X) centralizes s and E = [E, O(X)]. Let α ∈ A(s). Then sα = z and Eα = [O(X)α∗ , Eα] ≤ O(z)CS (O(z)), so Eα induces inner automorphisms on O(z). Therefore (Eα)# ⊆ D∗ (z) by 3.3.5, so Eα ≤ O(z) by 7.2.17. Then as the relation D∗ is symmetric, s ∈ D ∗ (z), a contradiction.  Lemma 7.2.23. (1) Z = z E . (2) E is isomorphic to G2 [m] or M12 . (3) F = Fz , NF (W ) .

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Proof. By 7.2.22 and 3.3.6, and adopting the notation of 7.1.2, Z = Zout ∪ Zin . Arguing as in 7.1.12: (a) If s ∈ Zout then s inverts W . Then arguing as in 7.1.13: (b) Suppose s ∈ Z − E centralizes E and set A = Es . Then s inverts W , 7.1.13.2 holds, and A# ⊆ Z. Suppose s ∈ Zout . Then by (a), (b), and 7.1.13.2, there is h ∈ M with sh ∈ O(z), so s ∈ z E . On the other hand if s ∈ Zin then s = z or s ∈ D ∗ (z) by 3.3.5 and 7.2.17, and hence again s ∈ z E by 7.2.17. This proves (1). Suppose E ∼ = L3 (2)/E8 or E = FT (M ). Then Z = O2 (E)# by (1), so O2 (E)  F by 14.1 in [Asc11]. But then 7.2.3.3 holds by Theorem 2, contrary to Remark 7.2.16. Together with 7.2.20, this proves (2). It remains to prove (3). Recall the subsystem E(τ ) from Definition 3.4.2, and set Y = E(τ ). In our setup, Y = Fz , NF (W ) , so we may assume Y = F, and it remains to produce a contradiction. By the Alperin-Goldschmidt Fusion Theorem 2.10, there exists R ∈ F e with AutF (R) = AutY (R). As in section 3.4 let G be a model for NF (R), S0 = S ∩ G, and G∗ = G/R; then G∗ has a strongly embedded subgroup M0∗ with S0 ≤ M0 . As z ∈ Z(S), z ∈ R and then U = z G ≤ Z(R). As Fz ≤ Y and AutG (R) = AutY (R), we have U = z = Z. By 3.4.5: (c) For each P ∈ Δ, |P ∩ R| ≤ 2. By 7.2.22, U ≤ NS (K), so from an earlier remark: (d) z G − {z} ⊂ Zin ∪ Zout . Next [S, E] = Z ≤ R, so E ≤ G. Claim: (e) E ≤ R. Assume otherwise and set H0 = E G . As [R, E] ≤ Z, it follows from 2.4.3 in [Asc19] that H0 ∼ = S4 , U = O2 (H0 ), and CS (U ) = R. Let u ∈ U − Z; then CS (u) = CS (U ) = R, so by (c), CK (u) = Z and hence u ∈ Zout by (d). Similarly u ˜ but now u centralizes E, contradicting [U, E] = Z. This completes is outer on K; the proof of (e). Let v ∈ V of order 4. Then [S, v] = E ≤ R, so v ∈ G and then by (c): (f) v ∈ G − R. ¯ = R/U . Then [R, ¯ v] ≤ E ¯ of order at most 2, so if H1 Set H1 = v G and R ¯ ¯ ¯ H1 ∼ ¯1 ) = does not centralize R then by 2.4.1 in [Asc19], U1 = E = E4 and AutH1 (U h h ¯ GL(U1 ). But then for e ∈ E − Z there is h ∈ H1 with [¯ e , v] = e¯, so e does not act on K, contrary to 7.2.22. Thus (g) [H1 , R] ≤ U so [H1 , U ] = 1. By (g) and (d), [U, v] = Z, so by 2.4.1 in [Asc19], m(U ) = 2 and AutH1 (U ) = GL(U ). Then as [H1 , R] ≤ U it follows that U = [H1 , R] and then G = H1 ×CG (H1 ) with H1 ∼ = S4 . Now E centralizes the Sylow group v, U of H1 , so there is e ∈ CE (H1 ) − Z. Then [R, e] ≤ Z ∩ CR (H1 ) = 1, so e ∈ Z(R). As Fz ≤ Y, AutCG (H1 ) (R) ≤ AutY (R), so AutH1 (R) ≤ AutY (R). But z ∈ eNF (W ) and Fz and NF (W ) ≤ Y, so Fe ≤ Y. Therefore as RH1 centralizes e, we have AutH1 (R) ≤ AutFe (R) ≤ AutY (R), a contradiction.  Lemma 7.2.24. F = E. Proof. By 7.2.13, E is G2 [m] or M12 . In the first case the proof of 7.1.23.1 shows that E = F, while this follows from the proof of 7.1.28 when E ∼ = M12 . Of

7.2. A(z) = ∅

191

course the proof uses 7.2.23.3 in place of 7.1.5 and 7.1.9, and also makes use of  7.2.21 and the fact that H  Fz and FT (M )  NF (W ). Observe that 7.2.16, 7.2.23.2, and 7.2.24 contradict the choice of τ as a counter example and complete the proof of Theorem 7.2.3. Remark 7.2.25. Observe that Theorem 3 follows from Theorem 7.2.3. Theorem 7.2.26. Assume τ = (F, Ω) is a quaternion fusion packet and Theorem 1 holds for each subpacket τ (K) of τ with K ∈ Ω and z(K) ∈ F f . Then the conclusions of Theorem 4 hold for τ ◦ . Proof. We may assume τ = τ ◦ . Suppose first that A(τ ) = ∅. Then by Theorem 7.2.3, τ = τ0 ∗ τ∗ where τ0 = τ1 ∗ · · · ∗ τr with τi = (Fi , Ωi ) satisfying one of the conclusions of 7.2.3.3 and with μ(τi ) ∼ = D12 for each 1 ≤ i ≤ r. By 6.1.2, D(τi )c is connected. Therefore it remains to show that τ∗ satisfies the conclusions of Theorem 4, so, replacing τ by τ∗ , we may assume A(τ ) = ∅. Now by 6.6.6.1, τ = ρ1 ∗ · · · ∗ ρn where ρi = (Yi , Γi ) and Yi = [Γi ]Yi is transitive on Γi . Hence the theorem holds unless conclusion (4) of Theorem 4 fails to hold for some i ∈ I. However ρi = τ (K) for K ∈ Ωi with z(K) ∈ F f , so by hypothesis, τi satisfies one of the conclusions of Theorem 1. Now 6.6.6.2 completes the proof.  Corollary 7.2.27. Theorem 1 implies Theorem 4.

Coconnectedness

CHAPTER 8

τ ◦ not coconnected Let τ = (F, Ω) be a quaternion fusion packet. Given Theorem 3, we may assume that A(τ ) = ∅. We will prove Theorem 1, and many of our other theorems, by induction on the order of τ , so with little loss of generality we may assume the following Inductive Hypothesis : Theorem 1 holds in all proper subpackets ρ of τ satisfying the hypothesis of Theorem 1 and of the form (Y, Γ) with Y ≤ F and Γ ⊆ Ω. We assume the Inductive Hypothesis repeatedly during the remainder of the paper. In any event if A(τ ) = ∅ and the Inductive Hypothesis holds, then by 6.6.8 if ρ = ρ◦ = (Y, Γ) is a proper subpacket of τ with Y ≤ F and Γ ⊆ Ω, then ρ is a central product of subpackets, as described in section 1.1 after the statement of Theorem 4; that is to say Theorem 4 holds in ρ. At many points in the paper we encounter the following setup: {Ω1 , . . . , Ωr } is a nontrivial partition of Ω, ρi = (Bi , Ωi ) is a quaternion fusion packet, and F contains a central product B = B1 ∗ · · · ∗ Br . Under suitable extra conditions we wish to show that B  F; then if F = F ◦ is transitive on Ω, we have B = F ◦ , contradicting r > 1 and F transitive on Ω. In Chapter 8 we consider various hypotheses building on this setup, and show that each such hypothesis implies that B = F ◦ . This is accomplished by showing that Bi is tightly embedded in F, and appealing to the theory of tightly embedded subsystems from [Asc19]. Among the various such hypotheses, Hypothesis 8.1.2 is the most general. For example from 8.1.1 we find that Hypothesis 8.1.8 implies Hypothesis 8.1.2. Then later hypotheses are shown to reduce to 8.1.8. 8.1. Dc disconnected In this section we assume the following hypothesis: Hypothesis 8.1.1. τ = (F, Ω) is a quaternion fusion packet satisfying the Inductive Hypothesis. Sometimes we also assume the following hypothesis:  Hypothesis 8.1.2. (1) Ω = Γ1 Γ2 is a nontrivial partition of Ω, and for i = 1, 2, ρi = (Bi , Γi ) is a quaternion fusion packet such that Bi = Bi◦ and F contains a central product B = B1 ∗ B2 . Let Ti be Sylow in Bi and assume T1 ∈ F f . (2) For each involution x ∈ T1 with x ∈ B1f , and for α ∈ A(x), B2 α∗  Fxα . ◦ ◦ = (Fxα , CΩ (xα)) not contained in Further if ρ is a coconnected component of τxα   ρ2 α then ρ = ρ α for some coconnected component ρ of (CB1 (x)◦ , CΓ1 (x)). 195

196

8. τ ◦ NOT COCONNECTED

(3) Z(F) ∩ T1 = 1. (4) For α ∈ A(T2 ), B1 α∗  NF (T2 α). (5) If ρ is a coconnected component of B2 with Sylow group T then T1 ∩ T = 1, and if φ ∈ homF (T, T1 ) then either ρφ = ρ1 or ρ = (OT , T ) for some T ∈ Γ2 and B1 = O(z1 ) for some z1 ∈ ZS . Remark 8.1.3. The condition in 8.1.2.3 that Z(F)∩T1 = 1 insures that, in the setup of 8.1.2.2, τxα is proper in τ , and hence Theorem 4 holds in τxα by Hypothesis 8.1.1. This in turn insures that the notion of a coconnected component of τxα in 8.1.2.2 makes sense. Lemma 8.1.4. Assume Hypothesis 8.1.2. Then (1) T2 ∈ F f and Bi  NF (T3−i ) for i = 1, 2.  (2) For each 1 = P ∈ C1f and α ∈ A(P ), O 2 (CB1 (P ))α∗ and B2 α∗ are normal in NF (P α). (3) AutF (T1 ) ≤ Aut(B1 ). (4) For each involution u ∈ T1 , uF ∩ T1 = uAutF (T1 )B1 . Proof. Suppose that B2  NF (T1 ). Then NS (T1 ) ≤ NS (T2 ). Let α ∈ A(T2 ). Then as T1 ∈ F f by 8.1.2.1, also T1 α ∈ F f and there exists β ∈ A(T1 α) such that T1 ξ = T1 , where ξ = αβ. As B2  NF (T1 ), T2 ζ = T2 , so NS (T2 α)β ≤ NS (T2 ), proving that T2 ∈ F f . Then by 8.1.2.4 applied to the identity map in A(T2 ), B1  NF (T2 ). Thus (1) will follow once we show that B2  NF (T1 ). Choose P and α as in (2), set Y = NF (P α), X = CF (P α), and A = B2 α∗ . Let x be an involution in P ∩ Z(NT1 (P )) and θ ∈ AB1 (x). Replacing x, P by xθ, P θ, we may assume x ∈ B1f . Let β ∈ A(xα), and γ = αβ. Then A ≤ X ≤ CF (xα), so X β ∗ ≤ CF (xγ). Let δ ∈ A(x) with xδ = xγ; then ζ = δ −1 γ ∈ homFxγ ((P CS (P ))δ, CS (xγ)). By 8.1.2.2, B2 δ ∗  Fxγ , so B2 γ ∗ = (B2 δ ∗ )ζ ∗ = B2 δ ∗  Fxγ . Then as X β ∗ ≤ Fxγ , A  X by 1.2.6. In particular, choosing P = T1 and α the identity map, B2  CF (T1 ). Then as Ω = Γ1 ∪ Γ2 with Γ1 ⊆ T1 but no member of Γ2 is contained in T1 as T1 centralizes T2 , it follows that B2  NF (T1 ), completing the proof of (1). As B2 δ ∗  Fxγ , ζ acts on T2 δ. Thus ζδ −∗ acts on T2 , so as B1  NF (T2 ), we have P γδ −1 = P ζδ −∗ ≤ CT1 (x). Therefore P γ ≤ CT1 (x)δ. Suppose A is not normal in Y and let R be Sylow in Y ◦ . By an earlier reduction, A  X , so there exists a coconnected component ρ = (C, Γ) of ρ2 with Sylow group T , and φ ∈ AutY (R) such that ραφ is not contained in ρ2 α. Therefore ραφβ is not contained in ρ2 γ = ρ2 δ. Then as ρ is coconnected, ραφβ is contained in a coconnected component θ of τxδ not contained in ρ2 δ by 6.6.11, and by 8.1.2.2, θ = ρ δ for some coconnected component ρ of (CB1 (x), CΓ1 (x)). In particular T αφβ ≤ CT1 (x)δ. Now by 8.1.2.5, either T αφβδ −1 = T1 or C = OT with T ∈ Γ2 and B1 = O(z1 ) for some z1 ∈ ZS . Suppose the former. Then as T αφβδ −1 ≤ CT1 (x), we have x ∈ Z(T1 ), so T αφβ = T1 δ, and by the previous paragraph, P γ ≤ T1 δ, so P α ≤ T1 δβ −1 = T αφ. Thus P α = P αφ−1 ≤ T α ≤ T2 α, so 1 = P ≤ T1 ∩ T , contrary to 8.1.2.5. So assume the latter and let z2 = z(T ). As Q = T αφβδ −1 ≤ T1 and B1 = O(z1 ), we have Q ∈ Ω(z1 ), so z2 αφβδ −1 = z1 . Next Qγ centralizes B2 γ ∗ = B1 δ ∗  Fxγ , so Qγ ∈ CΓ1 (x)δ, and hence z1 γ = z1 δ. Therefore z2 αφ = z1 δβ −1 = z1 γβ −1 = z1 α. As B1 = O(z1 ), T1 = O(z1 ). Also Ω(z2 ) ⊆ Γ2 , so O(z2 )α = O(z2 α), so as φ acts on R, O(z2 )αφ = O(z2 αφ) = O(z1 α). Therefore P α ≥ Z(O(z1 α)) = z1 α , so

8.1. D c DISCONNECTED

197

P = z1 . By 8.1.2.5, T ∩T1 = 1, so z2 = z1 , contradicting z2 αφ = z1 α ∈ P α ≤ T1 α and φ acts on P α. This finally proves that A = B2 α∗  Y.  We now complete the proof of (2). Let L = O 2 (NB1 (P )). Then Lα∗ ≤ CY (A) ≤ CY (T2 α). Let ϕ ∈ A(T2 α) with T2 ψ = T2 , where ψ = αϕ. But B1  NF (T2 ) by (1), so Lψ ∗ = O 2 (NB1 (P ψ))  NNF (T2 ) (P ψ), and hence Lα∗  NY (T2 α) by 2.2 in [Asc10]. Then as Lα∗ ≤ CY (A) ≤ CY (T2 α), Lα∗  CY (A) by 2.6. As Lα∗ is normal in ACY (A) and NY (T2 α), we conclude from 7.4 in [Asc11] that Lα∗  Y. This completes the proof of (2). By (1), B2  NF (T1 ), so T2  NNF (T1 CS (T1 )) (T1 ). Hence for each ϕ ∈ AutF (T1 ), ϕ extend to ψ ∈ NF (T1 T2 ) acting on T2 . Then as B1  NF (T2 ) by (1), ϕ ∈ AutF (B1 ). Therefore (3) holds. It remains to prove (4), so let ui , i = 1, 2 be an involutions in T1 conjugate in F; we must show there is ϕ ∈ AutF (T1 ) and φ ∈ homB1 (u1 ϕ, T1 ) with u1 ϕφ = u2 . Let ψ ∈ homF (u1 , T1 ) with u1 ψ = u2 . Let β ∈ AB1 (u1 ). If β −1 ψ = ϕ0 φ0 then ψ = βϕ0 φ0 = ϕ0 · βϕ∗0 φ0 , and as AutF (T1 ) ≤ AutF (B1 ) by (3), we have βϕ∗0 φ0 a B1 -map. Thus we may assume ui ∈ B1f for i = 1, 2. f By (2) and 3.1.5 in [Asc19], there is v ∈ uF i ∩ T1 ∩ F ; let γi ∈ A(ui ) with ui γ = v. If γi = ϕi φi , then arguing as in the previous paragraph, (4) holds. Thus it suffices to assume u ∈ B1f , and to show that if γ ∈ A(u) with uγ = v then γ = ϕφ with ϕ ∈ AutF (T1 ) and φ ∈ homB1 (uϕ, T1 ). Choose such a γ. Applying 8.1.2.2 to u, γ and v, id in the role of x, α, we conclude B2 γ ∗ and B2 are normal in Fv , and if ρ is a coconnected component of ρ2 with Sylow group T such that ργ is not contained in ρ2 , then it follows from 8.1.2.5 that either ργ = ρ1 or B1 = O(z1 ) and ρ = (OT , T ) for some z1 ∈ ZS and T ∈ Γ2 with T γ ∈ Γ1 . In the former case T γ = T1 , so as v ∈ T1 we have tγ = v = uγ for some t ∈ T , and then u = t ∈ T ∩ T1 , contrary to 8.1.2.5. In the latter case, setting z2 = z(T ), as T ∈ Γ2 we have z2 γ = z1 . Then O(z2 )γ = O(z2 γ) = O(z1 ) = T1 , so as u centralizes O(z2 ), also v = uγ centralizes T1 . Thus v ∈ Z(T1 ) = z1 , so uγ = v = z1 . But now u = z2 contradicting Ω = Γ1 ∪ Γ2 a partition. Therefore B2 γ ∗ = B2 and hence T2 γ = T2 . By (1), B1  NF (T2 ), so γ = ϕφ,  where ϕ ∈ AutF (T1 ) and φ ∈ homB1 (uϕ, T1 ), completing the proof of (4). Lemma 8.1.5. Assume Hypothesis 8.1.2. Then B1 is tightly embedded in F. Proof. Observe that conditions (T1)-(T3) of Definition 3.1.2 of [Asc19] follow from parts (2)-(4) of 8.1.4.  Lemma 8.1.6. Assume Hypothesis 8.1.2, B1 = OK for any K ∈ Ω, and either ρ1 is coconnected or B1 = O(z1 ) for some z1 ∈ ZS . Then (1) B1 is subnormal in F. (2) F ◦ = B1 ∗ B2 . (3) The coconnected components of τ are the coconnected components of ρ1 and ρ2 . Proof. Let K ∈ Γ1 and z = z(K). By 8.1.5, B1 is tightly embedded in F. By hypothesis B = OK for any K ∈ Ω, so m2 (T1 ) > 1. Also either ρ1 is coconnected and hence B1 = FT1 (T1 ), or B1 = O(z). As K ≤ T1 , T1 is not dihedral. Thus if B1 = FT1 (T1 ), then (1) follows from Theorem 3.0.3 in [Asc19]. So assume that B1 = O(z) = T1 ; in this case we verify that none of the exceptional cases (2)-(8) of Theorem 3.0.2 of [Asc19] are satisfied, and then (1) follows from that theorem.

198

8. τ ◦ NOT COCONNECTED

As B1 = OK , n = |Ω(z)| > 1 and T1 is a central product of n quaternion groups. As m2 (T1 ) > 1, case (3) of Theorem 3.0.2 in [Asc19] does not hold. As T1 is not dihedral, case (4) does not hold, and as K ≤ T1 , Φ(T1 ) = 1 so case (2) does not hold. As T1 has no abelian subgroup of index 2, cases (5), (7) and (8) are not satisfied. As T1 = Ω1 (T1 ), case (6) does not hold. This completes the proof of (1). Let B0 = [T1 ]F , Γ0 = ΓF 1 , Γ3 = Γ0 − Γ1 , and Γ4 = Γ2 − Γ3 . Observe that CF (T1 )◦ = B2 , since B2◦ = B2  NF (T1 ) by 8.1.4.1, and since Γ2 = Ω − Γ1 . By (1) and Theorem 1 in [Asc19], B0 is the central product of the conjugates of B1 . Let B1 = A1 , . . . , An and Γ1 = Σ1 , . . . , Σn be the conjugates of B1 and Γ1 , respectively, and set ξi = (Ai , Σi ). Then ξ2 ∗ · · · ∗ ξn ≤ ρ2 , so as ρ2 satisfies Theorem 4, the coconnected components of each ξi are components of ρ2 . Let ρ be the product of the remaining components of ρ2 ; then ρ = ([Γ4 ]B2 , Γ4 ). In particular Γ4 centralizes B0 , so [Γ4 ]F ≤ CF (B0 ), and hence F ◦ = B0 ∗ CF (B0 )◦ with CF (B0 )◦ = [Γ4 ]F . Then as CF (B0 )◦ ≤ CF (T1 )◦ = B2 , (2) and (3) follow.    Hypothesis 8.1.7. r > 1 is an integer, Ω = Ω1 · · · Ωr is a partition of Ω, and for 1 ≤ i ≤ r, τi = (Ci , Ωi ) is a quaternion fusion packet such that τi is coconnected a central product C = C1 ∗ · · · ∗ Cr . Let Si be Sylow in  and F contains Ci , Si = j =i Sj , and Ci = j =i Cj . Assume S1 ∈ F f . Hypothesis 8.1.8. Hypothesis 8.1.7 holds, as do the following conditions: (1) τ1 is minimal among the τi , 1 ≤ i ≤ r; that is τi is not conjugate in F to a proper subpacket of τ1 . (2) For each involution x ∈ S1 with x ∈ C1f , for each 1 < i ≤ r, and for α ∈ A(x), τi α is a coconnected component of τxα = (CF (xα), CΩ (xα)), and any other coconnected component is a subpacket of (CC1 (x), CΩ1 (x))α. (3) For α ∈ A(S1 ), τ1 α is a coconnected component of τ1 = (CF (S1 α), Ω1 α). (4) For 1 < i ≤ r, S1 ∩ Si = 1. Lemma 8.1.9. Assume Hypothesis 8.1.8. Then (1) For α ∈ A(S1 ), C1 α∗  NF (S1 α). (2) For each involution x ∈ S1 with x ∈ C1f , and for α ∈ A(x), C1 α∗  Fxα . Proof. In (1) take P = S1 and in (2) take P = x . In each case let α ∈ A(P ), Y = NF (P α), and X = CF (P α). Suppose first that P = x . Then for i > 1, Ci centralizes P as C is a central product. Then C1 α∗ ≤ X . By 8.1.8.2, τi α is a coconnected component of τP α for each i > 1. Suppose C1 α∗ is not normal in Y; then there exists i and φ ∈ homY (Si , CS (P α)) such that Ci α∗ φ∗ = Cj α∗ for any j. Then by 8.1.8.2, τi αφ = ρα for some coconnected component ρ of (CC1 (P ), CΩ1 (P )). Then by minimality of τ1 in 8.1.8.1, ρ = τ1 , so Si αφ = S1 α. In particular, P α ≤ Si αφ, so P α = P αφ−1 ≤ Si α, and hence 1 = P ≤ S1 ∩ Si , contrary to 8.1.8.4. Thus (2) holds. Now assume P = S1 . Then by 8.1.8.3, τ1 α is a coconnected component of τ1 . As Ω(τ1 ) = Ω1 α = Ω(τ1 α), it follows that τ1 α is the unique such component. Thus C1 α∗  NF (S1 α), establishing (1).  Lemma 8.1.10. Assume Hypothesis 8.1.8 with C1 = OK for K ∈ Ω1 and S1 ∩ Z(F) = 1. Then (1) C1 is tightly embedded in F. (2) C1 is subnormal in F.

8.1. D c DISCONNECTED

199

(3) F ◦ = C and {τi : 1 ≤ i ≤ r} is the set of coconnected components of τ ◦ . Proof. We verify that Hypothesis 8.1.2 is satisfied with ρ1 = τ1 , B2 = C1 , and Γ2 = Ω − Ω1 ; then we verify the extra condition in the hypothesis of 8.1.6, and appeal to 8.1.5 and 8.1.6 to complete the proof. Condition (1) of Hypothesis 8.1.2 holds by Hypothesis 8.1.7. Indeed by 8.1.7, ρ1 = τ1 is coconnected, verifying the condition in 8.1.6. The first constraint in 8.1.2.2 holds by 8.1.9.2, while the second holds by 8.1.8.2. Condition 8.1.2.3 holds by assumption, and 8.1.2.4 follows from 8.1.9.1. The first constraint in 8.1.2.5 holds by 8.1.8.4, and the second follows from 8.1.8.1.  Lemma 8.1.11. Assume Hypothesis 8.1.7 and choose notation so that τ1 is minimal among the τi . In addition assume (a) {D(τi )c : 1 ≤ i ≤ r) are the connected components of D(τ )c . (b) S1 ∩ Z(F) = 1. (c) For 1 < i ≤ r, S1 ∩ Si = 1. (d) C1 = OK for K ∈ Ω1 . Then F ◦ = C and {τi : 1 ≤ i ≤ r} is the set of coconnected components of τ . Proof. We show that Hypothesis 8.1.8 is satisfied, and then appeal to 8.1.10 and (b) and (d) to complete the proof. Observe hypotheses 8.1.8.1 holds by the choice of τ1 and 8.1.8.4 holds by (c). Thus it remains to verify conditions (2) and (3) of Hypothesis 8.1.8. For 1 ≤ j ≤ r, set Zj = Z(τj ). Assume either (i) P ∈ C1f is of order 2 and i > 1, or (ii) P = S1 and i = 1. Let α ∈ A(P ), and set Y = NF (P α) and ρ = (Y, NΩ (P α)). By (a), Zi is a connected component of D(τ )c , so as P centralizes Ci , Zi α is contained in a connected component Ui of D(ρ)c , and then Ui is contained in a connected component Z(τj(i) ) of D(τ )c for some j(i). Suppose first that (ii) holds. As Z ⊆ T = S1 S1 , Zα = Z and then {Zj α : 1 ≤ j ≤ r} is the set {Zk : 1 ≤ k ≤ r} of connected components of D(τ )c . Then Z1 α is the unique coconnected component of ρ of order greater than 1, so C1 α∗  Y, verifying hypothesis 8.1.8.3. So assume that (i) holds. Then Zi α ⊆ Zj(i) , so Ci α ⊆ Cj(i) . Proceeding recursively, define j n (i) = j(j n−1 (i)) whenever j l (i) = 1 for 1 ≤ l < n. If j(i) = 1 then Ci α = C1 by minimality of τ1 , and in particular τi α is a coconnected component of ρ. Similarly if j(i) > 1 then either Ci α∗ = Cj(i) and τi α is a coconnected component of ρ, or Ci α∗ < Cj(i) . Assume the latter; then by minimality of τ1 , j n (i) = 1 for any n. Now there exists m such that Cj m (i) α∗ < Cj m+1 (i) , and for each n > m, Cj n (i) α∗ = Cj n+1 (i) . Let I = {j n (i) : n ≥ m}. Claim j is a permutation of I. For if not there are distinct k, l ∈ I with j(k) = j(l), and we may assume Ck α∗ = Cj(k) . Thus Cl α∗ ≤ Ck α∗ , so Cl ≤ Ck , contradicting Ωl not contained in Ωk . This establishes the claim. Now as Cj m (i) α∗ = Cj m+1 (i) , j m (i) is not a fixed point of j, so for some n > m, Cj n (i) α∗ = Cj m (i) , a contradiction. We have shown that for each i > 1, τi α = τj(i) is a coconnected component of ρ, verifying the first condition of 8.1.8.2. Then all other coconnected components of ρ are contained in τk , where {k} = {1, . . . , r} − {j(i) : 1 < i ≤ r}, completing the proof. 

200

8. τ ◦ NOT COCONNECTED

Hypothesis 8.1.12. Hypothesis 8.1.1 holds with F transitive on Ω. Let η ∈ G Assume η(τ ), W = η , G a model for NF (C S (W)), and M = K  .  (1) For some r > 1, η = η1 · · · ηr , Ω = Ω1 · · · Ωr , Ki ∈ Ωi , and M = M1 ∗ · · · ∗ Mr is a central product, where Mi = KiM , Ωi = KiM ∩ S, and ηi = (Ki ∩ W )M . Further Dη is a set of 3-transpositions of μ(τ ). f ◦ ◦ (2) Let K = K1 and z = z(K) ∈ F . Then Fz is a central product Fz = Cz 1 1, Si = S ∩ Mi is Sylow in Ci and Ci = Si Oi , FSi (Mi ) , where  Zi = Z(O(τi )) and Oi = O(τi ) if m > 8 while if m = 8 then O(τi ) = O 2 (Oi ) and Oi = O(τi ) or O 2 (CCi (Zi )) acts on each member of Ωi . Lemma 8.1.13. Assume Hypothesis 8.1.12. Then (1) NG (O(τ )) is transitive on Ω and G is transitive on {M1 , . . . , Mr }. Set T = S ∩ M , C1 = C2g and τ1 = τ2g for g ∈ NG (T ) with M2g = M1 . (2) C = C1 ∗ · · · ∗ Cr ≤ F with τi a coconnected component of C and AutF (T ) ≤ AutF (C) is transitive on Λ = {τi : 1 ≤ i ≤ r}. (3) S1 ∈ F f . Proof. As F is transitive on Ω, G is transitive on the Mi and NG (O(τ )) is transitive on Ω by 6.1.6, and hence (1) holds. Observe that by 8.1.12.3, C1 = S1 O1 , FS1 (M1 ) , where O1 = O2g  NF (Z1 ). Now for i = j, Mi centralizes Sj by 8.1.12.1. Also if m > 8 then Oi = O(τi ), and Sj centralizes Si and hence also Oi by 2.6.12. If m = 8 then we work in a model Gi for NF (Si CS (Si )), where by 8.1.12.3 there is a normal subgroup Hi modeling Oi with Sylow group O(τi ) and [Sj , Hi ] ≤ CHi (Si ) = Zi , so again Sj centralizes Oi . Therefore as Ci = Si Oi , FSi (Mi ) by 8.1.12.3, we conclude that Ci centralizes Sj . Hence by 2.3 in [Asc11], C = C1 ∗ · · · ∗ Cr ≤ F. Let ρ = (C, Ω), so that ρ is a quaternion fusion packet and T is Sylow in C since Si is Sylow in Ci for each i. Then by construction, Λ is the set of coconnected components of ρ. As AutF (T ) permutes the Mi and Oi transitively, it permutes Λ transitively, and then AutF (T ) ≤ Aut(C), completing the proof of (2). As z ∈ F f and S ∈ Syl2 (G), the transitivity of G on the Mi implies that S1 ∈ F f , proving (3).  Hypothesis 8.1.14. Hypothesis 8.1.12 holds. Choose g as in 8.1.13.1 and assume for each involution x ∈ S1 − z C1 with x ∈ C1f , for each 1 < i ≤ r, and for α ∈ A(x), τi α is a coconnected component of τxα = (CF (xα), CΩ (xα)), and any other coconnected component is a subpacket of (CC1 (x), CΩ1 (x))α. Lemma 8.1.15. Assume Hypothesis 8.1.14 with M1 = K and S1 ∩ Z(Ci ) = 1 for i > 1. Then F ◦ = C and Λ is the set of coconnected components of τ . Proof. We verify that Hypothesis 8.1.8 is satisfied and appeal to 8.1.10. Hypothesis 8.1.7 is satisfied by parts (2) and (3) of 8.1.13. By the transitivity of AutF (C) on Λ, τ1 is minimal, so that 8.1.8.1 is satisfied. If Z = S1 ∩ Z(F) = 1 then as Aut(T ) is transitive on Λ, Z ≤ S1 ∩ Z(Ci ) for i > 1, contrary to assumption. Similarly S1 ∩ Si ≤ S1 ∩ Z(Ci ), so S1 ∩ Si = 1, establishing 8.1.8.4. Observe 8.1.8.3 follows from the transitivity of AutF (T ) on Λ in 8.1.13.2, and the fact that τ2 is a coconnected component of Fz . Thus it remains to verify 8.1.8.2.

8.1. D c DISCONNECTED

201

Let x ∈ C1f be an involution. If x ∈ / z C1 then the conditions in 8.1.8.2 follow C1 from 8.1.14. If x ∈ z then we may take x = z, and then the first condition in 8.1.8.2 follow from 8.1.12.2. If ξ = (X , Γ) is a coconnected component of Fz other than τi , then Γ ⊆ Ω1 by 8.1.12.2, and then [Γ]Fz◦ ≤ [Ω1 ]CF (S1 ) ≤ C1 from our verification of 8.1.8.3. This completes the proof.  Lemma 8.1.16. Assume Hypothesis 8.1.12 with M1 = K and C2 has one class of involutions. Then F ◦ = C and Λ is the set of coconnected components of τ . Proof. As C2 has one class of involutions, so does C1 , and that class is z C1 . Therefore the extra condition in Hypothesis 8.1.14 is vacuously satisfied. Similarly / Z(M1 ), so as C1 as one class of involutions, we conclude Z(C1 ) = 1, as M1 = K, z ∈  and hence S2 ∩ Z(C1 ) = 1. Therefore the lemma follows from 8.1.15. Hypothesis 8.1.17. (1) Ω = Ω1 ∪ Ω2 is a partition of Ω, and for i = 1, 2, τi = (Ci , Ωi ) is a quaternion fusion packet such that τi is coconnected, A(τi ) = ∅, and F contains a central product C = C1 ∗ C2 . Let Si be Sylow in Ci . (2) K ∈ Ω2 with z = z(K) ∈ F f and Fz◦ = C1 ∗ CC2 (z)◦ . (3) C2 = OK , |C1 | = |C2 |, and Z(τ1 ) ⊆ z F . Lemma 8.1.18. Assume Hypothesis 8.1.17. Then: (1) C1  Fz . (2) Conjugating in F, we may take S1 ∈ F f . (3) C2 = CF (S1 )◦  NF (S1 ). (4) S2 ∈ F f . (5) P = NS (S1 ) = NS (S2 ) and there is α ∈ AutF (P ) interchanging S1 and S2 and C1 and C2 . (6) Suppose x ∈ C2f is an involution and for β ∈ A(x), C1 β ∗ is a coconnected ◦ ◦ component of Fxβ . Assume S1 ∩ S2 = 1. Then Fxβ = (CC2 (x)◦ ∗ C1 )β ∗ . Proof. By 8.1.17.2, Fz◦ = CC2 (z) ◦ C1 , and by 8.1.17.1, τ2 is coconnected, while by 8.1.17.3, C2 = OK , so we conclude |CC2 (z)| < |C2 |. On the other hand by 8.1.17.3, |C1 | = |C2 |, so it follows that C1 is not fused into CC2 (z) in Fz , and then (1) follows. By (1), S1  CS (z), so (2) follows. Set P = NS (S1 ) and let t ∈ Z(τ1 ) ∩ C1f . By 8.1.17.3, t ∈ z F , so as z ∈ F f there is α ∈ A(t) with tα = z. Then C2 α∗ ≤ Fz◦ , so as |CC2 (z)| < |C2 | = |C1 | and as C2 is coconnected, it follows that C2 α∗ = C1 and S2 α = S1 . Observe C1 is transitive on Ω1 as C1 is coconnected, A(τ1 ) = ∅, and τ1 satisfies Theorem 4. Hence NC1 (O(τ1 )) is transitive on Ω1 . Then as P acts on Ω1 , and hence also O(τ1 ), it follows from a Frattini argument that P = S1 CP (t) and |P : CP (t)| = |S1 : CS1 (t)|. But we saw that CS (z) acts on S1 , so CS (z) = CP (z) and |P : CP (z)| ≥ |S2 : CS2 (z)| = |S1 : CS1 (t)| by the previous paragraph. Then as |P : CP (t)| = |S1 : CS1 (t)| and z ∈ F f , it follows that t ∈ F f . Then we have symmetry between z, C1 and t, C2 . By 2.5.2, ρ = (NF (S1 ), Ω) is a quaternion fusion packet. Also Ω2 = CΩ (S1 ) ⊆ CΩ (t) and C2 ≤ CF (S1 )◦ ≤ Ft◦ by 6.6.11, so C2 = CF (S1 )◦  NF (S1 ), proving (3). By (3), S2  P , and we saw earlier that S2 α = S1 , so P = NS (S2 ) and hence (4) holds. Then P α ≤ NS (S1 ) = P , so α ∈ AutF (P ). Further by symmetry between S1 and S2 , S1 α = S2 , establishing (5).

202

8. τ ◦ NOT COCONNECTED

Assume the setup of (6). Set b = xβ and suppose A = C1 β ∗ is not normal in Y = Fb . Then as A is a coconnected component of Y there is φ ∈ homY (S1 β, CS (b)), such that Aφ∗ is a second component. Then by (3), Aφ = CF (S1 β)◦ = C2 β ∗ , so bφ = b ∈ S2 β. Then as b = (bφ)φ−1 ∈ S2 βφ−1 = S1 β, we have x ∈ S1 ∩ S2 , contradicting S1 ∩ S2 = 1. Therefore A  Y. Then Y ◦ = X ∗ A, with X ≤ CF (S1 )◦ β ∗ = C2 β ∗ by (3), completing the proof of (6).  Hypothesis 8.1.19. Hypothesis 8.1.17 holds and for each involution x ∈ C2f not in z C2 and for β ∈ A(x), C1 β ∗ is a coconnected component of CF (xβ)◦ . Lemma 8.1.20. Assume Hypothesis 8.1.19 with S1 ∩ S2 = 1. Then F ◦ = C. Proof. We show that Hypothesis 8.1.8 is satisfied and then appeal to 8.1.10 to complete the proof. By 8.1.17.1 and 8.1.18.4, Hypothesis 8.1.7 is satisfied with r = 2. By 8.1.18.4, C1 and C2 are conjugate, so 8.1.8.1 holds. By 8.1.17.2, 8.1.18.6, and 8.1.19, 8.1.8.2 holds. By 8.1.18.3, 8.1.8.3 holds. Finally by 8.1.18.5, S1 and S2 are interchanged in Aut(NS (S1 )), so as S1 ∩ S2 = 1, 8.1.8.4 holds and S1 ∩ Z(F) = 1. This completes the proof.  Lemma 8.1.21. Assume Hypothesis 8.1.17 and C1 has one class of involutions. Then F ◦ = C. Proof. By 8.1.18.4, C1 and C2 are conjugate, so C2 has one class of involutions. Thus the extra condition in Hypothesis 8.1.19 is vacuously satisfied. If 1 = S1 ∩ S2 then there is an involution in Z(C1 ) ∩ Z(C2 ) so z ∈ Z(C2 ) as C2 has one class of involutions. This is a contradiction as τ2 is coconnected and C2 = OK by 8.1.17.3. Now the lemma follows from 8.1.20. 

Theorem 6

CHAPTER 9

Ω = Ω(z) of order 2 Let τ = (F, Ω) be a quaternion fusion packet with Sylow group S. The next four chapters consider the case where A(τ ) = ∅ and |Ω(z)| > 1 for z ∈ ZS (τ ). This thread culminates in a proof of Theorem 6 under the assumption that τ satisfies the Extended Inductive Hypothesis. Chapter 9 focuses on the case where |Ω(z)| = 2. We begin in section 9.1 with some basic notions and results for |Ω(z)| = 2. Then in section 9.2 we prove results on the normalizers of essential subgroups R of S such that AutF (R) = AutE(τ ) (R); such results will later allow us to pin down the generation of F when F = E(τ ). In sections 9.3 and 9.4 we assume the Inductive Hypothesis and that F ◦ is transitive on Ω; in the final section 9.5 of Chapter 9 these conditions follow from other parts of Hypothesis 9.5.1. Section 9.3 treats the case where Z(τ )∩O(z) = {z}. In section 9.4 we determine the packets with μ abelian. Finally section 9.5 deals with the case μ ∼ = S4 and Z(τ ) ∩ O(z) = {z}. Using inductive arguments, we remove the Inductive Hypothesis from our treatment in sections 9.3 and 9.4 when |Ω| = 2 in Theorems 9.4.34 and 9.4.35. Then in Theorem 9.5.34 we determine all packets τ with A(τ ) = ∅, Ω = Ω(z) of order 2, and F = F ◦ transitive on Ω. Another important byproduct of Chapter 9 is the description of the generation of F in these cases, appearing in 9.4.34, 9.4.35 and 9.5.9. 9.1. |Ω(z)| = 2 In this section we assume the following hypothesis: Hypothesis 9.1.1. τ = (F, Ω) is a quaternion fusion packet such that A(τ ) = ∅ and for all z ∈ ZS , |Ω(z)| = 2. Notation 9.1.2. Pick K ∈ Ω such that z = z(K) ∈ F f , and let Ω(z) = {K1 , K2 } with K = K1 . Define Do (z) to consist of those t ∈ Z such that t induces an outer automorphism on K, and set Dm (z) = {t ∈ Z : K1t = K2 }. For s ∈ Z and x ∈ {o, m}, write Dx (s) for the set of t ∈ CZ (s) such that tα ∈ Dx (sα) for α ∈ A(s). Observe that the definition is independent of α. Lemma 9.1.3. For each t ∈ CZ (z), one of the following holds: (1) t ∈ z ⊥ and t centralizes O(z). (2) t ∈ D∗ (z) − D(z) and t induces a nontrivial inner automorphism on Ki for i = 1, 2. (3) t ∈ Do (z), and for i = 1, 2, t induces an outer automorphism on Ki with CKi (t) = z , and t inverts the unique t-invariant cyclic subgroup of Ki of index 2. (4) t ∈ Dm (z) and K1t = K2 . 205

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9. Ω = Ω(z) OF ORDER 2

Proof. We may assume t = z. As t centralizes z, t permutes Ω(z) = {K1 , K2 }, so either t acts on K or K1t = K2 . In the latter case, (4) holds, so we may assume the former. Then by 3.3.6, one of cases (3)-(5) of that lemma hold. If 3.3.6.3 holds then t ∈ z ⊥ centralizes K, and then t centralizes O(z) by 3.3.4, so that (1) holds. In case 3.3.6.4, (2) holds by 3.3.5. Finally in case 3.3.6.5, t induces an outer automorphism on K, so t ∈ / D∗ (z), and hence t induces an outer automorphism on Ki for i = 1, 2, and hence (3) holds.  Lemma 9.1.4. For each t ∈ Z acting nontrivially on K, tz ∈ tK , so tz ∈ Z. Proof. Observe Ω1 (CK (t)t ) = z, t , so tu = tz for u ∈ NK (CK (t)) − CK (t).  Lemma 9.1.5. Let t ∈ Z and E = t, z . (1) If t ∈ D(z) then z ∈ D(t). (2) If t ∈ D∗ (z) − D(z) then z ∈ D∗ (t) − D(t) and AutF (E) = GL(E). Proof. From 3.1.13 the relations D and D∗ are symmetric, so the first remark in (1) and (2) holds. In (2), t is nontrivial on K, so by 9.1.4 there is k ∈ K with tk = tz. Then by symmetry there is β ∈ AutF (z, t with zβ = tz, so (2) follows.  Notation 9.1.6. For i = 1, 2, let vi be a NS (Ki )-invariant subgroup of Ki of C (z) order 4, and choose v2 ∈ v1 S if K2 ∈ K F . Set v = v1 v2 . Let η ∈ ηS (τ ), W = η , G a model for NF (CS (W )), and M = K G . Set μ = μ(τ ) and D = Dη . Lemma 9.1.7. (1) Each member of Do (z) inverts v1 , v2 . (2) Each member of Do (z) ∪ Dm (z) centralizes v, z . Proof. Suppose t ∈ Do (z). From 9.1.3, t inverts vi for i = 1, 2, so (1) holds and t centralizes v. On the other hand if s ∈ Dm (z) then v1s = v2 for some = ±1, so s centralizes v.  Hypothesis 9.1.8. (1) Hypothesis 9.1.1 holds. (2) F is transitive on Ω. (3) D ∗ (z) = D(z). Note that (by 3.3.14) Hypothesis 9.1.8 implies that |η(τ )| = 1, and hence that W is weakly closed in S with respect to F. Lemma 9.1.9. Assume Hypothesis 9.1.8. Then (1) Let {ηi : 1 ≤ i ≤ r} be the orbits of M on η, Ωi = {J ∈ Ω : J ∩ W ∈ ηi }, Mi = JiM for Ji ∈ Ωi , Wi = ηi , and Di = {dV : V ∈ ηi }. Then D is a set of 3-transpositions of μ, M/W = μ, Di is a conjugacy class of 3-transpositions of Mi /Wi , and M = M1 · · · Mr is a central product. Choose notation so that K ∈ Ω1 ; then K2 ∈ Ω1 . (2) NG (O(τ )) is transitive on Ω and {Ω1 , . . . , Ωr }, and G is transitive on {M1 , . . . , Mr } and on η. (3) τi = (FS∩Mi (Mi ), Ωi ) is a quaternion fusion packet with μi = μ(τi ) = Mi /Wi , Wi = W ∩ Mi , and μ = μ1 × · · · × μr . (4) For each 1 ≤ i ≤ r, μi ∼ = Weyl(Dn ) for some n ≥ 3. (5) Let Vi = W ∩ Ki and di = dVi ∈ D. Then d2 ∈ Vd1 in the 3-transposition groups μ and μ1 .

9.1. |Ω(z)| = 2

207

Proof. By 9.1.1, A(τ ) = ∅, so D is a set of 3-transpositions of μ by 3.1.25. Next M = M1 · · · Mr is a central product by 4.2.10.2, so as M is transitive on ηi , so is Mi . Then Di is a conjugacy class of 3-transpositions of μi by 4.2.8.1. Choose notation so that K ∈ Ω1 ; then (1) is established, modulo the assertions that K2 ∈ Ω1 , Wi = CMi (Wi ), and W = CM (W ), which we prove in a moment. For example by 3.1.19, μ = AutM (W ) = M/CM (W ), so if W = CM (W ) then μ = M/W . By 9.1.8.2, F is transitive on Ω, so NG (O(τ )) is transitive on Ω by 6.1.6.1. Then (2) follows from (1). By 2.5.2, τi is a quaternion fusion packet, and by definition μi = μ(τi ). We prove the remaining two assertions in (3) below. ⊥ By 3.1.21, d⊥ 1 = d2 in the 3-transposition group μ on D; that is d2 ∈ Vd1 in ∗ μ. By 9.1.8.3, D (z) = D(z), so M1 is nonabelian, and hence the groups Mi are nonabelian for each i by (2). Then as K centralizes Mi for i > 1 and d2 ∈ Vd1 , K2 also centralizes Mi , and hence K2 ∈ Ω1 as Mi is nonabelian and transitive on Di . This also shows d2 ∈ Vd1 in the 3-transposition group μ1 , completing the proof of (5). By (1), μi is transitive on Di , and by (5), d2 ∈ Vd1 in μ1 , so it follows from O (μ ) 4.1.7 that d2 ∈ d1 2 1 . Then by 4.3.5, μ1 ∼ = Weyl(Dn ) for some n ≥ 3. By (2), G is transitive on the τi , so μi ∼ = μ1 for each i, establishing (4). As M is a central product of the Mi , μ is a central product of the μi . By 4.3.8.4, CMi (Wi ) = Wi , so CM (W ) = W , completing the proof of (1). Also Z(Mi ) ≤ Wi , so μ = μ1 ×· · ·×μr . Let Wi = j =i Wj . Now W = W1 · · · Wr and Wi ≤ W ∩Mi = Ui , so Ui = Wi (Mi ∩ Wi ) ≤ Wi Z(Mi ), so as Z(Mi ) ≤ Wi , Ui = Wi , completing the proof of (3) and the lemma.  Lemma 9.1.10. Assume Hypothesis 9.1.8 and let t ∈ D∗ (z) − D(z). Then (1) t = k1 k2 c where ki ∈ Ki is of order 4 and c ∈ CS (O(z)) with c2 = 1. (2) vi ∈ W ans there exists φ ∈ homFz (k1 , k2 , t , W ) with ki φ = vi , cφ ∈ CS (O(z)), and tφ ∈ ZΔ . (3) Fz is transitive on D∗ (z) − D(z). (4) The following are equivalent: (a) v ∈ Z. (b) Z ∩ O(z) = {z}. (c) n = 3 and Z(M1 ) = 1, where Mi is as in 9.1.9.1 and n as in 9.1.9.4. Proof. Part (1) follows from 9.1.3. Let Vi = Ki ∩ W . As D∗ (z) = D(z), Ω2 (Vi ) is the unique W -invariant subgroup of Ki of order 4, ao vi = Vi ≤ W . As F is transitive on Ω and z ∈ F f , there is β ∈ A(t) with tβ = z. Then ki β ∈ Xi ∈ Δ(zβ), and as t ∈ / D(z), Xi is cyclic and nontrivial on K1 and K2 . Then there exists δ ∈ A(zβ) with zβδ = z. Set φ = βδ. Now k1 , k2 , t β ≤ W and W δ = W , so φ ∈ homFz (k1 , k2 , t , W ). As Ω(z) = {K1 , K2 } and vi is the subgroup of order 4 in Ki ∩ W , ki φ = vj(i) . Then conjugating in M1 we may take ki φ = vi , completing the proof of (2). By (2), there is s ∈ tFz such that s is in the set S of s ∈ ZΔ such that for  s ∈ V ∈ η, dV ∈ Ad1 . By 3.1.2.1, S = ZΔ ∩ D∗ (z) − D(z). But by 9.1.9.4 and 7.9 in [Asc97], Cμ1 (d1 ) is transitive on Ad1 , so CM (z) is transitive on S, completing the proof of (3). Suppose s ∈ O(z)∩Z −{z}. Then s induces a nontrivial inner automorphism on K, so by 9.1.3, s ∈ D∗ (z) − D(z). Then by (1) and (2), there is ζ ∈ homFz (s, W )

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with sζ = v1 v2 x for some x ∈ CS (O(z)). As O(z)  Fz and s ∈ O(z), also sζ ∈ O(z), so sζ = v or vz. Then by 9.1.4, v ∈ Z. Therefore (4b) implies (4a). Assume (4a). Then by 3.1.11.7, v ∈ ZΔ , and then by paragraph two of this proof, v ∈ V ∈ η such that dV ∈ Ad1 . By 9.1.9.4, μ1 ∼ = Weyl(Dn ), so by 4.3.8.2, M1 is an image of the universal ¯1 = ω ¯ (Dn , m) described in 4.3.1. Adopt the notation of 4.3.1, let z¯i be group M ¯ i , and zi the image of z¯i in M1 . As |Ω(z)| = 2, by 5.8.5 we may the involution in K take z = zn−1 = zn , so u = zn−1 zn = 1. As v ∈ ZΔ with dV ∈ Ad1 , we may take v = zn−2 . Then as v ∈ O(z), 5.7.4.3 says Z(M0 ) = 1, where M0 = K, K2 , J and J ∈ K M with z(J) = v; equivalently v1 v2 zn−2 = 1 in M0 and hence also in M1 . ¯ 1 ), so by 5.8.4 and its proof, n = 3. That is M1 = M0 , so Thus v¯1 v¯2 z¯n−2 ∈ Z(M Z(M1 ) = Z(M0 ) = 1. Therefore (4a) implies (4c). Finally if (4c) holds then from the discussion above, v1 v2 zn−2 = 1, so v ∈ ZΔ and hence (4b) holds, completing the proof of (4) and the lemma.  Lemma 9.1.11. Assume Hypothesis 9.1.8. (1) If m > 8 then OK = K or OK ∼ = SL2 [m]. (2) Assume m = 8 and if n = 3 assume v ∈ Z. Let G0 be a model for NFz (O(z)CS (O(z))). Recall W acts on O(z) by 3.1.12, and set H0 = W G0 O(z). Then either (a) CG0 (O(z)) ≤ NG0 (W ), so O 2 (H0 ) centralizes CS (O(z)) and is either trivial or of index 1 or 3 in SL2 (3) ∗ SL2 (3), or (b) n = 4 and O 2 (H0 ) centralizes CS (O(τ1 )) and is of index 3e in (SL2 (3) ∗ SL2 (3))2 for some 0 ≤ e ≤ 3. Proof. Part (1) follows from 2.5.4 as W O(z) is transitive on the Q8 -subgroups of K, using 4.2.5.1. Thus we may assume m = 8 and if n = 3 that v ∈ Z. Set S0 = CS (O(z)). Arguing as in the proof of 9.1.10, from 9.1.9.4, M1 is an image of ¯1 = ω M ¯ (Dn , m), described in 4.3.1. Adopt the notation of 4.3.1 and 4.3.3; in that notation, we may take K = Kn , take Ω(z) = {K, Kn−1 }, and let V1 ∈ η1 −{V1 } with z(V1 ) = z(V1 ). Set W− = W2 · · · Wr , A = V1 · · · Vn−3 W− , and B = Vn−2 Vn−1 Vn . If n > 3 set W0 = V1 A, while if n = 3 set W0 = A. Then W = AB with A ≤ S0 , so CW (O(z)) = ACB (O(z)). Set M0 = Kn−2 , Kn−1 , Kn ; then M0 is an image of ω ¯ (D3 , m), so from 5.7.5.3, CB (O(z)) = x0 , z , where x0 = zn−2 v. Then CW (O(z)) = Ax0 , z . Suppose n = 3. Then W0 = A = W− and by hypothesis v ∈ Z, so v = z  z1 by 9.1.10.4 and 5.7.4, and hence x0 ∈ z . Hence in this case CW (O(z)) = W0 z = W− z . So assume n > 3; then W0 = V1 A. Let L = Ω+ 2n (q) and define τL as in 5.3.4; then τL satisfies Hypothesis 9.1.8 with r = 1. Thus we can use the uniqueness of M1 , and can check in Ω+ 2n (q) (cf. 5.3.4) that x0 z ∈ W0 , so again CW (O(z)) = W0 z . Observe that W0 = η0 , where η0 = η ∩ S0 . Let Ω0 = Ω − {K, Kn−1 }, g ∈ G0 , and V0 ∈ η0 ; then V0 ≤ W0 ≤ S0 . Now Ω0 ⊆ S0 and G0 acts on Ω and Ω(z), and hence on S0 and Ω0 . Therefore if V0 ∈ Δ then as V0 ≤ S0 , also V0g ∈ Δ, so V0g ∈ η0 . / Δ then V0 ≤ K0 ∈ Ω0 and, as W0 ≤ S0 , either V0 is the unique S0 -invariant If V0 ∈ subgroup of K0 of order 4 or n = 4 and V0 ∈ {V1 , V1 }. In the first case V0g ∈ η0 , so if n = 4 then W0  G0 . If n = 4 let W+ = O(τ1 )W− . Thus O(τ1 ), W+ , and W− are normal in G0 . Now [W, S0 ] ≤ W ∩ S0 = CW (O(z)) = W0 z and W centralizes W0 z , so if W0  G0 then W ≤ O2 (W CG0 (O(z))) and hence CG0 (O(z)) ≤ NG0 (W ), so

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O 2 (H0 ) centralizes S0 , and hence CO2 (H0 ) (O(z)) ≤ Z(S0 ). Then as O(z) ∼ = Q28 and as AutF (O(z)) is transitive on Ω(z), (2a) holds in this case. So assume n = 4 and set S1 = CS (O(τ1 )). Here [W, S1 ] ≤ [W, S0 ] ≤ W0 z ≤ W+ and W centralizes W∗ = W− Z(O(τ1 )). Then as O(τ1 ) and W∗ are normal in G0 , O 2 (H0 ) centralizes S1 . Therefore each element of odd order in H0 is faithful on O(τ1 ). Then as O(τ1 ) = O(z) × O(z1 ) ∼ = Q28 × Q28 , as AutF (O(z)) is transitive on Ω(z), and as AutF (O(τ1 )) is transitive on Ω1 , either O 2 (H) = 1, and hence (2a) holds, or (2b) holds.  Definition 9.1.12. Given t ∈ Dm (z) and a Q8 -subgroup Q1 of K, set E(t, Q1 ) = CQ1 Qt1 (t)t ≤ S. ∼ E16 , t ∈ Dm (z) ∩ E, and Lemma 9.1.13. Assume z ∈ E ≤ S with E = NE (K) ≤ O(z). Let Q1 be the preimage of the projection of NE (K)/z on K/z . Then E = E(t, Q1 ) and Q1 = NK (E). ˜ = O/z ; by hypothesis E8 ∼ Proof. Set F = NE (K), O = O(z), and O =F ≤ ˜ 1 of F˜ on K ˜ is isomorphic O(z). As m(K) = 1, F ∩ K = z , so the projection Q ˜ t] = C ˜ (t) ∼ to F˜ ∼ = E4 . Hence Q1 ∼ = Q8 and then F ≤ Q = Q1 Qt1 . Also [Q, = E4 , Q ˜ ˜ ˜ ˜  so F = [Q, t], so Q1 ≤ NK (E). Then as NK˜ (F ) = Q1 , we have Q1 = NK (E). Lemma 9.1.14. Let t ∈ Dm (z), Q1 a Q8 -subgroup of K, Q = Q1 Qt1 , and E = E(t, Q1 ). (1) Q ∼ = E16 , and Q acts on E with QE/E ∼ = E4 . = Q28 , E ∼  Q (2) t is a basis for E (viewed as an F2 -space) of order 4 and s∈tQ s = z. (3) There exists γ ∈ A(E) with zγ = z and CQ (t)γ ≤ O(z). (4) Suppose F is transitive on Ω and A = AutF (E) is irreducible on E. Then either (a) A preserves a symplectic form f on E, E # = z A ⊆ Z, and A is of index at most 2 in O(E, f ) = Sp(E), or (b) A preserves a quadratic form q on E of sign −1, A is of index at most 2 in O(E, q) ∼ = O4− (2), and E ∩ Z is the set of singular points of E. Proof. Part (1) is straightforward and shows that |tQ | = |Q : CQ (t)| = 4. ˜ = X/z , and let tQ = {t1 , . . . , t4 } with t = t4 . Set Set F = CQ (t), X = QE, X Q ˜ = 1 and hence x = 1 or z. However if x = 1 then x = t1 · · · t4 . Then t˜ = t˜F˜ , so x t = t1 t2 t3 and E = T × z , where T = t1 , t2 , t3 = tQ is X-invariant. This is a contradiction as T ∼ / T , whereas z = Z(X). Similarly E = tQ , so (2) = T˜ , so z ∈ holds. ¯ centralizes zδ. Let ¯ = Eδ. As CE (Q) = z , NS (E) Let δ ∈ A(E) and E ¯ ≤ NS (Eζ), so as β ∈ A(zδ) such that zζ = z, where ζ = δβ. Then NS (E)β ¯ ∈ F f , also Eζ ∈ F f . Thus replacing E ¯ by Eζ, we may assume zδ = z, and hence E δ ∈ homFz (NS (E), CS (z)). As O(z)  Fz by 2.6.11, F δ = (E ∩ O(z))δ ≤ O(z), completing the proof of (3). Assume the setup of (4) and set O = O(z). Replacing E by Eδ, we may assume E ∈ F f . Let Γ = {Q1 , Qt1 } and observe that Γ consists of the subgroups NJ (E) for J ∈ Ω with z(J) ∈ E and NJ (E) nonabelian. For example given such a J, with j = z(J) = z, J centralizes O(z) but j does not. Therefore by 2.5.2, ρ = (NF (E), Γ) is a quaternion fusion packet. Then as A is irreducible on E it follows from Theorem 2 that NF (E)◦ is AE5 or AE6 . Note that conclusion (4) of

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Theorem 2 does not hold as Qe1 = Qt1 for e ∈ z NF (z) − Q. Hence (4) holds, modulo the statement in (4b) that E ∩ Z is the set of singular points of E, which we now prove. Suppose the statement fails; then E # ⊆ Z. Let e ∈ Q ∩ E − z and α ∈ A(e) with eα = z. Then e is nonsingular in E. As e ∈ D∗ (z), z ∈ D ∗ (e) by 9.1.5, so zα ∈ O, since D ∗ (z)−D(z) ⊆ O by 9.1.10.3. Then X = z CA (e) α ≤ O. But as e is nonsingular, m(X) = 3, so m(X) = m(Q) = 3 and hence X = Eα ∩ O. Let y ∈ Eα − X and set y0 = yα−1 . If y ∈ D(z) then AutO (Eα) contains a 4-group of transvections on Eα with center z, so CA (y0 ) contains a 4-group of transvections with center e, contradicting A ≤ O(E, q). Thus y ∈ / D(z), so as y centralizes X and D∗ (z) − D(z) ⊆ O(z), y ∈ Dm (z). Then by 9.1.13, Eα = E(y, P1 ) for some Q8 -subgroup P1 of K, so AutO (Eα) contains a 4-group free on Eα. But now CA (e) contains such a 4-group, contradicting e nonsingular. This completes the proof.  Lemma 9.1.15. Assume O(z) ∩ Z = {z} and either (a) |ZS | = 1, or (b) μ is abelian. Then (1) If t ∈ Dm (z) then z ∈ D m (t). (2) If t ∈ Do (z) then z ∈ Do (t) and AutF (z, t ) = GL(z, t ). Proof. If t ∈ Do (z) then by 9.1.5, z ∈ / D∗ (t), so by 9.1.3, z ∈ Do (t) ∪ Dm (t). Hence (2) follows from (1), using 9.1.4 to establish the last statement in (2), so it remains to prove (1). Assume t ∈ Dm (z), let Q1 be a Q8 -subgroup of K, Q = Q1 Qt1 , and E = E(t, Q1 ). Adopt the notation in the proof of 9.1.14 and assume z ∈ / Dm (t). Then o by 9.1.3 and 9.1.5, z ∈ D (t). Let α ∈ A(t) with tα = z; as z ∈ Do (t), we have zα ∈ Do (z). Set si = ti α for 1 ≤ i ≤ 4, and s = zα. By 9.1.14.2, t1 t2 t3 t4 = z, so s1 s2 s3 s4 = s. Hence as s ∈ Do (z), an even number r of the si are in Dm (s4 ), so r = 0 or 2. For i ≤ 3, tti ∈ F − z , so as Z ∩ O = {z}, tti ∈ / Z, and hence by 9.1.3 and 9.1.4, si ∈ D(s4 ) ∪ Dm (s4 ). Thus we may assume s3 ∈ D(s4 ). In particular |ZS | > 1, so the lemma holds when (a) is satisfied. Thus we may assume (b) is satisfied, so μ is abelian. Let δ ∈ A(s3 ). By 3.1.14, s3 centralizes O(s4 ), so s4 δ ∈ ZS while s3 δ ∈ ZS by 3.1.2.2. Next f = tt3 ∈ F , so f = u1 u2 with ui ∈ Ki of order 4. Also f centralizes each member of Ω0 = Ω−Ω(z). Let β ∈ A(f ) and γ ∈ A(f αδ) with f αδγ = f β = b. Now f αδ = s3 δs4 δ ∈ Z(O(τ )), so O(τ )γ = O(τ ) and hence b ∈ Z(O(τ )). Let ρ ∈ A(zβ) with zξ = z, where ξ = βρ. If zβ ∈ ZS then O(τ )ρ = O(τ ), so f ξ = f βρ ∈ Z(O(τ )). This is impossible as ξ is an Fz -map and O(z)  Fz but f ∈ / Z(O(z)). Therefore zβ ∈ / ZS . But ui β ∈ Yi ∈ Δ(zi β), so zβ ∈ ZΔ − ZS , contrary to (b) and 3.1.21.  Lemma 9.1.16. Let U = z, v and B = U, v1 Then (1) B  NFz (U ). (2) Suppose B ≤ P ≤ CS (U ) and α ∈ AutF (P ) is a 3-element with U = [U, α]. Then n = 3, v ∈ Z, and B ≤ [B, α] ∼ = Z24 . Proof. As U  CS (z), U ∈ Fzf . Then as z ∈ F f , also U ∈ Ff . Let ˜ = O(z)/z . Now NO(z) (U ) = O(z), so as O(z)  Fz , E = NFz (U ) and O(z) we have O(z)  E by 1.2.12. Therefore E permutes Ω(z), and hence also the set ˜ i . Therefore E permutes {v1 , v2 }, so (1) holds. {˜ v1 , v˜2 } of projections of v˜ on K

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Assume the setup of (2). Then α is transitive on U # , so v ∈ z F . Therefore N (U) n = 3 and Z(M1 ) = 1 by 9.1.10.4. By (1), A = v1 F = B · Bα · Bα2 ≤ O2 (CF (U )). By 3.1.3.2, v1 α ∈ D ∈ Δ(zα), so D ∈ η1 . Thus A ≤ W1 and as n = 3  and Z(M1 ) = 1, A = v1 , v1 α ∼ = Z24 by 5.7.4, and A = [A, α] = [B, α].

9.2. Generation when |Ω(z)| = 2 In this section we assume the following hypothesis: Hypothesis 9.2.1. τ = (F, Ω) is a quaternion fusion packet such that A(τ ) = ∅, F is transitive on Ω, and z ∈ ZS ∩ F f with |Ω(z)| = 2. Assume R ∈ F e with z ∈ Z(NS (R)) and let G be a model for NF (R), and set Z = z G and G∗ = G/R. After the first lemma we assume in the remainder of the section that AutF (R) = AutE(τ ) (R). Notation 9.2.2. We adopt the notation of section 9.1, particularly Notation 9.1.2 and 9.1.6. Set H = v1G ; this definition makes sense as v1 ∈ G from 9.2.3.3. Lemma 9.2.3. (1) Z ≤ Z(R). (2) v ∈ R. (3) v1 ∈ NS (R). (4) If AutE(τ ) (R) = AutF (R) then v1 ∈ / R. Proof. As z ∈ Z(NS (R)), (1) holds. Further [R, v] ≤ [CS (z), v] = z , so v ∈ NS (R) ≤ G. Suppose v centralizes Z. Then as [R, v] ≤ z ≤ Z and F ∗ (G) = R, (2) holds. Then [R, v1 ] ≤ [CS (z), v1 ] ≤ v, z ≤ R, so (3) holds. Part (4) follows from 3.4.5. Thus we may assume [Z, v] = 1 and it remains to derive a contradiction. Hence there exists g ∈ G such that t = z g does not centralize v. Set H = v G . As [R, v] = z , it follows from 2.4.3 in [Asc19] that H ∼ = S4 , G = H × CG (H), and R = CS (t). As [t, v] = 1, it follows from 9.1.3 and 9.1.7.2 that t ∈ D∗ (z) − D(z). Then by 9.1.10.1, t = k1 k2 c with ki ∈ Ki of order 4 and c ∈ CS (O(z)) with c2 = 1. As [t, v] = 1 there exists a unique i such that ki centralizes vi , say i = 1. Then v1 ∈ CS (t) = R, so v1g ∈ R. But then as v, z  CS (z), t = (v1g )2 centralizes v, a contradiction.  Lemma 9.2.4. (1) [H, Z] = 1. (2) R = CS (Z). (3) v1 ∈ O2 (CG (z)). Proof. Let Y = CNF (R) (z). As z ∈ Z(NS (R)), z ∈ NF (R)f , so Y = NFz (R). Observe CG (z) is the model for Y in G. By 2.6.11.1 and 1.2.12, NO(z) (R)  NFz (R), so the model X for NO(z) (R) is normal in CG (z). Note NO(z) (R) is Sylow in X. As F ∗ (X) = O2 (X) and [v1 , O(z)] = z , it follows that v1 ∈ O2 (X), so as X  CG (z), (3) follows. By 9.2.3.4, H ∗ = 1. Suppose (1) fails. Then H ≤ CG (z), so v1 ∈ O2 (H) by (3). Thus v1 ∈ R as H  G, contrary to 9.2.3.4. This establishes (1). Now (2) follows from (1) and 3.4.9.2 applied to Z in the role of Q. 

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Lemma 9.2.5. Assume Z ≤ NS (K). Then (1) Z = z, t for some t ∈ Do (z), O2 (H) = CH (Z), and H ∗ = GL(Z). (2) O2 (H) = U ∼ = E16 with U = Z × E, where E = v H does not act on K and H/U ∼ = S3 . (3) R0 = CR (U ) is of index 2 in R, with E  = E r the third irreducible for H on U , for r ∈ R − R0 . (4) R0 = U × CR (H). (5) There exists y ∈ O(z) inducing a transvection on U with center v. (6) Let GU be a model for the stabilizer in NF (R0 ) of U containing H, and set HU = H GU , y ≤ GU . Then U = O2 (HU ), HU R0 = HU × CR (H), and there is an HU -invariant quadratic form q on U of sign +1 such that HU /U is the subgroup of O(U, q) generated by transvections. Further GU = HU R, GU /R0 = O(U, q), and z HU , v GU is the set of singular, nonsingular points in U , respectively. (7) If J ∈ Ω then t ∈ / D ∗ (z(J)). Proof. As Z acts on K, and H does not centralize Z by 9.2.4.1, v1 induces a transvection on Z with center z. Then by 2.4.1 in [Asc19], Z ∼ = E4 and H/CH (Z) = GL(Z). By 9.2.4.3, v1 ∈ O2 (CG (z)), so [v1 , CH (Z)] ≤ O2 (CH (Z)) ≤ R by 9.2.4.2, so v1 centralizes CH (Z)∗ . Then as H = v1H we have CH (Z)∗ ≤ Z(H ∗ ). But by 9.2.4.2, CH (Z)∗ is of odd order, so as H ∗ /CH (Z)∗ ∼ = S3 , we have O2 (H) = R ∩H = CH (Z). Therefore H ∗ = H/CH (Z). Let t ∈ Z − z , and recall t ∈ Z ≤ NS (K) and Z # ⊆ Z H . If t ∈ D ∗ (z) then by 9.1.10.1, t = k1 k2 c with ki ∈ Ki of order 4 and c ∈ CS (O(z)). But then ki ∈ CS (Z) = R by 9.2.4.2, contrary to 3.4.5. Thus t ∈ D o (z) by 9.1.3, completing the proof of (1). As t ∈ Do (z), there is y ∈ O(z) − v1 , v2 with [t, y] = v. If R acts on K then [R, v1 ] = z , so H ∼ = S4 and G = H × CG (H) by 2.4.3 in [Asc19]. In particular H acts on each overgroup of Z in R. In this case set U = Z, v , so that HR acts on U . Suppose on the other hand that R does not act on K. Then for r ∈ R−NR (K), [v1 , r] = v or vz. Set G+ = G/Z and E = v H . Then v1 induces a transvection on R+ with center v + , so E + ∼ = E4 and H induces GL(E + ) on E + . In this case set U = ZE. As H is transitive on E +# , as v ∈ E is an involution, and as Z ≤ Z(U ), it follows that U ∼ = E16 . Let H0 = v1 , v1h for h ∈ H of order 3. As |U : CU (v1 )| = 4 = |R : CR (v1 )|, it follows that R = U CR (H0 ) and then H = H0 U with H0 ∼ = S4 . Notice that we have established (2) in the cae R ≤ NS (K). Now remove the assumption that R does not act on K. By construction, HR acts on U . By 9.2.4.2, CS (U ) ≤ CS (Z) = R, so H acts on CS (U ) and hence we, replacing U by a fully normalized conjugate, we may assume U is fully normalized. Now pick GU to be the stabilizeer of U in a model for NF (CS (U )). We may choose GU so that H ≤ GU . Suppose U = Z, v . Then y acts on U as [t, y] = v. Set Y = HU, y ≤ GU and U0 = Z(HU ). Now as G = H × CG (H), we have HU ∼ = Z2 × S4 and CU (y) = v, z ≥ U0 , so Y /U is the stabilizer in GL(U ) of U0 . Further R acts on HU and hence on U0 with O2 (Y ) = [O2 (Y ), H] and H is irreducible on O2 (Y )/U . Thus [R, O2 (Y )] ≤ U ≤ R, so O2 (Y ) ≤ NS (R) ≤ G, and then as O2 (Y ) = [O2 (Y ), H], O2 (Y ) ≤ O2 (H) ≤ R. This is a contradiction as O2 (Y ) does not act on Z, and this contradiction completes the proof of (2). Set u = v h . Then U = u, v, Z and as E does not act on K, we have K u = K2 . −1 . Next as [t, y] = v, y = y1 y2 Now [u, v1 ] ∈ E ∩ v, z , so [u, v1 ] = v and viu = v3−i

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with yi ∈ Ki and [t, yi ] = vi . Thus y1u = y2 v2β for some β, so y u = yv β . If β ≡ 0 mod 4 then u centralizes y, so y centralizes Ez and (5) holds in this case. If β ≡ 2 mod 4 then replacing y by yz and y1 by y1 z, we reduce to the previous case. Finally suppose β ≡ ±1 mod 4. If β = −1, then replacing y by yz and y1 by y1 z, we reduce to the case β = 1, where y centralizes z, v, tu = E  z . Thus in each case, Y = H, y ≤ GU and (5) holds. Moreover y centralizes F z for some F ∈ {E, E  }. Set R0 = CR (U ). Now H has three irreducibles Z, E, and E  = (vz)H on U , so either R = R0 and R = U × CR (H), or |R : R0 | = 2, R0 = U × CR (H), and E r = E  for r ∈ R − R0 . Also [E, v1 ] ≤ Z so E does not act on K. As y centralizes F and H acts on F , Y acts on F . If R = R0 then R is Sylow in CGU (U ), so y acts on R and hence R  Y , so Y ≤ G, a contradiction as y does not act on Z. Therefore R = R0 and in particular E r = E  for r ∈ R − R0 , so B is irreducible on U . Hence (3) and (4) hold. Set X = Y, R , B = AutX (U ), and D = αB , where α is the transvection induced by y on U . As B is irreducible on U , it follows from [McL69] that either D is irreducible on U and B = D = GL(U ), Sp(U ), or O(U, q) for some quadratic form q on U of sign −1, or D is not irreducible and B = O(U, q) with q of sign +1. Further setting Σ = AutF (U ), as AutH (Z) = GL(Z), AutB (Z) = AutΣ (Z) = GL(Z). Also O2 (NΣ (Z)) = AutR (U ) ∼ = Z2 with Z = CU (R). We conclude that B = O(U, q) = Σ = GU /R0 with (U, q) of sign +1 and Z is a totally singular line in this orthogonal space. Therefore (6) holds. Finally suppose J ∈ Ω, j = z(J), and t ∈ D ∗ (j). As t ∈ Do (z), j = z, so O(z) centralizes J. Now there is x ∈ J of order 4 centralized by t. Then x ∈ CS (Z) = R, contrary to 3.4.5. This establishes (7).  Lemma 9.2.6. Dm (z) = ∅. Proof. If Z ≤ NS (K) then there is g ∈ G with z g ∈ D m (z), so we may assume Z ≤ NS (K). Thus the hypotheses of 9.2.5 are satisfied; adopt the notation of that lemma. By 9.2.5.2, U does not act on K, so by 9.2.5.6, there is g ∈ GU with z g ∈ / NU (K), so z g ∈ D m (z).  Lemma 9.2.7. Assume Z ≤ NS (K) and set U = z, v . Then (1) Either (a) [Z, v1 ] = U , or (b) Z = Z0 × CZ (H); O 2 (H) = XZ0 ∼ = Z3 ; Z0 = v G ; = A4 , where X ∼ G m 2 Z for t ∈ z ∩ D (z); H = O (H)w , where w = v1 t inverts CZ (H) = v1 v1t ∼ = 2 X and w2 = v1 v1t . (2) Assume [Z, v1 ] = U . Then CH (Z) = R ∩ H and O 2 (G∗ ) = O 2 (H ∗ ). Moreover one of the following holds: (i) m(Z) = 4 and H ∗ ∼ = D10 , A5 , or Z2 /E9 . If H ∗ ∼ = A5 then Z is the A5 -module or L2 (4)-module for H ∗ , and in the latter case v ∈ Z. (ii) m(Z) = 5 or 6 and Z/CZ (H) is the L2 (4)-module for H ∗ ∼ = L2 (4). (3) Suppose m(Z) = 4 and NZ (K) ≤ O(z). Let Q1 be the preimage of the projection of NZ (K)/z on K/z and t ∈ z G − NZ (K). Then Z = E(t, Q1 ) is the A5 -module for H ∗ ∼ = A5 and Q1 = NK (Z). (4) If Z is the A5 -module for H ∗ ∼ = A5 then z is singular in the othogonal space Z and NZ (K) ≤ O(z).

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∼ A5 . Then CH (Z) = Z (5) Assume [Z, v1 ] = U and Z is the A5 -module if H ∗ = and there is a complement L to Z in H, G = ZNG (L), and R = Z × CR (H). In particular Φ(CS (Z)) ≤ CR (H). (6) Assume m(Z) = 4. Then m = 8 and for t ∈ z G − NZ (K) and a ∈ NZ (K) − U there is x ∈ CO(z) (t) − U such that a = xc for some c ∈ CS (O(z)) centralizing Z. (7) Suppose Z/CZ (H) is the L2 (4)-module for H ∗ ∼ = L2 (4). Then Z = O2 (O 2 (H)) 2 and H = O (H)v1 . Proof. Sete G+ = G/Z. As Z ≤ NS (K) there is g ∈ G such that t = z g ∈ Dm (z). Thus [v1 , t] = v ∈ Z and [v1 , CH (Z)] ≤ U ≤ Z by 9.1.16.1, so H centralizes CH (Z)+ . In particular O 2 (CH (Z)) = 1, so CH (Z) = O2 (H) = R ∩ H. Hence H ∗ = H/CH (Z) and CH (Z)+ ≤ Z(H + ). Suppose that [v1 , Z] = v . Then v1 induces a transvection on Z with center v, so by 2.4.1 in [Asc19], Z = v, z, t ∼ = S3 , and as an H-module, = E8 , AutH (Z) ∼ Z = v G × CZ (H), with CZ (H) = v1 v1t as v1 v1t = z[v1 , t]. Let H0 = O 2 (H)v1 . As CH (Z)+ = Z(H + ) we conclude O 2 (H) = XZ0 , where Z0 = v G . Now O 2 (H) is transitive on U G , and hence as NG (U ) = CG (U ) and v1t = v2 , O 2 (H) is also transitive on v1 G , using 9.1.16.1. Now (1b) holds and the proof of (1) is complete. So assume that [Z, v1 ] = U . Then by 2.8.15, H ∗ and its action on Z are described in that lemma. We claim (*) There exists no 3-element g ∈ G centralizing v1∗ with U = [U, g]. For if such a g exists then [v1 , g] ∈ R ≤ CG (Z). We apply 9.1.16.2 to P = v1 R and α = cg , to conclude that B = U v1 ≤ [B, g]. This is a contradiction as B ≤ R ≥ [P, g]. By (*), H ∗ , Z does not appear in case (5) of 2.8.15. Similarly: (**) In cases (1), (2), and (4) of 2.8.15, there is no element g ∗ of order 3 in CG∗ (H ∗ ) with Z = [Z, g]. Suppose that case (1) of 2.8.15 holds with H ∗ ∼ = S3 . Then H ∗ = Aut(H ∗ ) so ∗ ∗ ∗ ∗ G = H × CG∗ (H ). As G has a strongly embedded subgroup, CG∗ (H ∗ ) is of odd order, and then CG∗ (H ∗ ) = 1 by (**). Therefore G∗ = H ∗ . But now for each x ∈ Z # with CG∗ (x) of even order, xG ∼ = E4 , contradicting Z = z G of rank 4. We’ve shown that either m(Z) = 4 and case (1), (2), or (3) of 2.8.15 holds with H∗ ∼ = D10 , A5 , or Z2 /E9 ; or case (4) of 2.8.15 holds. This establishes (2). Assume the setup of (3). Then Z = E(t, Q1 ) and Q1 = NK (Z) by 9.1.13, and then (3) follows from comparing the description of AutF (Z) in 9.1.14.4 to that in (2). Conversely assume the setup of (4). As z ∈ Z(NS (R)), z is singular in Z. Then Z0 = v CH (z) is a hyperplane of Z, and as v ∈ O(z) and O(z)  Fz , also Z0 ≤ O(z). Thus Z0 = NZ (K), proving (4). Let H0 = O 2 (H)v1 . As H ∗ has a strongly embedded subgroup, O 2 (H ∗ ) is transitive on the involutions in H ∗ . By (2) and (**), CG∗ (v1∗ ) acts on [Z, v1 ] = U as the 2-group AutNS (R) (U ), so as z ∈ Z(NS (R)), we have CG (v1∗ ) ≤ CG (z). Then by 9.1.16.1, CG (v1∗ ) permutes {v1 , v2 }. Hence as t ∈ Z with v1t = v2 , it follows that H0 is transitive on v1 G . Therefore G = NG (v1 )H0 , so H = v1G = v1 H0 = H0 . Recall that CH (Z)+ ≤ Z(H + ). Therefore if H ∗ is D10 or Z2 /E9 then O 2 (H) = XZ, where X is a Hall 2 -subgroup of H; thus H = XZv1 and Z = CH (Z). If Z is the A5 -module for H ∗ ∼ = A5 then O 2 (CH (z))∗ ∼ = A4 , so as O 2 (CH (z)) acts 2 on KH = K ∩ O2 (CH (z)) we have v1 ∈ KH ≤ O (CH (z)), so H = O 2 (H). As

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∼ A5 or SL2 (5), and the former holds as v + is an (R ∩ H)+ ≤ Z(H + ) we have H + = 1 + involution in H . We next prove (5), so assume the setup of (5). By the previous paragraph, CH (Z) = Z, so H ∗ ∼ = H + . Moreover Z is a projective H ∗ -module, so there is a complement L to Z in H, and Z is transitive on complements to Z in H. As [v1 , R] = Z1 ≤ Z, v1 centralizes R+ , and hence H = v1G centralizes R+ . Then as H is transitive on complements to Z in H, by a Frattini argument, G = ZNG (L) and as H centralizes R+ , R = Z ×CR (H). By 9.2.4.2, R = CS (Z) so as R = Z ×CR (H) and Φ(Z) = 1, it follows that Φ(CS (Z)) ≤ CR (H), establishing (5). We next prove (6), so assume m(Z) = 4 and let O = O(z) and a ∈ NZ (K) − U . Then CO (t) = z × X where X = {x1 xt1 : x1 ∈ K} ∼ = Dm/2 . As U = [Z, v1 ], a  inverts v1 . Thus if m > 8 then for x1 ∈ K of order 8 with x21 = v1 , xa1 = x−1 1 z . t a −1 Set x = x1 x1 ; then x is of order 4 with x = x , so x induces a transvection on Z with center v = v1 v1t and axis U t . But now (2) supplies a contradiction, since G∗ has a strongly embedded subgroup and H ∗  G∗ . Therefore m = 8. Suppose next that a induces an outer automorphism on K. Then for x1 ∈ K − v1 and x = x1 xt1 , [x, u] = v, so again x induces a transvection on Z with center v and axis U t , for the same contradiction. Hence a induces an inner automorphism on K inverting v1 so a = x1 c1 for some x1 ∈ K − v1 and c1 ∈ CS (K). Then as a centralizes t and K t = K2 , a = xc where x = x1 xt1 and c ∈ CS (O) centralizes t and c2 = 1. Further c centralizes U, a, t = Z. This completes the proof of (6). Finally assume the setup of (7). Then as v1+ is an involution and CH (Z)+ ≤ Z(H + ), it follows that H + = O 2 (H + )v1= is isomorphic to L2 (4), L2 (4) × Z2 , or SL2 (5) ∗ Z4 . In the first two cases, (7) holds. In the third case, setting SH = S ∩ H, + ∼ we have SH = Q8 ∗ Z4 does not centralize v1+ , contrary to 9.1.16.1 and the fact that  SH centralizes U . This completes the proof of (7) and the lemma. 9.3. |Ω(z)| = 2 and Z ∩ O(z) = {z} In this section we assume the following hypothesis: Hypothesis 9.3.1. (1) τ = (F, Ω) is a quaternion fusion packet such that A(τ ) = ∅ and for z ∈ ZS , |Ω(z)| = 2. (2) F ◦ is transitive on Ω. (3) For z ∈ ZS , Z ∩ O(z) = {z}. (4) The Inductive Hypothesis holds: Theorem 1 holds in each proper subpacket of τ satisfying the hypotheses of Theorem 1 and of the form (Y, Γ) such that Y ≤ F and Γ ⊆ Ω. Lemma 9.3.2. Hypothesis 9.1.8 is satisfied. Proof. Hypothesis 9.1.1 holds by 9.3.1.1. Condition (2) of 9.1.8 holds by 9.3.1.2. Let z ∈ ZS with z ∈ F f and K ∈ Ω(z). By 9.3.1.3, there is u ∈ Z ∩ O(z) − {z}. Then u induces a nontrivial inner automorphism on K, so by 9.1.3,  u ∈ D∗ (z) − D(z), verifying condition (3) of 9.1.8. Notation 9.3.3. Adopt Notation 9.1.2 and 9.1.6, plus the notation in 9.1.9. Set O = O(z).

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Lemma 9.3.4. (1) M = M1 × · · · × Mr with Z(Mi ) = 1. (2) μ = μ1 × · · · × μr with μi = Mi /Wi ∼ = S4 . Proof. By 9.1.9, M = M1 · · · Mr is a central product, μ = μ1 × · · · × μr , and μi = Mi /Wi ∼ = Weyl(Dn ) for some n. By 9.3.1.3 and 9.1.10.4, n = 3 and Z(Mi ) = 1, so the central product M = M1 · · · × Mr is direct. Finally Weyl(D3 ) = Weyl(A3 ) ∼  = S4 . Lemma 9.3.5. Fz is transitive on O ∩ Z − {z} = D∗ (z) − D(z). Proof. By 9.1.10.3, Fz is transitive on D∗ (z) − D(z), so as O(z)  Fz , it  follows from 9.3.1.3 and 9.1.3 that D∗ (z) − D(z) = O ∩ Z − {z}. Recall from 9.1.9 that Wi = ηi = W ∩ Mi , and let Si be Sylow in Mi and Ji ∈ Ω ∩ Mi . Lemma 9.3.6. Suppose m = 8. Then (1) W1 ∼ = Z24 and μ1 = M1 /W1 ∼ = S4 . (2) O ∩W1 is of index 2 in W1 and w ∈ W1 −O induces an outer automorphism on Ki for i = 1, 2. (3) NS1 (K) = OW1 and each involution in NS1 (K) is in O. (4) Each involution s in S1 with K1s = K2 is conjugate in M1 to an element of O − W1 . ∼ Z2 from 5.7.4, which says that Proof. By 9.3.4.1, Z(M1 ) = 1, so W1 = 4 3 W1 ∼ = Z4 /Z4 . By 9.3.4.2, μ1 = M1 /W1 ∼ = S4 , completing the proof of (1). Next O1 = O ∩ W1 = v1 , v2 ∼ = Z2 × Z4 , so O1 is of index 2 in W1 by (1). From 4.2.5.1, w ∈ W1 − O induces an outer automorphism on Ki for i = 1, 2, proving (2). Next from (1) and (2), NS1 (K) = OW1 and |OW1 : O| = 2. For ki ∈ Ki − W1 , ai = ki v is an involution in ki W1 and [W1 , ai ] = [W1 , ki ] = vi is the subgroup of W1 inverted by ai , so the involutions in ki W1 are the members of ai vi ⊆ ki O ⊆ O. On the other hand k = k1 k2 is an involution centralizing Ω1 (W1 ) and inverting v1 with y1k = y2 , where yi , i = 1, 2, are the cyclic subgroups of order 4 of W1 containing v. Thus v1 , v2 is the subgroup of W1 inverted by k, so the set of involutions in kW1 is kv1 , v2 ⊆ O. This establishes (3). Choose s as in (4). By (3), NS1 (K) = OW1 , so as μ1 ∼ = S4 , s ∈ O2 (M1 ) − W1 and then sg ∈ kW1 for some g ∈ M1 . But we just saw that all involutions in kW1 are in O, so (4) holds.  Lemma 9.3.7. (1) CS1 (O) = CW1 (O) is cyclic of order m/4, |W1 : W1 ∩ OCS1 (O)| = 2, and w ∈ W1 − OCS1 (O) induces an outer automorphism on Ki for i = 1, 2. (2) |S1 : OW1 | = 2, NS1 (K) = OW1 and each involution in NS1 (K) is in OCW1 (O). (3) Each involution s in S1 with K1s = K2 is conjugate in M1 to an element of O − W1 . (4) m(S1 ) ≤ 4. ¯ 1 be the universal group ω Proof. Let M ¯ (A3 , m) of type M1 described in 4.3.1 ¯ 1 → M1 and, and 5.7.4; from 5.7.4 there is a surjective homomorphism π : M modifying the notation in 4.3.1 and 5.7.4 slightly so that 3 indexes the middle node

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¯ 1 = V¯1 × V¯2 × U ¯ , where V¯i = K ¯i ∩ W ¯ 1 for i = 1, 2, of the Coxeter diagram, W ¯ ∈ η¯ with involution z¯3 . Let y¯i be the canonical generator of V¯i and and V¯3 = U m/8 ¯ π, etc. Set M1∗ = M1 /W1 , so that v¯i = y¯i . Write Vi , U , etc. for V¯i π, U ∗ ∼ M1 = μ1 = S4 . ¯ 1 ) = ¯ v1 v¯2−1 z¯3 by 5.7.4.2, we conclude that By 9.3.4.1, Z(M1 ) = 1, so as Z(M −1 2 m/8 = 1. (y1 y2 y3 ) Now E4 ∼ = O ∗ = CM1∗ (O ∗ ) so CS1 (O) ≤ OW1 . As O ∩ W1 = CO (O ∩ W1 ), we have CS1 (O) = COW1 (O) ≤ COW1 (O ∩ W1 ) = W1 , so CS1 (O) = CW1 (O). By 5.7.5.3, CW1 (O) = x0 , z , with x ¯0 = y¯1 y¯2 y¯32 , so as (y1 y2−1 y32 )m/8 = 1 it follows that x0 = x0 , z is of order m/4. Set W0 = OCS1 (O). As Z(O) = z , |W0 | = |x0 ||O|/2 = m3 /16 = |OW1 |/2, so |OW1 : W0 | = 2 and hence |W1 : W1 ∩ W0 | = 2. From 4.2.5.1, u ∈ U − Φ(U ) induces an outer automorphism on Ki , completing the proof of (1). Now if m = 8 then the remainder of the lemma follows from 9.3.6; for example m(S1 ) ≤ m(W1 ) + m(M1∗ ) = 4. Thus we may assume m > 8. ¯1 ∼ As W = Z3m/2 is of rank 3, so is W1 , and hence Ω1 (W1 ) = z, v, w , where m/16 −m/8

w ¯ 2 = v¯1 v¯2−1 z¯3 , so we may take w = (y1 y2−1 y32 )m/16 = x0 y2 ∈ W0 . That is m/16 m/8 y2 = w, so W∗ = W1 k Ω1 (W1 ) ≤ W0 . Also setting k = k1 k2 ∈ O, wk = x0 is of 2-rank 3, and hence m(S1 ) ≤ m(W∗ ) + m(F ∗ /W∗ ) = 4 for each 4-subgroup F ∗ of S1∗ , proving (4). Let ki ∈ Ki −Vi ; then ai = ki v is an involution in ki W1 with I(ki ) = I(ai ) = Ii , where I(x) is the subgroup of W1 inverted by x. Hence the set of involutions in ¯ 1 ,we find ki W1 is ai Ii . Calculating using 5.7.4.2 and the defining relations for M I1 = V1 w , where w is defined above, so that a1 I1 ⊆ W0 . Similarly a2 I2 ⊆ W0 . On the other hand k = k1 k2 is an involution inverting V1 V2 = [W1 , k], so the involutions in kW1 are in kV1 V2 ⊆ O. This completes the proof of (2). Finally choose s as in (3). Then sh ∈ kW1 for some h ∈ M1 , and we saw earlier  that all involutions in kW1 are in O, so (3) holds. Lemma 9.3.8. One of the following holds: (1) m > 8, OK = K, and Z ∩ O = {z, v, vz}. (2) m > 8, OK ∼ = SL2 [m], and Z ∩ O is the set of involutions in O. (3) m = 8, O 2 (AutF (O)) = 1, and Z ∩ O = {z, v, vz}. (4) m = 8 and there is O  Fz of index 3 in SL2 [8] ∗ SL2 [8] with Sylow group O and O ∩ Z = E0# for some E8 ∼ = E0  Fz . (5) m = 8 and there is O  Fz isomorphic to SL2 [8] ∗ SL2 [8] with Sylow group O and O ∩ Z is the set of involutions in O. Proof. Suppose m > 8. Then by 9.1.11.1, OK = K or OK ∼ = SL2 [m]. In the first case O  Fz , so v, z = Ω1 (Z2 (O))  Fz . Hence (1) holds in this case by 9.3.5. Similarly if OK ∼ = SL2 [m] then all involutions in O − z are fused in O(z), so (2) holds. So assume m = 8. Then O = O(z)  Fz . From 9.3.6.2, vi is the unique NS (K)-invariant subgroup of Ki of order 4, so if A = AutF (O) is a 2-group then v1 , v2  Fz , and then (3) holds by 9.3.5. Thus we may assume A is not a 2-group. Adopt the notation of 9.1.11.2. By that lemma, O 2 (H0 ) is of index 1 or 3 in SL2 (3) ∗ SL2 (3), and centralizes S0 = CS (O). Then by 2.3 in [Asc11], E = O ∗ CF (O) is a subsystem of Fz , where O = FO (O 2 (H0 )). Further G0 is a model for NFz (OS0 ) = NFz (S0 ), so as O 2 (H0 )  G0 , NE (S0 )  NFz (S0 ), and hence

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E  Fz by 1.5.1 in [Asc19] applied to F2 , CF (O) in the role of F, F0 . Then as Fz = CS (z)E and CS (z) acts on O 2 (H0 ), O  Fz . Now (4) or (5) holds for |O 2 (H0 )|3 = 3 or 9, respectively.  Notation 9.3.9. If m > 8 set O = O(z). If m = 8 and AutF (O) is a 2-group, set O = O. Finally if m = 8 and AutF (O) is not a 2-group, define O as in part (4) or (5) of 9.3.8. Then from 9.3.8, O  Fz with Sylow group O, and O = O or O∼ = SL2 [m] ∗ SL2 [m], or m = 8 and O is of index 3 in SL2 [8] ∗ SL2 [8]. Further O centralizes CS (O). In the next lemma, two statements are proved by induction on the order of τ . The order of τ is defined in section 1.3. I suppose technically this process involves proving results first for small packets, then second for larger packets, and so on. But of course reading is much easier when the proof has a different organization. In each of the two results, ρ = (Y, Γ) is a proper subpacket of τ with Γ ⊆ Ω, satisfuing Hypothesis 9.3.1 By induction on the order of τ , ρ satisfies one of the conclusions of Theorem 9.3.24, and has the generation property of 9.3.20. Actually we could avoid the appeal to induction (as we will see in a moment), but I think this approach is easier and more attractive. To avoid the inductive appeal: in the first case by 9.3.1.4, ρ satisfies one of the conclusions of Theorem 1; then working with that list of conclusions, we could show that ρ appears in 9.3.24. For example μ(ρ) ∼ = S4 provides a first sieve. In the second case as ρ appears in 9.3.24, we can check the generation property directly; for example it is trivial in 9.3.24.1 and in 9.3.24.3 it holds by 9.3.20.2. Lemma 9.3.10. Suppose r > 1. Then: (1) Fz◦ = O(z) × C2 × · · · × Cr where ρi = (Ci , Ωi ) is a quaternion fusion N (W ) packet, Si is Sylow in Ci , ηi ∈ η(ρi ), and Mi is a model for Ji Ci i for Ji ∈ Ωi . Moreover (OKj , Kj ), j = 1, 2, and ρi , 2 ≤ i ≤ r, are the coconnected components of τz◦ . (2) One of the following holds for each 2 ≤ i ≤ r: (a) O = O and Ci = FSi (Mi ). (b) O ∼ = SL2 [m] ∗ SL2 [m] and Ci ∼ = L+ 4 [m]. (c) m = 8, O is of index 3 in SL2 [8] ∗ SL2 [8], and Ci ∼ = AE6 . (3) Ci = Si Oi , FSi (Mi ) , where Oi  NCi (zi ), zi = z(Ji ), and Oi ∼ = O. Proof. By 9.3.1.4 and 6.6.8 applied to τz in the role of (Y, Γ), τz◦ has coconnected components ξj = (Fj , Γj ), 1 ≤ j ≤ a. We may take Fj = OKj and Γj = {Kj } for j = 1, 2. Further FSi (Mi ) is coconnected for 2 ≤ i ≤ r and centralizes z, so, using 6.6.11.5, FSi (Mi ) ≤ Fj(i) for some j = j(i) > 2. Let zi = z(Ji ) for Ji ∈ Ωi . Then by induction on the order of τ (cf. section 1.3 for the definition of the order of τ , and apply 9.3.24) in an inductive context to ξj ), as ξj is coconnected and Z(ξj ) ∩ O(zj ) = {zj }, it follows that μ(ξj ) ∼ = S4 , ηi ∈ η(ξj ), and Mi is NFj (Wi ) ∼ L+ [m], or a model for Ji . Moreover by induction, Fj = FSi (Mi ) or Fj = 4 ∼ AE6 . In particular Ωi = Γj , so the map i → j(i) is a bijection m = 8 and Fj = of {2, . . . , r} with {3, . . . , a}, so we may relabel so that ξ1 , ξ2 , ρ2 , . . . , ρr are the coconnected components of Fz and (1) holds. ∼ We next prove (2). Suppose C2 ∼ = L+ 4 [m]. Then if m > 8, O(z2 ) = SL2 [m] ∗ 2 SL2 [m], while if m = 8 then |O (AutC2 (O(z2 )))|3 = 9. It follows from 9.3.9 that O∼ = SL2 [m] ∗ SL2 [m].

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Conversely suppose O ∼ = SL2 [m] ∗ SL2 [m]. By 9.3.9, O centralizes S2 · · · Sr , so as F is transitive on Ω, O(z2 ) ∼ = SL2 [m] ∗ SL2 [m] when m > 8, and when m = 8 we have |O 2 (AutFz (O(z2 ))|3 = 9. We conclude, using our proof of (1), that C2 ∼ = L+ 4 [m]. Therefore we may assume O is a 2-group or m = 8 and O is of index 3 in SL2 [8] ∗ SL2 [8], and we may also assume that C2 is FS2 (M2 ) or AE6 . Suppose C2 ∼ = AE6 . Then m = 8 and |O 2 (AutC2 (O(z2 )))|3 = 3, so O is of index 3 in SL2 [8] ∗ SL2 [8]. Conversely if O is of index 2 in SL2 [8] ∗ SL2 [8] then, as in the previous paragraph, O 2 (AutFz (O(z2 ))) = 1, so C2 ∼ = AE6 . This completes the proof of (2). Part (3) follows by induction on the order of τ . That is we apply 9.3.20.1 to ρi in an inductive setting, observing in each case that CCi (zi ) = Si Oi amd  NCI (Ei ) = FSi (Mi ) Lemma 9.3.11. r = 1. Proof. Assume r > 1 and adopt the notation of 9.3.10. We verify Hypothesis 8.1.14, and then appeal to 8.1.15 to conclude that C2 is subnormal in F and F ◦ is the direct product of the conjugates of C2 under F. (The constraint in 8.1.15 that Z(Ci ) = 1 holds as Z(Mi ) = 1 by 9.3.4.1 and as Si is Sylow in Ci by 9.3.10.1.) Then we have a contradiction to the transitivity of F ◦ on Ω. Hence it remains to verify Hypothesis 8.1.14. First, Hypothesis 8.1.1 holds by conditions (1) and (4) in Hypothesis 9.3.1. Then condition 8.1.12.1 holds by 9.3.2 and 9.1.9.1. Condition 8.1.12.2 holds by 9.3.10.1, and condition 8.1.12.3 holds by parts (1) and (3) of 9.3.10. Therefore Hypothesis 8.1.12 holds, so it remains to check the condition in 8.1.14: setting C1 = C2g for g ∈ G with z2g = z, if x ∈ S1 is an involution in C1f , then for each 1 < i ≤ r and α ∈ A(x), ρi α is a coconnected component of τxα = (CF (xα), CΩ (xα)), and any other coconnected component is a subpacket of (CC1 (x)α∗ , CΩ (x)α). The first constraint follows from the argument in paragraph one of the proof of 9.3.10, given that x centralizes Ci for i > 1. Further the same argument shows that for R = S2 · · · Sr , CF (R)◦ = C1 . Then if ξ = (E, Γ) is any other coconnected component, we have Γ ⊆ Ω(z)α and E centralizes Rα, so for β ∈ A(Rα) with Rαβ = R, Eβ ∗ ≤ CF (R)◦ = C1 . As ζ = αβ ∈ homNF (R) (CS (x), NS (R)) and C1  NF (R), Eα−∗ = Eβ ∗ ζ −∗ ≤ C1 , so E is a subsystem of CC1 (x)α∗ , completing the proof.  Notation 9.3.12. Set T = S ∩ M and E = z, v . Let G be a model for NF (CS (W )). Lemma 9.3.13. (1) Ω = {K1 , K2 } so D(z) = ∅, M = M1 , W = W1 , and μ∼ = S4 . (2) z ∈ Z(S) and O, T , and E are normal in S. (3) W and FT (M ) are normal in NF (E). Proof. By 9.3.11, r = 1, so as Ω is partitioned by the Ωi , 1 ≤ i ≤ r, by 9.1.9.1, we have Ω = Ω1 . By 9.3.4.2, μ = μ1 ∼ = S4 , so Ω = Ω1 is of order 2 by 4.2.8.6, and hence Ω = {K1 , K2 }. By 9.3.4.1, M = M1 × · · · × Mr , so as r = 1, M = M1 and W = W1 , completing the proof of (1). By (1) Ω = {K1 , K2 }, so O = K1 K2  S. Then as z = Z(O), z ∈ Z(S). As S acts on W , S ≤ G, so as M  G by 9.1.6, S acts on S ∩ M = T . From the proof of 9.3.8, E  CS (z), so E  S, completing the proof of (2).

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Let Y = NF (E) and ξ = (Y, Ω); by 2.5.2, ξ is a quaternion fusion packet. Then μ(ξ) ∼ = E4 , = S4 with Z(ξ) ∩ O = {z} and z ∈ E  Y, so by Theorem 2 and as E ∼ Y ◦ = FT (M ), proving (3).  Lemma 9.3.14. Assume s ∈ Do (z) ∪ Dm (z). Then (1) D∗ (s) ∩ E = ∅. (2) s ∈ Dm (z). (3) s ∈ T . (4) s is conjugate in M to a member of O − W . Proof. Suppose that D ∗ (s) ∩ E = ∅. By hypothesis, s ∈ / D∗ (z). Thus we may ∗ g g take v ∈ D (s). Let g ∈ M with z = v. By 9.3.5, s ∈ O in M , so s ∈ COg (z) ≤ T , proving (3). As s ∈ T , 9.3.7.2 implies that s ∈ / Do (z), so (2) holds. Then (4) follows from 9.3.7.3. Observe that we’ve shown that (1) implies (2)-(4). It remains to prove (1), so assume D∗ (s) ∩ E = ∅. By 9.1.7.2, s centralizes E. Let α ∈ A(s), so that sα = z. Suppose m > 8 and s ∈ Dm (z). Then CO (s) = z × X where X ∼ = Dm/2 and Z(X) = x for some x ∈ {v, vz}. As X ∼ = Dm/2 and m > 8, xα ∈ Φ(Xα) ∩ Z ⊆ D∗ (z) and hence x ∈ D∗ (s), contrary to assumption. Thus: (a) If m > 8 then s ∈ Do (z). Next as the graph D∗ is F-invariant, Eα ∩ D∗ (z) = ∅, so Eα ∩ OCS (O) = 1 by 9.1.3. Thus, using 9.1.3.3, Eα = y1 , y2 with y1 ∈ Do (z) and y2 ∈ D m (z), so by 9.1.7.2, Eα centralizes E. Hence E, Eα = E × Eα ∼ = E16 . Also [y1 , v1 ] = z and [y2 , v1 ] ∈ vz , so [Eα, v1 ] = E. Let β ∈ A(Eα) with Eζ = E, where ζ = αβ. If zζ = z then as z ∈ Z(S) we have NS (Eα) = CS (Eα), and we adjust β by composing with a member of AutM (CS (E)) / D∗ (z), s0 = sζ ∈ / D ∗ (z). to set zζ = z. Then ζ ∈ homFz (CS (s), CS (z)), so as s ∈ ∗ ∼ Set A = EEβ, so that A = E × Eβ = E16 . Let γ = cv1 β ∈ AutF (A) = Σ, where cv1 is the automorphism induced on EEα by v1 . Hence Eβ = [EEα, v1 ]β = [A, γ]. Suppose there is y ∈ A − E with y ∈ D∗ (z). Then y ∈ O, so |CO (s0 )| ≥ 8 and hence s0 ∈ D m (z) by 9.1.3.3. Then by 9.1.13, A = E(s0 , Q1 ) for some Q8 subgroup Q1 of K. Set Q = Q1 Qs10 . By 9.1.14.3 we may assume A ∈ F f . Then ρ = (NF (A), {Q1 , Q2 }) is a quaternion fusion packet by 2.5.2. Claim that Σ is irreducible on A; assume otherwise and let B be a proper nonzero Σ-submodule of A. Then z = CA (Q) ≤ B. If x ∈ z Σ − Q then A = xQ ≤ B, contrary to the choice of B. Thus z Σ ⊆ Q. Hence there is σ ∈ Σ with z = zσ ∈ Q, so zσ ∈ D ∗ (z). Then as W (ρ) ≤ W , we have E ≤ B, so as [E, γ] = [A, γ] = Eβ, we have A ≤ B, contrary to the choice of B. This completes the proof of the claim. By the claim and 9.1.14.4, and as v ∈ A ∩ Z, Σ is of index at most 2 in Sp(A). This is a contradiction as CΣ (zβ) acts on Eβ = (O ∩ EEα)β. Therefore D∗ (z) ∩ A ⊆ E. In particular (b) Eβ = a, b with a ∈ Do (z) and b ∈ D m (z). By 9.3.7.2, a ∈ / T , so as we showed that (1) implies (3), it follows that D∗ (a) ∩ / D ∗ (e), an argument above E = ∅. Let e ∈ E − z and g ∈ M with z g = e. As a ∈ g ∗ using the action of A on O shows that D (e) ∩ A ⊆ E. Hence (c) For each e ∈ E # , D∗ (e) ∩ A ⊆ E. Suppose m > 8. Then by (a), E ∩ D ∗ (b) = ∅, contrary to the previous paragraph. Hence (d) m = 8.

9.3. |Ω(z)| = 2 AND Z ∩ O(z) = {z}

221

By (c), for c ∈ Eβ # , E ∩ D ∗ (c) = ∅, so c centralizes no element of order 4 in W . Thus (e) For 1 = c ∈ Eβ, CW (c) = E. Set G+ = G/E and P = O2 (M ). By 9.1.3.3, CO (a) = E, so it follows from 9.3.6.3 that m(CP (a)) < 4. In particular A ≤ P . As M is irreducible on W + , CS (E) centralizes W + , so W acts on A. From (e), for each x ∈ W −E, CA (x) = E. Thus AutW (A) ∼ = E4 and Σ0 = AutW (A), γ ≤ Σ is irreducible on A, as E = [c, W ] for each c ∈ A − E and [E, γ] = Eβ. Further E = A ∩ O, so CΣ (z) ≤ NΣ (E). It follows from the list of irreducible subgroups of GL(A) that L2 (4) ∼ = Σ0  Σ and A is the L2 (4)-module for Σ0 . In particular there is a subgroup of order 3 in NΣ0 (E) which is induced by X1 of order 3 in NG (A). Then X1 centralizes a Sylow 3-subgroup X2 of M . Set X = X1 X2 and X0 = CX (W + ). As [X2 , CS (E)] = P and A ≤ P , we have X1 = X2 , so X ∼ = E9 . Now by 5.7.4.4, P + ∼ = E16 and M has three irreducibles W + and Pi+ , i = 1, 2 + on P , and M is also irreducible on E. By parts (4) and (5) of 5.7.4, Pi ∼ = E16 . For each of these four irreducibles P0 , EndM (P0 ) = F2 , so there is X0 ∼ = Z3 centralizes M/P and W , and then also centralizes each of the four irreducibles, so X0 centralizes M . Thus AutX1 (P ) = AutX2 (P ). Then A = [A, X1 ] centralizes Pi+ for i = 1, 2, so A centralizes P + and hence P acts on A with AutP (A) = [AutP (A), X1 ] ≤ Σ0 from the definition of X1 . Therefore P = W CP (A) and then CP (A) = Pi for i = 1 or 2, contradicting m(CP (a)) < 4. The proof is at last complete.  Lemma 9.3.15. (1) Do (z) = ∅. (2) Z ⊆ T . Proof. Part (1) follows from 9.3.14.2. By 9.3.13.1, D(z) = ∅. By 9.3.5, D∗ (z) ⊆ O ≤ T , while by 9.3.14.3, Dm (z) ⊆ T . Now (2) follows from 9.1.3.  Lemma 9.3.16. The following are equivalent: (1) F ◦ = FT (M ). (2) E # = Z. (3) D m (z) = ∅. Proof. If (1) holds then Z ⊆ W ≤ NS (K), so (1) implies (3). Assume (3). Then by 9.1.3 and 9.3.15.1, Z = {z} ∪ D∗ (z), and then by 9.3.5, Z ⊆ O. Suppose s ∈ Z −E and let g ∈ M −CM (z). Now s = k1 k2 with ki of order 4 / W for i = 1, 2. Therefore s ∈ CT (E) = O2 (M ) in Ki . By 9.3.8 we may choose ki ∈ g with K1gs = K2g , so sg ∈ O2 (M ) with K1s = K2 . Hence sg ∈ Dm (z), contrary to (3). Thus (3) implies (2). Assume (2) holds. Then E is a strongly closed abelian subgroup of S, so by  14.1 in [Asc11], E  F. Thus FT (M )  F by 9.3.13.3, so (1) follows.

In light of 9.3.16, during the remainder of the section, except in Theorem 9.3.24, we assume the following hypothesis: Hypothesis 9.3.17. Hypothesis 9.3.1 holds with D m (z) = ∅.

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Lemma 9.3.18. Either (1) O ∼ = SL2 [m] ∗ SL2 [m] and O ∩ Z is the set of involutions in O, or (2) m = 8, O is of index 3 in SL2 [8] ∗ SL2 [8], and O ∩ Z = E0# for some E8 ∼ = E0  Fz . Proof. By 9.3.17 there exists s ∈ D m (z). Then by 9.3.14.4, s is conjugate to  some x ∈ O − E. Hence Z ∩ O = E # , so the lemma follows from 9.3.8. Lemma 9.3.19. T O  Fz . Proof. As CFT (M ) (z) = T , we conclude from 9.3.13.3 and 1.2.12 that T  NFz (E). Next if m > 8 then E = Ω1 (Z2 (O)), so NFz (O) ≤ NFz (E) and NO (O) = O. Thus T NO (O) = T  NFz (O). Then T O  Fz by Theorem 1.5.2 in [Asc19]. Therefore we may assume m = 8. Let S0 = CS (O) and F0 = CF (O); by Theorem 4 in [Asc11], F0  Fz . By 1.3.2 in [Asc19], Fz = F1 , F2 , where F1 = SF0 and F2 = NFz (S0 ). Set E1 = T and E2 = T O. As T  NFz (E) with E  F1 , we have T  F1 . Further from the proof of 9.3.8, W O = FT (O 2 (H0 )W )  FS (G) = F2 , so E2  F2 . We have verified Hypothesis 1.4.7 in [Asc19], so by 1.4.8 in [Asc19], T O = E1 , E2  Fz , completing the proof.  Lemma 9.3.20. (1) F = Fz , NF (E) . ˜ etc. Then F˜ = S˜O, ˜ F ˜ (M ˜ ) . (2) Let F˜ = L+ ˜ the Lie packet for F, 4 [m], τ S Proof. We first prove (1). Let E = Fz , NF (E) . We verify that E satisfies conditions (a) and (b) of 2.1.8, and then appeal to that lemma to complete the proof. By definition of E, 2.1.8.a holds, so it remains to verify 2.1.8.b. Let s ∈ Z − {z} and Ss = CS (s). Suppose first that s ∈ D∗ (z). By 9.3.5 there is α ∈ AFz (s) with sα = v. Then Ss α ≤ CS (E) ≤ O2 (SM ) = Q in G, and there is g ∈ M with v g = z. Then γ = αcg ∈ homE (Ss , S) with sγ = z, so 2.1.8.b holds in this case. Therefore we may assume s ∈ / D∗ (z). Then by 9.1.3, 9.3.14.1, and 9.3.15 we ∗ may assume D (s) ∩ E = {v}. Therefore s ∈ CS (E) ≤ Q. Then Ssg ≤ Q with sg ∈ D∗ (z), so by the previous paragraph there is δ ∈ homE (Ssg , S) with sg δ = z. Now β = cg δ ∈ homE (Ss , S) with sβ = z, completing the proof of (1). Assume the setup of (2); observe that τ˜ satisfies Hypothesis 9.3.1 (except possibly the Inductive Hypothesis), so the proof of (1) (based on 2.1.8) works here too, once we make several observations that follow from the description of F˜ as ˜ for L ˜ ∼ ˜ and FS˜ (L) = Lπ4 (q) ∼ = P Ωπ6 (q) in sections 5.7 and 5.3. First, F˜z = S˜O ∗ m m ˜ ˜ NF˜ (E) = FS˜ (M ). Second, CZ˜ (z) = {z} ∪ D (z) ∪ D (z). Third, if s ∈ D (z) then ˜ and a generator for N ˜ (Ls ) is in D∗ (s), where s ∈ Ls by 9.1.7.2, s centralizes E E ˜ and Ls is a fundamental subgroup of L.  Lemma 9.3.21. Assume m = 8 and O is of index 3 in SL2 [m] ∗ SL2 [m]. Then F◦ ∼ = AE6 . Proof. Recall from 9.3.6 that W ∼ = Z24 and E = Ω1 (W ). By 9.3.18, O ∩ Z = + ∼ for some E8 = E0  Fz . Set M = M/E, P = O2 (M ), and let X ∈ Syl3 (M ). Observe that E0 ≤ CT (E) = P . Next by 5.7.4.4, P + = [P + , X] ∼ = E16 , so for each Z2 ∼ = U + ≤ P + , U +X ∼ = E4 . Let A be the preimage in P of E0+X . Then A is the union of conjugates of E0 , so as E0# ⊆ Z, A# ⊆ Z. Now D ∗ (z) = E0 − z and E0#

9.4. |Ω(z)| = 2 AND D ∗ (z) = D(z)

223

by 9.3.14, Z = {z} ∪ D∗ (z) ∪ Dm (z). Then by 9.3.14.4, A# = Z. Therefore A is a strongly closed abelian subgroup of S, so by 14.1 in [Asc11], A  F. Then the  lemma follows from Theorem 2 and the fact that E16 ∼ = A and μ ∼ = S4 . Lemma 9.3.22. Assume O ∼ = SL2 [m] ∗ SL2 [m] and set E = T O, FT (M ) T , regarded as a fusion system on T . Then E ∼ = L+ 4 [m]. Proof. By 9.3.4.1, Z(M ) = 1, so by 5.7.4.5 and 4.3.7.3, the pair M, Ω is deter˜ mined up to isomorphism. Let F˜ = L+ 4 [m]. Then by 5.3.5.4 and 5.3.8, F satisfies the Hypotheses 9.3.1 and 9.3.17 of F (except possibly the Inductive Hypothesis) ˜ so we may take M ˜ = M , S˜ = T , and Ω ˜ = Ω. Further ˜ and F˜z˜ = S˜O, with S˜ ≤ M ˜ ˜ ˜ by 9.3.20.2, F = T O, FT (M ) , so to show that F = E, it suffices to show X˜ = X , ˜ But this follows from Theorem 2.7.3 applied to the where X = T O and X˜ = T O. pairs (X , Ω) and (X˜ , Ω).  Lemma 9.3.23. Assume O ∼ = SL2 [m] ∗ SL2 [m]. Then F ◦ ∼ = L+ 4 [m]. Proof. Let E1 = T O, E2 = FT (M ), E = E1 , E2 , F1 = Fz , and F2 = NF (E). We verify Hypothesis 1.4.7 in [Asc19], and then appeal to 1.4.8 in [Asc19] to show that E  F. Then as F ◦ = [O]F = E, the lemma follows from 9.3.22. Thus it remains to establish Hypothesis 1.4.7 in [Asc19]. By 9.3.20, Hypothesis 1.4.1 in [Asc19] holds. By 9.3.13.3 and 9.3.19, Ei  Fi for i = 1, 2. As z = Z(T ) and E = Z2 (T ) are characteristic in T , AutF (T ) = AutFi (T ). Finally by 9.3.22, E is saturated, so Hypothesis 1.4.7 in [Asc19] is  satisfied, so as E = O 2 (E), 1.4.8 in [Asc19] shows E  F.  Theorem 9.3.24. Assume Hypothesis 9.3.1, let η ∈ η(τ ), W = η , G a model for NF (W ), and M = K G for K ∈ Ω. Set z = z(K). Then μ(τ ) ∼ = S4 , F = Fz , NF (W ) , and one of the following holds: (1) F ◦ = FS∩M (M ). (2) m = 8 and F ◦ ∼ = AE6 . (3) F ◦ ∼ = L+ 4 [m]. Proof. By 9.3.4.2 and 9.3.11, μ ∼ = S4 . If D m (z) = ∅ then (1) and the theorem hold by 9.3.16, so we may assume Hypothesis 9.3.17 is satisfied. Therefore F = Fz , NF (W ) by 9.3.20. If 9.3.18.2 holds, then so does (2) by 9.3.21. Thus by 9.3.18, we may assume that 9.3.18.1 holds. Therefore (3) holds by 9.3.23, completing the proof. 

9.4. |Ω(z)| = 2 and D∗ (z) = D(z) In this section, except in Theorems 9.4.34 and 9.4.35, we assume the following hypothesis: Hypothesis 9.4.1. (1) τ = (F, Ω) is a quaternion fusion packet such that A(τ ) = ∅ and for z ∈ ZS , |Ω(z)| = 2. (2) F ◦ is transitive on Ω. (3) For z ∈ ZS , D∗ (z) = D(z). (4) The Inductive Hypothesis is satisfied.

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9. Ω = Ω(z) OF ORDER 2

Remark 9.4.2. Observe that Hypothesis 9.4.1.1 is just a restatement of Hypothesis 9.1.1, so we can appeal to the lemmas in section 9.1 that do not assume Hypothesis 9.1.8, and we adopt Notation 9.1.2 and 9.1.6. Similarly later, after we show |Ω| = 2 in 9.4.8 and essential subgroups enter the picture, such subgroups will satisfy Hypothesis 9.2.1 by parts (1) and (2) of 9.4.1, so we can appeal to results in section 9.2. Set O = O(z). Lemma 9.4.3. Assume Ω(z) ⊆ Γ ⊆ Ω such that ρ = (Y, Γ) is a proper coconnected subpacket of τ . Then (1) If Dρo (z) = ∅ then Y ∼ = P Sp4 [m] or AE5 . o (2) If Dρ (z) = ∅ then Y ∼ = L− 4 [m]. Proof. Observe that ρ satisfies Hypothesis 9.4.1; most of the conditions in 9.4.1 inherit immediately, while the transitivity of Y on Γ follows from 9.4.1.4 and 6.6.6. Then the lemma follows by applying Theorem 9.4.33 in an inductive context in the proper subpacket.  Lemma 9.4.4. Let u = z be an involution in O. Then (1) uFz ∩ O ∩ F f = ∅. (2) If u ∈ F f and ρ = (Y, Γ) is a coconnected subpacket of τu◦ such that Ω(t) ⊆ Γ for some t ∈ ZS − {z}, then ρ is a coconnected component of τz◦ . Proof. As u = z is an involution in O, u = u1 u2 with ui ∈ Ki of order 4. Let Σ = Ω − Ω(z) and α ∈ A(u). Then ui α ∈ Di ∈ Δ(zα) centralizes Σα, so by 9.4.1.3, Di ∈ Ω − Σα and Q = D1 D2 = O(zα). Let β ∈ A(zα) with zζ = z, where ζ = αβ. As Q is the unique member of {O(t) : t ∈ ZS } containing uα, uζ = (uα)β ∈ F f . As ζ ∈ homFz (CS (u), CS (z)), (1) holds. Assume the hypothesis of (2) and let T be Sylow in Y and U = CO (u). Then U is nonabelian with z ∈ [U, U ]. As U centralizes O(t), U acts on Y, and hence on T . Indeed T acts on CUT (O(t)) = U t , using 9.4.3, and then acts on Φ(U t ) = Φ(U ). Hence T centralizes z, so T acts on U = O ∩ U t , so [T, U ] ≤ T ∩ U = 1. Then by 2.3.1 in [Asc19], z ∈ [U, U ] centralizes Y. As U  CFz (u), also U centralizes Y. Indeed if P = CO (Y) then P  = NO (P )  NFz (P ), and as ρ is a coconnected component of CF (P ) with P  centralizing O(t), P  acts on Y and then as P   NFz (P ) it follows that P  centralizes Y. That is O = P  centralizes Y. Then T ≤ CFz (O(z)) by 2.6.12, so Y ≤ [T ]Fz ≤ CFz (O(z)). Then as CFz (O(z)) ≤ CFz (u), ρ is a coconnected component of CFz (O(z)), and hence of  τz◦ . Lemma 9.4.5. z ∈ / Z(F ◦ ). Proof. If z ∈ Z(F ◦ ), then as F ◦ is transitive on Ω, Ω = Ω(z), so F ◦ = Fz◦ = O(z). This is a contradiction as |Ω(z)| = 2 and F ◦ is transitive on Ω.  Lemma 9.4.6. If each coconnected component of Fz is of the form ρJ = (OJ , J) for some J ∈ Ω, then Ω = Ω(z). Proof.  Assume otherwise and let Γ1 = Ω(z), Γ2 = Ω − Γ1 , ρ1 = (O(z), Γ1 ), and B2 = t∈ZS −{z} O(t). We show Hypothesis 8.1.2 is satisfied by ρ1 and ρ2 = (B2 , Γ2 ). Then by 8.1.6, ρ1 is a coconnected component of τ , so as F = F ◦ and F is transitive on Ω, τ = OK , contrary to 9.4.5. It remains to verify Hypothesis 8.1.2. First, Hypothesis 8.1.1 holds by 9.4.1.4 and 6.6.6 (cf Remark 6.6.10). By construction, 8.1.2.1 holds. Next T1 = O and

9.4. |Ω(z)| = 2 AND D ∗ (z) = D(z)

225

if x = z is an involution in B1f then by 9.4.4.1 we may take x ∈ F f . Then by 9.4.4.2, B2  Fx , and as Γ2 = CΩ (x), B2 contains all coconnected components of Fx . Therefore 8.1.2.2 holds. Next 8.1.2.3 follows from 9.4.5. As in the proof of 8.1.4.1, T2 ∈ F f , and as F is transitive on Ω and ρJ is a component of τz for each J ∈ Ω, we have B1 = O(z)  NF (T2 ). That is 8.1.2.4 holds. Assume the setup of 8.1.2.5. Then T ∈ Ω − Γ1 so T1 ∩ T = O ∩ T = 1. Hence 8.1.2.5 holds, completing the proof of the lemma.  Lemma 9.4.7. Let ZS = {z1 , . . . , zr } with z = z1 , and assume r > 1. For 1 ≤ i ≤ r, set Ωi = Ω(zi ), Oi = O(zi ), Q=

r 

NS (Oi ), and A = AutF (Q).

i=1

Then Q = CS (WS ) and: (1) For each i > 1, there exists a coconnected component τi = (Ci , Ωi ) of τz such that Ci ∼ [m], P Sp4 [m], or AE5 . Let Si be Sylow in Ci . = L− 4 (2) Let Pi = k =i Ok , C1 = CF (P1 )◦ , and S1 Sylow in C1 . Then A is transitive on Ω, on S = {S1 , . . . , Sr }, and on {Ck : 1 ≤ k ≤ r}. (3) F contains the central product C = C1 ∗ · · · ∗ Cr . Proof. Set Oi = O(zi ). By 9.4.3 applied to the components of τz◦ , for each i > 1 there is a quaternion fusion packet τi = (Ci , Ωi ) such that either τi is a coconnected component of Fz with Oi ≤ Ci , or Ci = Oi . Then Si centralizes Oj for each i = j. As r > 1, 9.4.6 says there exists a j > 1 such that Cj = Oj . Next as F is transitive on Ω, A is transitive on Ω, using 3.1.5. Let 1 < i ≤ r and choose α ∈ A with zj α = zi . Then for some k = j, zk α = z. As zk centralizes Cj , z centralizes Cj α and then Cj α ≤ Ci , so Ci = Oi . Then by symmetry between i and j, there is β ∈ A with Ci β ≤ Cj , so Cj α = Ci and hence τj α = τi . Now (1) follows from 9.4.3. As z ∈ F f , also O1 and P1 are in F f . Then by 2.5.2, τ1 = (C1 , Ω1 ) is a quaternion fusion packet. For i > 1, Ci centralizes Pi , and as A is transitive on Ω, also A is transitive on {Pk : 1 ≤ k ≤ r}. Thus for β ∈ A with Pi β = P1 , we have Ci β ∗ ≤ C1 , so as Ci is transitive on Ωi , C1 is coconnected and described in 9.4.3. Further there is γ ∈ A with O2 γ = O1 , so as C1 is coconnected, we have C1 γ ∗ ≤ Ck for some k > 1, and hence C1 = Ci β ∗ , so that also Si β = S1 . Therefore (2) holds. Next for i > 1, Q acts on Oi and hence on Si , so Si  Q. Therefore by transitivity of A on S, S1 Q. Then [S1 , Si ] ≤ S1 ∩ Si = 1 as zj = Z(Sj ) for each 1 ≤ j ≤ r. Thus S0 = k Sk is a central product by the transitivity of A on S.  Set T1 = i>1 Si . Claim that C1 centralizes T1 . Suppose that Si centralizes C1 for each i > 1. Now C1  N = NF (P1 ) and T1 ≤ CN (C1 ), establishing the claim. Therefore it suffices to show that C1 centralizes Si for each i > 1, or equivalently as A is transitive on S, for all i = j, Cj centralizes Si . Once this claim is established, 2.3 in [Asc11] says that F contains a central product of C1 , . . . , Cr , completing the proof of (3) and the lemma. Suppose first that r > 2 and for 1 ≤ i ≤ r set Bi = {β ∈ A : zβ = zi }. Then / {1, j}, Bi−1 = {αi ∈ A : zi αi = z}. Let j > 1. Then Cj centralizes Si for each i ∈ / {1, l}, where so C1 = Cj αj centralizes Si αj and hence C1 centralizes Sk for k ∈

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S1 αj = Sl . Therefore the claim holds unless for all j > 1, S1 αj = Sl for some fixed l; that is unless for all j > 1, Bj−1 = Bl . But as r > 2, there exists j > 1 with j = l, so Bj−1 = Bl = Bl−1 , and hence Bj = Bl , so that zj = zβ = zl for β ∈ Bj , a contradiction. Therefore we may assume r = 2, and it suffices to show that S1 centralizes C2 . Recall that O centralizes C2 and S1 centralizes S2 . If m > 8 set Oi = O(zi ). If m = 8 then from (1), CCi (zi ) has a normal subsystem Oi of index 1 or 3 in SL2 [8] ∗ SL2 [8] with Sylow group Oi . In either case as S1 centralizes O2 , it centralizes O2 by 2.6.12 and its proof. So S1 centralizes S2 O2 . Next (cf. 9.4.28 and 9.4.31.3) Ci = Si Oi , NCi (Ri ) : Ri ∈ Ri for a suitable subset Ri of Cie , and Si = Oi Vi , where Vi = Ui : Ri ∈ Ri and Ui = O2 (O 2 (NCi (Ri ))). Thus S1 = O1 V1 , and as O1 centralizes C2 , it remains to show that each U1 centralizes C2 . Then as S1 centralizes S2 O2 , it suffices to show that U1 centralizes NC2 (R2 ). Let G2 be a model for the stabilizer in NF (P ) of R2 , where P = R2 CS (R2 ). As r = 2, O is weakly closed in P with respect to G2 , so O  G2 . Also as C2  Fz , NC2 (R2 )  NFz (R2 ), so there is a normal subgroup H2 of G2 which is a model for NC2 (R2 ). Then [P, O 2 (H2 )] = U2 , so O 2 (H2 ) acts on U = U1 U2 . Thus we may regard H2 as a subgroup of a model GU for the stabilizer of U in NF (CS (U )). By symmetry there is a corresponding subgroup H1 of GU with [U, H1 ] = U1 . Set H = H1 , H2 . Then (cf. the list of the Ri in 9.4.22 and 9.4.27, and the structure of the corresponding Hi in 9.4.21. Note that case (2) of 9.4.22 does not occur here o 2 ∼ as Si is Sylow in Ci ∼ = L− 4 [m] when D (z) = ∅ by 9.4.3.) O (Hi ) = A4 or AE5 , so m(U ) = 4 or 8, and setting Yi = Ui ∩ Oi , Yi is a hyperplane of Ui such that Yi = [Ui , NOi (Ri )]. From 9.4.4, for each yi ∈ Yi# , CH (yi ) ≤ NH (U3−i ) ≤ NH (H3−i ). Set H ∗ = H/CH (U ), Li = O 2 (Hi ), and Wi = CU (L3−i ). As Hi centralizes the hyperplane Y3−i of U3−i , it centralizes U/Ui , and then as Ui is a projective Li module, U = Ui × W3−i with Y3−i a hyperplane of W3−i . As Si centralizes U3−i , Yi = [NOi (Ri ), Wi ]. In particular if W0 = W1 ∩ W2 = 1, as NOi (Ri ) acts on L1 and L2 it acts on W0 and then as Yi = [Wi , NOi (Ri )], it follows that W0 ≤ Y1 ∩ Y2 = 1, a contradiction. Therefore W1 ∩ W2 = 1. Claim H is irreducible on U . For suppose 1 = U0 < U is H-invariant. As Hi is irreducible on Ui , each proper nontrivial Hi -submodule is either contained in Wi or contains Ui . Then as U1 ∩ U2 = W1 ∩ W2 = 1, it follows that U0 = U1 or U2 , say the former. Set Q1 = CH ∗ (U1 ) ∩ CH ∗ (U/U1 ). Then Q1 ≤ CH ∗ (Y1 ) ≤ NH ∗ (U2 ), so Q1 centralizes U2 and hence Q1 = 1. Thus X = CH ∗ (U/U1 ) is faithful on U1 and H1∗ ≤ X. If m(U ) = 4 then H1∗ = GL(U1 ) so H1∗ = X  H ∗ and then H1  H. If m(U ) = 8 then ρ1 = (FS∩L1 (L1 ), {NK1 (R1 ), NK2 (R1 )}) is a quaternion fusion packet with U1 = O2 (L1 ) so L1  H. Thus in any event, L1  H, so W1  H, and hence H2 acts on W1 , so U2 = U1  H. Thus Ui = Wi centralizes H3−i , completing the proof in this case. So the claim is established. Suppose m(U ) = 8 and set Γ = {NJ (U ) : J ∈ Ω}). As in the previous paragraph, ρ = (FS∩H (H), Γ) is a quaternion fusion packet with z, z2 ∈ U , μ(Γ) is abelian, and H irreducible on U , so Theorem 2 supplies a contradiction as |Γ| = 4. Therefore m(U ) = 4 and H ∗ is an irreducible subgroup of GL(U ) = L4 (2). Then from the list of such subgroups, and as S3 ∼ = H2∗  CH ∗ (Y1 ), we conclude that + ∗ H ≤ O4 (2) preserves an orthogonal space structure on U , and Y1 is a nonsingular point in that orthogonal space. But now L∗1 and L∗2 are the normal subgroups of

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order 3 centralizing the two definite lines Ui = [Li , U ], so indeed U3−i centralizes  Li , and hence also Hi = Li NSi (Ri ), completing the proof. Lemma 9.4.8. Ω = Ω(z) is of order 2 so D∗ (z) = ∅. Proof. Assume otherwise and adopt the notation of 9.4.7; in particular, in that notation, r > 1, and Hypothesis 8.1.7 is satisfied by 9.4.7. We will verify Hypothesis 8.1.8. Then by 8.1.10.3, F = F ◦ = C, a contradiction as F is transitive on Ω and r > 1. Thus it remains to verify Hypothesis 8.1.8. As A is transitive on S, condition (1) of 8.1.8 is satisfied. For each involution x ∈ C1f , x ∈ O, so by 9.4.4.1 we may take x ∈ F f . Then by 9.4.4.2, τi is a coconnected component of τx . Further if x = z then τx has no other components, while if x = z then the remaining components are OKi for i = 1, 2. Therefore 8.1.8.2 holds. As in the proof of 8.1.4.1, S1 ∈ F f . Then as C1  Fz and A is transitive on ZS , C1  NF (S1 ), verifying 8.1.8.3. Finally as Z(C1 ) = 1, 8.1.8.4 holds. This completes the proof.  Lemma 9.4.9. O  S and z ∈ Z(S). Proof. By 9.4.8, O = O(τ ), so O  S. Then as z = Z(O), z ∈ Z(S).



Lemma 9.4.10. Assume t ∈ Dm (z), let Q1 be a Q8 -subgroup of K, and set Q = Q1 Qt1 , E = E(t, Q1 ), and U = Z ∩ E. (1) U = {z} ∪ tQ is of order 5. (2) CZ (E) ⊆ E. (3) AutF (E) acts transitively as Alt(U) or Sym(U) on U. (4) Z ∩ tCS (Q) = {t}. (5) t centralizes CS (Q). (6) Z ∩ tCS (Q)Q ⊆ E. Proof. Set F = E ∩ Q. We first prove (1). By 9.4.1.3, Z ∩ F = {z}, so U − {z} ⊆ E − F . By 9.1.14.2, Q has two orbits tQ and (tz)Q on E − F of order 4, so we may assume tz ∈ Z and it remains to derive a contradiction. Let α ∈ A(t); thus tα = z. Set tQ = {t1 , . . . , t4 } with t = t4 . For 1 ≤ i ≤ 3, tti and tti z are in F − z , so neither is in Z. Hence by 9.1.4, neither ti α nor (ti z)α acts on K, so ti α and (ti z)α are in Dm (z). Therefore zα = (ti α) · (ti z)α acts on K, contrary to 9.1.15.1. Thus (1) holds. Suppose s ∈ CZ (E) − E. If s ∈ Do (z) then v, z = CO (s), contradicting F ≤ CO (s). Therefore s ∈ Dm (z), so by 9.1.13, E  = F, s = E(s, Q1 ), and then in particlar Qs1 = Qt1 . Now st ∈ NS (Q1 ) centralizes F , so [Q, ts] ≤ z , and hence ts = f c for some f ∈ F and c ∈ CS (Q). Next let β ∈ A(s). As s ∈ Dm (z), zβ ∈ Dm (z) by 9.1.15.1. If Uβ ⊆ Dm (z), then by 9.1.14.2, zβ = i ti β acts on K, contradicting zβ ∈ Dm (z). Therefore we may assume tβ ∈ Do (z), so by 9.1.4, ts ∈ Z. Then as ts ∈ F CS (Q), ts induces an inner automorphism on O, contradicting 9.4.1.3. This completes the proof of (2). By (1), E = U and by 9.4.8, Z ⊆ Do (z) ∪ D m (z). Hence by 9.1.7.2, Eα centralizes v, so Eα centralizes r = (zα)v. As t ∈ Dm (z), we have z ∈ Dm (t) by 9.1.15.1; hence zα ∈ Dm (z). Then by (1) applied to zα in the role of t, either zαv or zαvz is in Z. Thus replacing v by vz if necessary, r ∈ Z. By 9.1.14.3 we may assume E ∈ F f , so there is γ ∈ A(Eα) with αγ ∈ AutF (E). Then rγ ∈ CZ (E), so

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rγ ∈ E by (2), and hence r ∈ Eα. Thus v = r(zα) ∈ Eα, so [Eα, v1 ] ≤ z, v ≤ Eα, so v1 induces ξ ∈ AutF (Eα). Then ζ = ξα−∗ ∈ AutF (E) with 1 = [z, ζ], so as AutQ (E) is regular on U − {z}, (3) follows. Suppose u ∈ Z ∩tCS (Q)∩CS (t). Then u ∈ CZ (E), so by (2), u ∈ tCS (Q)∩E = tCE (Q) = tz . Therefore u = t by (1), so {t} = Z ∩ tCS (Q) ∩ CS (t). Now if t = w ∈ Z ∩ tCS (Q) then 1 = tw ∈ CS (Q), so the involution d ∈ tw is in CS (Q), and td is a conjugate under t, w of t or w, so t = td ∈ Z ∩ tCS (Q) ∩ CS (t), a contradiction. This completes the proof of (4). As CS (Q) acts on Z ∩ tCS (Q), (4) implies (5). It remains to prove (6), so suppose t = s ∈ tCS (Q)Q ∩ Z. Set y = st and let y0 be the involution in y . As above, ty0 is conjugate to t or s, so ty0 ∈ Z. / CS (Q), so if |y| > 2 then y induces an outer automorphism on Hence by (4), y0 ∈ Q as Φ(Inn(Q)) = 1, contrary to the choice of s. Therefore y = y0 so [t, s] = 1. Next y = cx with c ∈ CS (Q) and x ∈ Q, and s, t centralizes c by (5), so s, t centralizes x = c−1 y. Therefore x ∈ CQ (t) = E ∩ Q so s = cxt ∈ CZ (E) ⊆ E by (2), completing the proof.  Notation 9.4.11. Set R = {R ∈ F e : AutF (R) = AutE(τ ) (R)}, and for R ∈ R, let GR be a model for NF (R), ZR = z GR , and HR = v1GR . Set T = O, D m (z) . Lemma 9.4.12. (1) E(τ ) = Fz . (2) R = ∅ and F = Fz , AutF (R) : R ∈ R . (3) For each R ∈ R, z ∈ Z(R), v ∈ R, and v1 ∈ NS (R) − R. (4) Dm (z) = ∅. Proof. Part (1) follows from the definition of E(τ ) in 3.4.2, and from 9.4.8; for example by 1.3.2 in [Asc19], Fz = SO(z), NF (O) . Then (1), 9.4.5, and the Alperin-Goldschmidt Fusion Theorem imply (2). Let R ∈ R. By 9.4.9, z ∈ Z(S), so z ∈ Z(R). Then 9.2.3 completes the proof of (3). Part (4) follows from (2), (3), and 9.2.6.  Lemma 9.4.13. (1) If m > 8 then OK ∼ = SL2 [m]. (2) If m = 8 then O 2 (AutF (O)) = 1. Proof. By 9.4.12.4 there is t ∈ Dm (z). Next for each Q8 -subgroup Q1 of K, setting E = E(t, Q1 ), it follows from 9.4.10.3 that AutNF (E) (Q1 ) contains an automorphism of order 3. Hence (2) follows and (1) follows from 2.5.4.  Lemma 9.4.14. Either (1) O is transitive on Dm (z), so that T = Ot for t ∈ Dm (z) and |T : O| = 2, or (2) O has two orbits on Dm (z) with representatives t1 and t2 , so that T = / Qt11 O for Q1 a Q8 Ot1 , t2 with T /O ∼ = E4 . Moreover if m > 8 then Qt12 ∈ subgroup of K. Proof. If O is transitive on Dm (z) then (1) holds, so assume s and t are in distinct orbits of O on Dm (z). Let Q1 be a Q8 -subgroup of K and set Q = Q1 Qt1 . / tO Suppose Qt1 = Qs1 . Then st acts on Q1 and hence also on Q as t inverts st. As s ∈ we conclude from 9.4.10.6 that s ∈ / tCS (Q)Q, so ts induces an outer automorphism on Q1 and Qt1 . Now Σ = Aut(Q) is the split extension of I = Inn(Q) ∼ = E16 by O4+ (2), so there are two orbits αiI , i = 1, 2, of I on the set of involutions in a Sylow 2-subgroup Λ of Σ which do not act on Q1 , and α1 and α2 are fused in Λ.

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Suppose m = 8. Then O = Q and we may take ct = α1 and cs = α2 , where cx is the automorphism induced by x on Q via conjugation. In particular if r ∈ Dm (z) then cr ∈ αiI for i = 1 or 2, so r ∈ sO ∪ tO by 9.4.10.4, and hence O has two orbits on Dm (z). Further s acts on tO , so s actx on Ot , so T /O ∼ = E4 and (2) holds in this case. Therefore we may assume m > 8. Then AutO (Q)t is Sylow in Σ, so by an earlier observation, s ∈ rCS (Q) for some r ∈ tNO (Q) , contrary to an earlier reduction. Thus Qt1 = Qs1 . However if r ∈ Dm (z) then as K2 has two orbits 2 for some x ∈ {t, s}, so replacing x by a suitable on its Q8 -subgroups, Qr1 ∈ QxK 1 conjugate under K2 , Qr1 = Qx1 , and hence r ∈ xO . Thus O has two orbits on Dm (z).  Again s acts on Ot , so T /O ∼ = E4 , completing the proof of the lemma. Lemma 9.4.15. CS (O) = CS (T ). Proof. This follows from 9.4.10.5.



Lemma 9.4.16. Suppose t ∈ Do (z) and set Z = z, t and R = CS (Z); then Z ∈ F f so there is a model GZ for the stabilizer of Z in NF (R). Set HZ = v1GZ . (1) Set U = O2 (HZ ). Then U ∼ = E16 with HZ /U ∼ = S3 . Set R0 = CS (U ) and embed GZ in a model GU for the stabilizer of U in NF (R0 ). (2) There exists y ∈ O − v, z with [t, y] = v such that y induces a transvection on U with center v. Set Y = HZ , y ≤ GU . (3) GU preserves a quadratic form q on U of sign +1 such that Y /U is the subgroup of O(U, q) generated by transvections, z Y = Z ∩ U is the set of singular points of U , and v Y ∪ (vz)Y is the set of nonsingular points of U . Either R = R0 and GU = Y R or |R : R0 | = 2 and GU /R0 = O(U, q). (4) Do (z) ∩ U and Dm (z) ∩ U are of order 4 and U = Dm (z) ∩ U ≤ T . (5) Do (z) ⊆ T . (6) O has two orbits on D m (z), so T /O ∼ = E4 . (7) U = CT (t). (8) O is transitive on Do (z). Proof. Let α ∈ A(Z); as v1 ∈ NS (Z) we have v1 α ∈ D ∈ Δ, so zα = (v1 α)2 = z and hence α is an Fz -map and tα ∈ Do (z). Therefore, replacing Z by Zα, we may assume for the moment that Z ∈ F f , and postpone the proof that z, t ∈ F f for all t ∈ D o (z) until later. Much of the rest of the proof is similar to that of 9.2.5. By 9.1.15.2, AutF (Z) = GL(Z). Then as v1 induces a transvection on Z with center z, there is h ∈ HZ of order 3 with z h = t and AutHZ (Z) = GL(Z). Further [R, v1 ] ≤ z, v and [NR (K), v1 ] = z , so, as in the proof of 9.2.5, either (a) R acts on K, HZ ∼ = S4 , and GZ = HZ × CGZ (HZ ), or (b) U = O2 (HZ ) = Z × E ∼ = E16 , where E = v, v h does not act on K, E  HZ , and HZ /U ∼ = S3 . Next as t ∈ Do (z) there is y ∈ O − v1 , v2 with [t, y] = v. Suppose (a) holds and set U = Zv and form GU and Y = HZ , y ≤ GU as in (1). Then CU (y) = v, z ≥ Z(HZ U ) = U0 , so AutY (U ) is the stabilizer in GL(U ) of U0 . But then no Sylow 2-subgroup of Y centralizes z, impossible as v1 ∈ Y , while by 9.4.8 Δ(τ ) = Ω, so by 3.1.3.2, z is in the center of any Sylow group of Y containing v1 . Therefore (b) holds, as does (1).

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Set u = v h . Then U = u, v, Z , and as E does not act on K, we have K u = K2 . −1 Now [u, v1 ] = v so viu = v3−i . Next y = y1 y2 with yi ∈ Ki and y1u = y2 v2β for some β, so y u = yv β . Therefore (z, t, v, u)y = (z, tv, v, uv β ). If β is even then u centralizes y, so y centralizes Ez and (2) holds in this case. If β is odd then y centralizes z, v, tu and again (2) holds. Suppose β is even. Then Y acts on E and y centralizes E and U/E, so cy ∈ O2 (AutY (U )). Then as HZ is irreducible on U/E it follows that O2 (AutY (U )) does not centralize z. Let v1 ∈ SY ∈ Syl2 (Y ) and g ∈ GU with SYg ≤ S. Then v1g ∈ D ∈ Δ, so z = (v1g )2 = z g by 9.4.8. Therefore z g ∈ Z(SYg ), a contradiction. Therefore β is odd, so y acts on E and centralizes the third irreducible E  = zv, tu for HZ on U . Hence Y acts on E and E  , so Y preserves a quadratic form q on U of sign +1 with Y /CY (U ) the subgroup of O(U, q) generated by transvections. As [y, R0 ] ≤ CO (U ) ≤ CO (t) = z, v ≤ U , we have U = CY (U ). Set SU = S ∩ GU . As z ∈ Z(SU ) and U = z Y , we have R0 = O2 (GU ). Thus by [McL69], either GU acts on E and E  , R = R0 , and GU = Y R, or GU is irreducible on U and either GU /R0 = O(U, q) with |R : R0 | = 2, or GU /R0 = GL(U ) or Sp(U ). In the first two cases (3) holds, while in the last two cases GU is transitive on U # so that v ∈ z F , a contradiction. This completes the proof of (3). Further, writing z ⊥ for the subspace of U orthogonal to z with respect to q, ⊥ z = NU (K), so Do (z) ∩ U = NU (K) − (O ∩ U ) and Dm (z) ∩ U = z G ∩ U − NU (K) are of order 4. In particular U = Dm (z) ∩ U , so U ≤ T by definition of T . That is (4) holds. By (4), t ∈ T , so (5) follows from the symmetry among the members of Do (z). Next if O had just one orbit on Dm (z), then by 9.4.14, |T : O| = 2, so O = NT (K) and hence NZ (K) = {z}, a contradiction. Now 9.4.14 completes the proof of (6). As CO (t) = v, z and |U : U ∩ O| = |T : O| = 4, (7) follows. As t Ki is semidihedral for i = 1, 2, O is transitive on the involutions in tO, so (8) follows from (5). Also (8) implies that for each s ∈ Do (z), we have z, s ∈ F f . This completes the proof of the lemma.  ∼ E4 . Then Lemma 9.4.17. Suppose T /O = (1) Do (z) = ∅. / tO , [s, t] = 1, and ts inverts v1 . Set (2) There exists t, s ∈ Dm (z) such that s ∈ A = z, v, t, s . (3) There exists w ∈ Do (z) ∩ A and for each such w, A = CT (w). Proof. Let t ∈ Dm (z), α ∈ A(t), and T˜ = T /z . Then Ot is of index 2 in T and O is transitive on tO ∩ Z by 9.4.14, so by a Frattini argument there is / O. a ∈ NT (K) ∩ CT (t) = Tt such that a ∈ Next the map ϕ : y → yy t is a Tt -equivariant homomorphism of K into CO (t) ˜ →X with image X such that CO (t) = z ×X, and ϕ induces an isomorphism ϕ˜ : K ˜ ∼ via ϕ˜ : y˜ → yϕ. Therefore X ∼ =K = Dm/2 . Observe v = v1 v1t generates Z(X) if m > 8. We first prove (2). To do so we wish to show: (a) We may choose a to be an involution. For if a is an involution, then as O has two orbits on involutions in taO with representatives ta and taz, and as one of these orbits is Z ∩ taO, replacing a by az

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if necessary, we have s = ta ∈ Z, so that s ∈ Dm (z) − tO . Then if a inverts v1 we have established (2); that is: (b) If (a) holds and a inverts v1 then (2) holds. As |NT (K) : O| = 2, a2 ∈ z X. Thus a2 ∈ CXz (a). Let b be the involution in X ∩ Φ(CT (t)). Now bα acts on K while zα ∈ Dm (z) by 9.1.15.1, so (zb)α does not act on K. Therefore as (aα)2 acts on K, we have (aα)2 ∈ Xα, so a2 ∈ X. Thus if m = 8 then Xa ∼ = D8 , so we may choose a to be an involution. As a induces an outer automorphism on K, Ka ∼ = SD16 , so a inverts v1 . Thus (2) holds by (b) in this case, so we may assume m > 8 during the remainder of the proof of (2). t = OK2 it follows from 2.12 in [Asc11] Then by 9.4.13, OK ∼ = SL2 [m], so as OK that COz (t) = z × X , where X is Sylow in X ∼ = OK /z ∼ = L2 [m]. Let s ∈ Dm (z) − tO , v0 the cyclic subgroup of K of index 2, k1 ∈ K − v0 , k = k1 k1t , Q1 a Q8 -subgroup of K, and E = E(t, Q1 ). By 9.4.14.2, Qs1 = Qty 1 for y ∈ K, so st, and hence also a, interchanges the two classes of non-2-central ˜ Hence applying the isomorphism ϕ: involutions of K. ˜ (c) a interchanges the two classes of non-2-central involutions of X. Write I(X) for the set of involutions of X. Then v ∈ I(X) with tzv = tv1 ∈ Z, so as X is transitive on I(X): (d) {tzi : i ∈ I(X)} = Z ∩ tzX is of order m/4. Set v+ = v0 v0t , v− = v0 v0−t , r+ = tk, r− = tz, V = v , and X = v , r . Then r ∈ I(X ) inverts v , so (e) X ∼ = Dm/2 . k1 k1 = r+ , so Also v− = v+ and r− (f) k1 interchanges X+ and X− , v ∈ X+ , and vz ∈ X− . Next k ∈ X inverts v0 , so j = kv0 is an involution which inverts v0 , v0t such j that r− = r− v− z and j −1 2 r+ = (tk)kv0 = (tk)v0 = (tv− )(kv02 ) = tkv− v0 = tkv+ = r+ v+ .

Also (X+ X− )j is of order m2 /2 = |CTt (v)|, so (g) CTt (v) = (X+ × X− )j , where j ∈ O is an involution inverting v with rj = r v . (h) Z ∩ Tt = Z− ∪ Z+ ∪ {z} is of order (m/2) + 1, where Z = v( )I(X ) is of order m/4, v(−) = v, and v(+) = vz. V j . Further v(+) = vz Namely I(X+ ) = {v} ∪ r++ ∪ (r+ v+ )V+ with r+ v+ = r+ j = (vzr+ )j ∈ centralizes X+ and vzr+ = vztk = tv1 k1 ∈ Z, so also vzr+ v+ = vzr+ Z. Therefore Z+ ⊆ Z is of order m/4, so by (f), also Z− ⊆ Z is of order 4. Then as |Z ∩ Tt | = |tO | + 1 = (m/2) + 1, (h) holds. (i) Z = {r ∈ tO : rv(− ) ∈ Z}. This is because for i ∈ I( ), v(− )i ∈ I( ). (j) X  CT (v). By (i) and the fact that CT (v) centralizes v, z = v(+), v(−) , CT (v) acts on Z , and hence also on I(X ) = v( )Z . ¯  of By symmetry between t and s and by (j), we have normal subgroups X ¯  . Then X ∩X− = CT (v) defined with respect to sO rather than tO . Set Y = X X V , and r¯ interchanges the two classes of non-2-central involutions of X by (c). Therefore Y = r , r¯ ∼ = Dm . Further Z(Y ) = v(− ) , so Y+ ∩ Y− = 1 and hence Y+ , Y− = Y+ × Y− is of order m2 . Therefore as |CT (v)| = m2 we conclude: (k) CT (v) = Y+ × Y− with Y ∼ = Dm .

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Next t, tvz = tv1 ∈ Z, so t ∈ Z− by (i). Thus t = vi for some i ∈ i(X− ). Now ¯ + ), vzi ∈ sO and t = vi centralizes vzi by (k). Thus we may take for i ∈ I(X  s = vzi , so that [s, t] = 1 and (a) holds. As s ∈ Z¯+ , sv ∈ Z by (i), so svz ∈ / Z and hence v1s = v1t , so a = ts inverts v1 . Therefore (b) finally completes the proof of (2). Notice (3) implies (1), so it remains to prove (3); thus we choose A as in (2). If w ∈ Do (z) ∩ A then A = CT (w) by 9.4.16.7, so we may assume Do (z) ∩ A = ∅. Let Σ = AutF (A), U = Z ∩ A, V = A# − U, U0 = {t, tzv, s, sv}, V0 = {v, vz} ∪ zU0 , and W = {st, stz, stv, stvz}. Then {z} ∪ U0 ⊆ U as we’ve seen that tv1 = tvz, and / Z, then as ts inverts v1 , sv1 = sv. Also O ∩ Z = {z} and for r ∈ Dm (z), rz ∈ so V0 ⊆ V. Next W ⊆ NT (K), so if w ∈ W ∩ U then w ∈ Do (z), contrary to assumption. Therefore: (l) U = {z} ∪ U0 is of order 5. Let ζ ∈ Σ be the automorphism cv1 induced on A by v1 and S the set of u ∈ U such that u is the unique fixed point of a Sylow 2-subgroup of CΣ (u). By Gleason’s Lemma: (m) Σ is transitive on S. As [t, v1 ] = vz and [s, v1 ] = v: (n) CA (v1 ) = [A, v1 ] = v, z , so z is the unique fixed point of ζ on U and hence z ∈ S. Let γ ∈ A(A). As v1 ∈ NS (A) it follows from 9.4.1.3 and 9.4.8 that zγ = (v1 γ)2 = z. Thus, replacing A by Aγ: (o) We may assume A ∈ F f . (p) CZ (A) = U. Namely CZ (A) ⊆ CT (t, z ) = z × Xa and CX (a) = v so CT (A) = A. Indeed there exists x ∈ X with [a, x] = v, so writing ξ for cx : (q) ξ ∈ Σ is a transvection with center v and axis z, t, v . (r) Σ is not transitive on U. Suppose Σ is transitive on U. By (l), U = {z} ∪ U0 is of order 5. Then as ζ, ξ ≤ Σ, Σ acts faithfully on U as Sym(U). But now NA (K) = [NA (K), CΣ (z)], so as v ∈ NA (K), we have NA (K) ≤ O, contradicting a ∈ NA (K) − O. We now obtain a contradiction, establishing (3) and completing the proof of the lemma. By (r) and (m), we may assume t ∈ U − S. Let α ∈ A(t). As t ∈ Dm (z), zα ∈ Dm (z) by 9.1.15.1, so r = (zα)cv1 = zαv or zαvz. If v ∈ / Aα then r ∈ CZ (Aα) − Aα. By (o), there is β ∈ A(Aα) with Aαβ = A. Then rβ ∈ CZ (A) − A, / S, contrary to (p). Thus v ∈ Aα, so v1 acts on Aα. Let θ = cv1 α−∗ ∈ Σ. As t ∈ θ = (z, u) is a transposition. Then as Λ = ζ, θ is not transitive on U, {z, t, tzv} is an orbit of Λ. Thus we have shown that if u ∈ U − S then u ∈ z Σ , so Σ is transitive on U by (m), contradicting (r) and completing the proof of the lemma.  Definition 9.4.18. Set Z1 = {z, t, v : t ∈ D m (z)}, let Q(K) be the set of Q8 -subgroups of K, define Z2 = {E(t, Q1 ) : t ∈ Dm (z) and Q1 ∈ Q(K)}, and set Z3 = {z, t : t ∈ D o (z)} and Z4 = {CT (t) : t ∈ Do (z)}. Set Ri = {CS (Z) : Z ∈ Zi }. Lemma 9.4.19. (1) O has |T : O|/2 orbits on Z1 . (2) If m = 8 and |T : O| = 2 then |Z2 | = 1. (3) If |T : O| = 4 then O is transitive on Z3 and Z4 .

9.4. |Ω(z)| = 2 AND D ∗ (z) = D(z)

Ff .

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(4) If i = 2 assume m = 8 and |T : O| = 2. Then Zi and Ri are contained in

Proof. By 9.4.14, O has |T : O|/2 orbits on D m (z). Also for t ∈ Dm (z), Z ∩ z, t, v = {z, t, tv1 }, so (1) follows. If m = 8 then Q(K) = {K}, and if |T : O| = 2 then O is transitive on Dm (z) by 9.4.14. Hence (2) holds. Finally if |T : O| = 4 then O is transitive on Do (z) by 9.4.16.8, so (3) follows. Choose i as in (4), let X ∈ Zi and α ∈ A(X). As v1 ∈ NS (X), the argument in the first paragraph of the proof of 9.4.16 shows that zα = z. Hence Xα ∈ Zi . Now by (1)-(3), Xα ∈ X S , so as Xα ∈ F f , also X ∈ F f . Similarly let Y = CS (X) and β ∈ A(Y ), Then β|X ∈ A(X) so there is s ∈ S with (Xβ)s = X. Then (Y β)s = Y ,  so Y ∈ F f , completing the proof of (4). Lemma 9.4.20. Assume |T : O| = 4 and let U ∈ Z4 . Then / R. (1) CS (U ) ∈ (2) If NS (U ) = NT (U )CS (U ) then R3 ∩ R = ∅. (3) If NS (U ) = NT (U )CS (U ) then R3 ⊆ R. Proof. As |T : O| = 4, there is t ∈ Do (z) by 9.4.17.1. Thus U = CT (t) ∈ Z4 and we adopt the notation of 9.4.16. By 9.4.16.3, GU preserves a quadratic form q on U of sign +1 and either (i) NS (U ) = NT (U )CS (U ), R = R0 , where R0 = CS (U ), GU = Y R, and GU /R is the subgroup of O(U, q) generated by transvections, or (ii) NS (U ) = NT (U )CS (U ), |R : R0 | = 2, and GU /R0 = O(U, q). Therefore R0 ∈ R4 and GU /R0 has no strongly embedded subgroup, so (1) / R, so (2) holds. On the other hand holds. Also R ∈ R3 and in case (i), R = R0 ∈ in case (ii), G/R ∼  = S3 and R ∈ F f by 9.4.19.4, so R ∈ R, establishing (3). Lemma 9.4.21. Let t ∈ Dm (z), Z = t, z, v , R = CS (Z), GR a model for NF (R), GZ = NGR (Z), and HZ = v1GZ . Then (1) HZ ∼ = S4 × Z2 , Z = O2 (HZ ), GZ = HZ CGZ (Z), and CGZ (Z)/R is of odd order. (2) R ∈ R and R ∈ R1 , so R1 ⊆ R. (3) Either (a) GR = GZ and R2 ∩ R = ∅, or (b) m = 8, |T : O| = 2, S = T CS (O), and for E ∈ Z2 , GE a model of NF (E), and HE = v1GE , we have HE ∼ = AE5 , R = ECS (O) = CS (E) ∈ R, and GZ ≤ GE . Proof. As in the proof of 9.2.7, HZ ∼ = S4 × Z2 with v = v1 v1t generating Z(HZ ), and GZ = HZ CGZ (Z). As Z  GZ , O2 (HZ ) = z HZ = Z. In particular (1) holds and GZ /R has a strongly embedded subgroup, so if Z  NF (R) then GR = GZ and R ∈ R. Hence (i) If Z  NF (R) then (2) holds. But, as in the proof of 9.4.17, CT (z, t ) = z, t ×Xs , where X = {x1 xt1 : x1 ∈ K} ∼ = Dm/2 , s = 1 if |T : O| = 2, and s ∈ Dm (z) with ts ∈ Do (z) if |T : O| = 4. Thus Z ≤ Z(CT (z, t ) so CT (Z) = CT (z, t ). Further Xz, t = E(t, Q1 ) : Q1 ∈ Q(K) , and if |T : O| = 4 then CT (Z) = U X, where U = CT (st) ∈ Z4 . Thus CT (Z) = CZ (Z)  NF (R). Further unless m = 8 and |T : O| = 2, Z = Z(CT (Z)), so indeed Z  NF (R). Therefore by (i): (ii) If m > 8 or |T : O| = 4 then (2) holds.

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Moreover it follows from 9.4.10.3 that if m > 8 or |T : O| = 4, then for E ∈ Z2 , AutF (E) ∼ = S5 has no strongly embedded subgroup, so R2 ∩ R = ∅. Thus (2) and (3a) hold is this case. Finally suppose m = 8 and |T : O| = 2. Then E = CT (Z) = E(t, K), so taking GE to be a model for NF (E), R0 = CS (E), and HE = v1GE , we have HE ∼ = AE5 A and R0 = ECS (O) ∈ R2 by 9.4.15. If S = T CS (O) then GE /R0 ∼ = 5 has a strongly embedded subgroup, so R0 ∈ R. Further R = R0 and GZ /R ∼ = S3 , so S (2) and (3b) hold in this case. Finally if S = T CS (O) then GE /R0 ∼ = 5 has no / R, but |R : R0 | = 2 and GZ /R ∼ strongly embedded subgroup, so R0 ∈ = S3 , so R ∈ R and (2) and (3a) hold.  Lemma 9.4.22. (1) If |T : O| = 4 and NS (U ) = NT (U )CS (U ) for U ∈ Z4 then R = R1 and O has two orbits on R. (2) If |T : O| = 4 and NS (U ) = NT (U )CS (U ) for U ∈ Z4 then R = R1 ∪ R3 and O has two orbits on R1 and is transitive on R3 . (3) If |T : O| = 2 and m > 8 then R = R1 and O is transitive on R. (4) If |T : O| = 2 and m = 8, then R2 = {E} is of order 1, O is transitive on R1 , and for each R ∈ R1 , NF (E) = NF (R), NFz (E) . Further either (a) S = T CS (O) and R = R1 , or (b) S = T CS (O) and R = R1 ∪ R2 . Proof. Let R ∈ R and adopt Notation 9.4.11. Suppose first that ZR ≤ NS (K). Then by 9.2.5.1, ZR = z, t for some t ∈ Do (z). Therefore |T : O| = 4 by 9.4.16.6, ZR ∈ Z3 , and R ∈ R3 by 9.2.4.2. Further GR is described in 9.2.5, and in particular U = O2 (HR ) ∈ Z4 by 9.4.16.7. By 9.4.20.2, NS (U ) = NT (U )CS (U ) and by 9.4.19.3, O is transitive on R3 so R3 ⊆ R. Conversely if |T : O| = 4 and NS (U ) = NT (U )CS (U ) for U ∈ Z3 , then R3 ⊆ R by 9.4.20.3. Therefore we may assume ZR ≤ NS (K). Then GR is discussed in 9.2.7, where there are two cases: cases (a) and (b) of 9.2.7.1. In case (b), ZR = z, v, t for some t ∈ Dm (z) and hence R = CS (ZR ) ∈ R1 by 9.2.4.2. Further by 9.4.21.2, R1 ⊆ R and by 9.4.19.1, O has |T : O|/2 orbits on R1 . So assume that case (a) holds. Then GR is described in 9.2.7.2. By 9.4.8, D∗ (z) = ∅, so neither case (i) nor (ii) of 9.2.7.2 holds, and in case (iii), ZR is not the L2 (4)-module. In case (iv), m(ZR ) > 4 and ZR = Z ∩ Zr , so ZR ≤ T , using 9.4.16.5. This contradicts m(T ) = 4. Therefore m(ZR ) = 4, so m = 8 by 9.2.7.6. Observe this completes the proof of (3). Suppose HR /ZR ∼ = Z2 /E9 . Then HR preserves a quadratic form q on ZR of sign +1 and z HR is the set of singular points of this form. In particular there is z = r ∈ z HR ∩ NZR (K), so r ∈ Do (z). Therefore |T : O| = 4 by 9.4.16.6 and ZR ∈ Z4 by 9.4.16.7, so R ∈ R4 , contrary to 9.4.20.1. Thus HR /ZR is not Z2 /E9 . Suppose next that HR /ZR ∼ = D10 . Then [ZR , v1 ] = z, v = Z0 and there are three hyperplanes of ZR containing Z0 , all conjugate under Aut(HR ). Therefore NZR (K) ∩ z HR = {z}, as this holds in AE5 which contains a copy of HR . Thus z HR − {z} ⊆ Dm (z). If |T : O| = 2 then NZR (K) ≤ O, so ZR = E(t, K) for t ∈ z HR − {z}, and hence AutF (ZR ) ∼ = A5 or S5 by 9.4.10, contradicting HR /ZR ∼ = D10 . Thus |T : O| = 4 and NZR (K) ≤ O, so ZR satisfies the hypothesis of the group A in 9.4.17. Hence ZR ∈ Z3 by 9.4.17.3. But then ZR has no automorphism of order 5 by parts (3) and (7) of 9.4.16. Thus HR /ZR is not D10 .

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235

We have eliminated all but one of the cases in 9.2.7.2, so we conclude from that lemma that HR /ZR ∼ = A5 and ZR is the A5 -module. Then by parts (3) and (4) of 9.2.7+QQ.7, ZR = E(t, K) for t ∈ z HR − {z}. As GR /R has a strongly embedded subgroup, GR = HR CGR (HR ). In particular T is Sylow in HR and |T : O| = 2. This completes the proof of (1) and (2), and (4) follows from 9.4.21.3.  Lemma 9.4.23. Assume m = 8 and |AutF (O)|3 = 3. Then |T : O| = 2 and F◦ ∼ = AE5 . Proof. Let t ∈ Dm (z) and E = E(t, K). By 9.4.10.3, Σ = AutF (E) is A5 or S5 .

Claim |T : O| = 2. If not then Σ ∼ = S5 and by 9.4.14 there is s ∈ D m (z) − tO . / O2 (CΣ (z)), so s ∈ / O2 (F). Set F = E(s, K) and Set X = NFz (E); then s ∈ Y = NFz (F ). As s ∈ O2 (Y) but s ∈ / O2 (X ), O 2 (AutX (O)) = O 2 (AutY (O)), contradicting |AutF (O)|3 = 3. This contradiction establishes the claim. By the claim and 9.4.19.2, Z2 = {E} so E  Fz . By 9.4.22.4, NF (R) ≤ NF (E) for each R ∈ R. Then by 9.4.12.2, F = Fz , NF (R) : R ∈ R ≤ NF (E), so E  F. Then the lemma follows from Theorem 2.  Because of 9.4.23, during the remainder of the section until Theorem 9.4.33 we assume: Hypothesis 9.4.24. Hypothesis 9.4.1 holds and if m = 8 then |AutF (O)|3 = 9. Lemma 9.4.25. T centralizes CF (O). Proof. As T = O, Dm (z) it suffices to show t ∈ Dm (z) centralizes Y = CF (O). By 9.4.15, t centralizes Y = CS (O), so t acts on R and NY (R) for each R ∈ Y e . Let G be a model for NYt (R); it suffices to show t ∈ Z(G). But [G, t] ≤ CY (R) = Z(R), so G acts on tZ(R), and then G centralizes t by 9.4.10.4,  since Y centralizes each Q8 -subgroup of K. Lemma 9.4.26. There exists a normal subsystem O of Fz with Sylow group O such that O ∼ = SL2 [m] ∗ SL2 [m] and T O  Fz . Proof. Suppose first that m > 8 and set O = O(z); then O ∼ = SL2 [m]∗SL2 [m] by 9.4.13, so it remains to show T O(z)  Fz . By 2.6.11, O  Fz . Applying Theorem 1.5.2 in [Asc19] to F0 = O and E = T , it suffices to show T NO (O)  NF (O). But O = NO (O) and Aut(O) is a 2-group, so NF (O) = SCF (O), and we must show T  SCF (O). As T = O, D m (z) , T  S, and by 9.4.25, T centralizes CF (O), completing the proof when m > 8. So assume m = 8, let S0 = CS (O), R = OS0 , G a model for NF (R), and H = T G . By 9.4.25, T centralizes B = CG (O) so H centralizes B. Set G∗ = ¯ i of G/B. By 9.4.24, O 2 (G∗ ) = G∗1 × G∗2 where G∗i ∼ = A4 and the preimage G ¯ i ); then Gt1 = G2 and G∗i in G centralizes K3−i . Let t ∈ D m (z) and Gi = O 2 (G H0 = [G1 G2 , t] ≤ H ≤ CG (B), so O = O2 (H0 ) and H0 /O ∼ = Z3 . Next let E ∈ Z2 , RE = CS (E), GE a model for NF (RE ), and HE = v1 GE . Then HE /E ∼ = A5 by 9.4.10 and HE has one class of complements to E, so RE = E × CRE (HE ). Then X = O 2 (CHE (z)) centralizes S0 as S0 centralizes T and then S0 = CRE (HE )z . Hence we may take X ≤ G and then X ≤ CG1 G2 (S0 ) = Y . We conclude that Y = H0 X = G1 ∗ G2 with Gi ∼ = SL2 (3). As T centralizes B, so does

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T Y = T [T, Y ]. Then as AutY (O) = O 2 (Aut(O)), G = SY B, and as T  S, we have T Y  G. Set O = FO (Y ), F1 = SCF (O), F2 = NFz (S0 ), E1 = T , and E2 = T O. Observe that this 4-tuple satisfies Hypothesis 1.4.7 in [Asc19]. For example Fz = F1 , F2 by 1.3.2 in [Asc19], E1  F1 by 9.4.25, and E2  F2 as T Y  G. Therefore  T O = E1 , E2  Fz by 1.4.8 in [Asc19], completing the proof. Definition 9.4.27. We define a subset R0 of R, depending on which case in 9.4.22 holds. In case (1), let R0 = {R1 , R2 } be representatives of the orbits of O on R, so that Ri ∈ R1 for i = 1, 2. In case (2) let R0 = {R1 , R2 , R3 } be representatives for the orbits on R1 for i = 1, 2, and R3 ∈ R3 . In cases (3) and (4) let R0 = {R1 } with R1 ∈ R1 . Set k = |R0 |, F0 = Fz , and for 1 ≤ i ≤ k set Fi = NF (Ri ) and Si = NS (Ri ). Set E0 = T O and T0 = T . For 1 ≤ i ≤ k let Ti = NT (Ri ), let Gi be a model for Fi , and set Hi = TiGi . We will see that Ti ∈ Syl2 (Hi ), so we can define Ei = FTi (Hi ) and we have Ei  Fi . Finally define E = E0 , AutEi (Ti ∩Ri ) : 1 ≤ i ≤ k T , regarded as a fusion system on T . Note E need not be E(τ ). Lemma 9.4.28. F = Fz , AutF (Ri ) : 1 ≤ i ≤ k . Proof. By 9.4.12.2, F = Fz , AutF (R) : R ∈ R . Therefore also F = Fz , AutF (Ri ) : 1 ≤ i ≤ l for any set {R1 , . . . , Rl } of representatives for the orbits of O on R. Hence the lemma follows from 9.4.22 and 9.4.27. For example in case (4b) of 9.4.22, NF (E) = NF (R), NFz (E) , so AutF (E) = AutNF (R) (E), AutFz (E) with AutNF (R) (E) induced in AutF (R) and AutFz (E) induced in Fz .  Lemma 9.4.29. (1) For 1 ≤ i ≤ k, Ti ∈ Syl2 (Hi ). (2) For 0 ≤ i ≤ k, Ei  Fi . Proof. By 9.4.26, E0  F0 , so we may take 1 ≤ i ≤ k, and from the discussion in 9.4.27, it remains to prove (1). Suppose Ri ∈ R1 . Then Ri = CS (Zi ) where Zi = z, v, ti for some ti ∈ D m (z). Suppose 9.4.21.2.a holds. Then by 9.4.21, T¯i = v1 , v, ti ∈ Syl2 (HZi ) and T¯i ≤ Ti . Then by 9.4.21, Si = Ri T¯i and Gi = HZi NGi (Si ). Finally from the proof of 9.4.21, Ti = Si ∩ Z , so Ti  NGi (Si ), and hence Hi = HZi Ti , so in particular Ti ∈ Syl2 (Hi ). So assume instead that 9.4.21.3.b holds. Then by 9.4.21, Hi ∼ = AE5 with Ti ∈ Syl2 (Hi ). This leaves the case R3 ∈ R3 in 9.4.22.2. In that event R3 = CS (Z) for Z = z, t ∈ Z3 , and from 9.4.16, U = CT (t) ∈ Z4 and, setting RU = CS (U ) and GU a model for NF (RU ), we have GU /RU = O(U, q) for a quadratic form q on U of sign +1, and with G3 = NGU (Z). Then from 9.4.16, H3 /U ∼ = S3 with T¯3 = U v1 ∈ Syl2 (H3 ) and T¯3 ≤ T3 . Further by 9.4.16.7, U = CT (Z) so T3 = T¯3 , completing the proof.  Lemma 9.4.30. (1) Assume |T : O| = 2; then: (1) E ∼ = P Sp4 [m]. (2) E = E0 , E1 .

9.4. |Ω(z)| = 2 AND D ∗ (z) = D(z)

237

Proof. As |T : O| = 2, case (3) or (4) of 9.4.22 holds, so from 9.4.27, R0 = {R1 } with R1 ∈ R1 . Thus E = E0 , AutE1 (T1 ∩ R1 ) . Let F˜ = P Sp4 [m] with Sylow ˜ group S˜ and Lie packet τ˜ = (F˜ , Ω). First, E0 = T O is determined up to isomorphism as E¯0 /Z(E¯0 ), where E¯0 = ˜ = |T | by 5.2.6 SL2 [m] wr Z2 . Therefore as F˜ satisfies our hypothesis with |S| ˜ (the Inductive Hypothesis is removed in Theorem 9.4.34), we may take T = S, ˜ ˜ ˜ ˜ ˜ Ω = Ω, and E0 = Fz = F0 . Using 9.4.28 applied to F, we conclude that F = ˜ 1 ) . Therefore to prove (1) it suffices to show that, when m > 8 F˜z , AutF˜ (R ˜ 1 ), while when m = 8 we have Aut ˜ (R ˜1) = we have AutE1 (T1 ∩ R1 ) = AutF˜ (R F AutE0 (T1 ∩ R1 ), AutE1 (T1 ∩ R1 ) . Pick t ∈ Dm (z). Then, in our identification of E0 with F˜z , we may choose t to be an image of an involution t¯ in a Sylow group T¯ of E¯0 interchanging the components C1 and C2 of E¯0 , while tz is the image of the element t¯z¯1 of order 4, where z¯1 = z(C1 ). m O ˜ We conclude that also t ∈ DF ˜ (z), and then that Z = t ∪ {z} = Z. Suppose m = 8 and set E = E(t, K). Then E = J(T ) so also E = EF˜ (t, K) ˜ 1 . Observe that F˜ satisfies ˜ Also E = R1 ∩ T1 = R and of course E ∩ Z = E ∩ Z. 9.4.21.3.b. Thus if F also satisfies 9.4.21.3.b then ˜ 1 ), AutE1 (E) = Alt(Z ∩ E) = Alt(Z˜ ∩ E) = AutF˜ (R proving (1) in this case. On the other hand if F satisfies 9.4.21.3.a, then at least ˜ 1 ) = AutE (T1 ∩ R1 ), AutE (T1 ∩ R1 ) by 9.4.22.4, so (1) holds in this AutF˜ (R 0 1 case too. Also E1e = {E}, so by the Alperin-Goldschmidt Fusion Theorem, E1 = AutE1 (T1 ), AutE1 (E) ≤ E since AutE1 (T1 ) ≤ AutE0 (T1 ). Thus (2) also holds when m = 8. So assume m > 8. Then R1 = CS (Z) where Z = z, v, t is a hyperplane ˜ 1 = z, t × X, where X = of E and v = v1 v1t . From the proof of 9.4.21, R t x : x ∈ K}. Let V = w be of index 2 in K, k1 ∈ K − V , Q1 = v1 , k1 , {x1 1 1 t 2 ˜ ˜ 1 )). Then Aut ˜ (E) = k = k1 k1 , and E = Ek1 = E(t, Q1 ). Set Σ = O (AutF˜ (R Σ 2 O (NAutF˜ (E) (Z)) centralizes v, tzk , as tzk and vtzk are the two singular points ˜ in the orthogonal space E distinct from the points z, t, tz in Z. Hence CR˜ 1 (Σ) ˜ is the subgroup of R1 generated by such 4-subgroups Ek1 as k1 varies over K − ˜ 1 )), C ˜ (Σ) is also generated by such 4V . Similarly setting Σ = O 2 (AutE1 (R R1 ˜ = C ˜ (Σ), so Aut ˜ (R ˜ 1 ) = AutE (R ˜ 1 ), and again E = F˜ subgroups. Hence C ˜ (Σ) R1

R1

F

and (1) is established. Similarly E1 ≤ E, so (2) holds.

1



Lemma 9.4.31. Assume |T : O| = 4. Then (1) E ∼ = L− 4 [m]. (2) E = Ei : 1 ≤ i ≤ k . (3) In case (2) of 9.4.22, E = E0 , AutEi (Ti ∩ Ri ) : i = 1, 2 . Proof. As |T : O| = 4, case (1) or (2) of 9.4.22 holds; in particular Ri ∈ R1 for i = 1, 2, and R3 ∈ R3 in case (2). This time set F˜ = L− 4 [m] with Sylow group ˜ Again E0 = T O is determined up to isomorphism: S˜ and Lie packet τ˜ = (F˜ , Ω). this time E0 = Eˆ0 /Z(Eˆ0 ), where Eˆ0 = SL2 [m] wr Z2 is E¯0 extended by an involution a ¯ commuting with t¯ and inducing an outer automorphism on the components Ci ¯ i ¯ ¯ i is Sylow in Ci . Therefore as F˜ such that K a is semidihedral for i = 1, 2 and K ˜ = |T | by 5.2.5.2 (again the Inductive Hypothesis is satisfies our hypothesis with |S|

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˜ Ω ˜ = Ω, and E0 = F˜z = F˜0 , and once removed in 9.4.34), we again may take T = S, ˜ i ) for 1 ≤ i ≤ k. again to prove (1) it suffices to show that AutEi (Ti ∩ Ri ) = AutF˜ (R As in the proof of the previous lemma, we may choose an identification of E m with F˜z so that Dm (z) = DF ˜ (z). Further, from the proof of 9.4.16.8, O is transitive o ˜ on the involutions in tsO, so Do (z) = DF ˜ (z), and hence Z = Z. ˜ Note that as S = T , for U ∈ (ZF˜ )4 we have NS˜ (U ) = NT (U ). Similarly if case (2) of 9.4.22 holds and R3 ∈ R3 then R3 = CS (Z3 ) for Z3 = z, t and t ∈ Do (z); and from 9.4.16, U = CT (t) ∈ Z4 with E3 a model for H3 and G3 = NGU (R3 ), where GU is a model for NF (U ). Then H3 ≤ HU = T3GU with HU = X1 × X2 and Xi ∼ = S4 . But R3 ∩ T3 = U and AutH3 (U ) ≤ AutHU (U ) = AutX1 (U ), AutX2 (U ) , while AutXi (U ) ≤ AutNF (Pi ) (U ) for Pi = CS (Zi ) ∈ R1 and Zi a suitable hyperplane of  U in Z1 . Then as Pi ∈ RO j(i) for some j(i) ∈ {1, 2}, AutH3 (H) ≤ AutE (U ), where   E = E0 , AutEi (Ti ∩ Ri ) : i = 1, 2 ; that is E = E , so (3) holds and it suffices to show AutEi (Ti ∩ Ri ) = AutF˜ (Ri ∩ Ti ) and Ei ≤ E for i = 1, 2. If m = 8 then NF (Ri ) ≤ NF (Ei ) for i = 1, 2, where t1 and t2 are representatives for the orbits of O on Dm (z), and Ei = E(ti , K). Indeed Ri ∩Ti = Ei ri for ri ∈ Ti inducing a transvection on Ei . Then we can argue as in the proof of the previous lemma to complete the proof. So assume m > 8. Recall Ri = CS (Zi ) with Zi = z, v, ti , and from 9.4.21.1, HZi ∼ = S4 × Z2 , while from the proof of 9.4.29, Hi = HZi Ti , so Ri ∩ Ti = CTi (Zi ). Further from the proof of 9.4.29, CTi (Zi ) = U Xi where Xi = E(ti , Q1 ) : Q1 ∈ Q(K) and U = CT (t1 t2 ). Arguing as in the proof of the previous lemma, the action of HZi on Ei = E(ti , Q1 ) is determined: O 2 (HZi ) fixes the two singular points of Ei not contained in Zi . Similarly the action of HZi on U is determined: HZi centralizes the definite line of U not contained in Zi . Therefore ˜ i Ti ), AutEi (Ri ∩ Ti ) = AutHi (Ri ∩ Ti ) = AutF˜ (R establishing (1). Also Ri ∩ Ti is the unique member of Eie and Ti = NEi (Ti ), so  Ei ≤ E, completing the proof. Lemma 9.4.32. F ◦ = E. Proof. We verify that the family Fi , Ei , 0 ≤ i ≤ k, satisfies Hypothesis 1.4.3 in [Asc19]. Then by 1.4.5 in [Asc19], E  F, so F ◦ = E. By 9.4.28, F = Fi : 0 ≤ i ≤ k , and then by construction of our tuple, Hypothesis 1.4.1 in [Asc19] is satisfied. By 9.4.29, Ei  Fi for each i. By 9.4.30  and 9.4.31, E is saturated and E = O 2 (E). As AutF (T ) permutes the orbits of O on R contained in R0 , AutF (T ) ≤ Aut(E) by 1.4.6 in [Asc19]. Finally we verify the extension condition in 1.4.3.3 in [Asc19]: In case (4b) of 9.4.22, T1 = T , so the lifting condition is trivially satisfied. In the remaining cases for Ri ∈ R0 , T = Ti O and Ki ∩Ti = vi , so AutF (Ti ) acts on v1 , v2 = X. But O  NF (X), so AutF (Ti ) lifts to OTi = T , completing the proof.  Theorem 9.4.33. Assume Hypothesis 9.4.1. Then either (1) |T : O| = 2, Do (z) = ∅, and F ◦ ∼ = AE5 or P Sp4 [m], or (2) |T : O| = 4, Do (z) = ∅, and F ◦ ∼ = L− 4 [m]. Proof. By 9.4.14, |T : O| = 2k, where k = 1 or 2 is the number of orbits of O on Dm (z). Then by 9.4.16.6 and 9.4.17, k = 1 iff Do (z) = ∅. By 9.4.23 we

9.4. |Ω(z)| = 2 AND D ∗ (z) = D(z)

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may assume that Hypothesis 9.4.24 is satisfied. By 9.4.32, F ◦ = E, and then F ◦ ∼ = [m] when k = 2 by 9.4.31.1.  P Sp4 [m] when k = 1 by 9.4.30.1, while F ◦ ∼ = L− 4 Theorem 9.4.34. Assume τ = (F, Ω) is a quaternion fusion packet such that for all z ∈ ZS we have |Ω(z)| = 2. Assume also that F ◦ is transitive on Ω and μ(τ ) ∼ = E4 . Then (1) F ◦ ∼ = AE5 , P Sp4 [m], or L− 4 [m]. (2) F = Fz , AutF (Ri ) : 1 ≤ i ≤ k , where the Ri are defined in 9.4.27 Proof. We verify Hypothesis 9.4.1; then the theorem follows from 9.4.33 and 9.4.28. By hypothesis, conditions (1) and (2) of 9.4.1 are satisfied; for example as μ∼ = E4 , 9.4.1.3 holds = E4 , μ has no D12 -subgroup, so A(τ ) = ∅ by 3.2.1. As μ(τ ) ∼ by 3.1.21. Suppose ρ = (Y, Γ) satisfies the hypothesis of 9.4.1.4. As μ(τ ) ∼ = E4 , we conclude that μ(ρ) ∼ = Z2 or E4 . In the first case, ρ satisfies one of the conclusions of Theorem 1 by 7.1.29. In the second, ρ satisfies the hypothesis of this theorem, so as ρ is proper in τ , also ρ satisfies one of its conclusions by induction. This completes the proof of the theorem.  Theorem 9.4.35. Assume τ = (F, Ω) is a quaternion fusion packet such that for z ∈ ZS , |Ω(z)| = 2 and Z ∩ O(z) = {z}. Assume μ(τ ) ∼ = S4 . Then F = Fz , NF (W ) for W ∈ W (τ ), and one of the following holds: (1) F ◦ = FS∩M (M ) for M ∈ M (τ ). (2) m = 8 and F ◦ ∼ = AE6 . (3) F ◦ ∼ [m]. = P Ω+ 6 Proof. We verify Hypothesis 9.3.1; then the theorem follows from 9.3.24. By hypothesis, μ = μ(τ ) ∼ = S4 , so A(τ ) = ∅. By 3.1.27, μ = μ(τ ◦ ). Let ◦ η ∈ η(τ ); as μ ∼ = S4 , μ is transitive on Dη , so by 4.2.8.7, F ◦ is transitive on Ω. Thus conditions (1)-(3) of 9.3.1 are satisfied. Let ρ = (Y, Ω) be as in the Inductive Hypothesis; we must show that ρ appears as a conclusion to Theorem 1. Now ν = μ(ρ) ≤ μ ∼ = S4 , so ν is Z2 , E4 , S3 , or S4 . If ν is Z2 or S3 then ρ appears in Theorem 1 by 7.1.29. If μ ∼ = E4 this follows from 9.4.34. Finally if μ is S4 then as W (ρ) ≤ W (τ ) and M (ρ) ≤ M (τ ), we conclude that M (ρ) = M (τ ), so O(z) ∩ Z(ρ) = ∅ by 9.1.10.4. Thus ρ satisfies the hypothesis of this theorem, so as ρ is proper in τ , it satisfies one of the conclusions of the theorem by induction, completing the proof.  The last two lemmas in this section will be applied in section 11.2. Lemma 9.4.36. Assume τ = (F, Ω) is the Lie packet of F ∼ = L− 4 [m]. Let z ∈ ZS , K ∈ Ω, v1 of order 4 in K with v1  NS (K), and set v = v1 v1s for some s ∈ S − NS (K). Let u1 = v and u2 = vz. Let t ∈ D o (z), U = CS (t), and GU a model for NF (U ). (1) U ∼ = E16 and G+ U = GU /U is the subgroup of O(U, q) generated by its transvections, for some quadratic form q on U of sign +1. Further v1+ inverts O(G+ U ). (2) For h of order 3 in GU with CU (h) = 1 we have GU = h, U NO(z) (U ) . (3) For i = 1, 2 let U (i) = z CGU (ui ) . Then F = CF (z), NF (U (i)) : i = 1, 2 .

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Proof. First τ satisfies the hypothesis of 9.4.34, and hence from the proof of 9.4.34, τ also satisfies Hypothesis 9.4.1. Then (1) follows from 9.4.16 and (1) implies (2). Part (3) follows from 9.4.22, 9.4.27, and 9.4.28, or see 9.4.34.2.  Lemma 9.4.37. Assume τ = (F, Ω) is the Lie packet of F ∼ = Ω7 [m] or Ω− 8 [m]. Let z ∈ ZS , K ∈ Ω, v1 of order 4 in K with v1  NS (K), and v = v1 v1s for some s ∈ S − NS (K), Set u1 = v and u2 = vz. Suppose E16 ∼ = U ≤ S such that U = z, zα, v, vα for some α ∈ AutF (U ) of order 3 with CU (α) = 1, zα ∈ Do (z), K vα = K, and cv1 inverts α. Set X = α, NO(z) (U )U UNO(z) U , and for i = 1, 2 let U (i) = z CX (ui ) . Set Y = U Oz , NO(z) (Ui )NX (Ui ) : i = 1, 2 . Then Y ∼ = L− 4 [m]. Proof. Regard F as FS (L) where L = Ωd (q), and let V be the orthogonal  space defining L. Set z1 = z, z2 = zα, Vi = [V, zi ], Yi = O p (CL (Vi⊥ )), Y = Y1 , Y2 , and V0 = V1 + V2 . By definition, Ozi = FO(zi ) (Yi ). As z2 ∈ Do (z), V0 is nondegenerate of dimension 6 and sign −π. Further Yi centralizes Vi⊥ , so ⊥ Y ≤ CL (V0⊥ ) ∼ = Ω−π 6 (q). Indeed Y = CL (V0 ) with T = U O(z) ∈ Syl2 (Y ), so we can apply 9.4.36 to Y0 = FY (Y) to conclude that Y = Y0 , completing the proof. 

9.5. |Ω(z)| = 2 and μ isomorphic to S4 In this section we assume the following hypothesis: Hypothesis 9.5.1. τ = (F, Ω) is a quaternion fusion packet such that (1) For z ∈ ZS , |Ω(z)| = 2 and Z ∩ O(z) = {z}. (2) μ ∼ = S4 . Notation 9.5.2. Observe that Hypotheses 9.1.1 and 9.1.8 are satisfied, and adopt Notation 9.1.2 and 9.1.6. For example as μ ∼ = S4 , F is transitive on Ω by 4.2.8.7. Let t ∈ z M − {z}, E = z, t , a = vzt, b = vt, and O = O(z). For c ∈ {a, b}, let Tc be the Sylow group of Fc◦ . The choice of a presupposes notation is chosen so that v = v1 v2 and Z(M ) = v1 v2 zt , as in the proof of 9.5.3. Lemma 9.5.3. (1) E # = ZΔ . (2) Z(M ) = a is of order 2. (3) b centralizes O and b ∈ O g for each g ∈ M − CM (z). Moreover M is transitive on {b, vz, v}. (4) a, b, z ∈ Z(S). (5) CS (v) = CS (t) = CS (E). (6) Let i be an involution in S. (i) If K i = K then i ∈ CS (E). (ii) If i ∈ Do (z) then i ∈ CS (E). (iii) If i ∈ Z − CZ (E) then i ∈ D∗ (z). (7) The Inductive Hypothesis is satisfied. (8) Ω = Ω(z). ∼ S4 , so |Ω| = 2 by 4.2.8.6. By 9.5.1.1, |Ω(z)| = 2, so Proof. By 9.5.1.2, μ = (8) holds. As μ ∼ = S4 , M is described in 5.7.4. In particular S ∩ M = CM (z) is of index 3 in M , so |z M | = 3. Applying 4.2.5 to K, K g for g ∈ M − CM (z), (1) follows.

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We conclude from 5.7.4.2 that Z(M ) = (¯ v1 v¯2−1 )m/8 t for suitable choice of genm/8 erators v¯i for Ki ∩ W and i = 1, 2. We choose notation so that vi = v¯i . By (8), −1 / O(z), v2 = v2 z so that Z(M ) is generated by v1 v2 zt = a. As Z ∩ O(z) = {z}, t ∈ so a = 1, establishing (2). By (2), b = az centralizes O. Similarly as M is transitive on E # , M is also transitive on aE # = {b, vz, v}. Then as v ∈ O, b ∈ O g for some g ∈ M − O, and then as O ≤ CS (b) is transitive on E − z , (3) follows. As Ω = Ω(z), z ∈ Z(S). As S acts on M , a ∈ Z(S) by (2). Then b = az ∈ Z(S), proving (4). By (4), vt = b and z are in Z(S), so (5) follows. Assume i is an involution in S. If K i = K then v1 v1i ∈ CS (i) with v1 v1i = vz  , so i centralizes v, and hence (6i) follows from (5). If i ∈ Do (z) then i inverts v1 and v2 , so i ∈ CS (v) = CS (E), proving (6ii). Finally assume i ∈ Z − CZ (E). Then by (6i), (6ii), and 9.1.3, i ∈ D∗ (z), completing the proof of (6). Assume the setup of the Inductive Hypothesis. Then, as in the proof of 9.4.35, ν is Z2 , E4 , S3 , or S4 , and when ν is Z2 or S3 then ρ satisfies one of the conclusions of Theorem 1 by 7.1.29. If ν is E4 the same holds by 9.4.34, so we may assume ν is S4 . Then ρ satisfies Hypothesis 9.5.1, so as ρ is proper in τ , it satisfies one of the conclusions of Theorem 9.5.33 by induction, and hence also appears in Theorem 1. Hence (7) holds.  For x an involution in S, we write Fx◦ for (Fx )◦ . Recall P Sp4 (q) ∼ = Ω5 (q) and ∼ = P Ω6 (q).

L4 (q)

Lemma 9.5.4. (1) τb = (Fb , Ω) is a quaternion fusion packet with μ(τb ) ∼ = E4 . (2) Either Fb◦ = O(z) or Fb◦ ∼ [m]. = AE5 , P Sp4 [m], or L− 4 Proof. From 9.5.3.3, b centralizes Ω and O  CM (b). Therefore τb is a quaternion fusion packet by 2.5.2, and as O  CM (b), μ(τb ) ∼ = E4 . That is (1) holds. By 9.5.3.8, S is transitive on Ω, so by 9.5.3.4, Fb is transitive on Ω. Suppose Fb◦ is transitive on Ω. Then (1) and Theorem 9.4.34 imply (2). So assume Fb◦ is not transitive on Ω. If z ∈ Z(Fb◦ then Fb◦ = O(z) and (2) holds, so assume otherwise. Let D = [K]Fb◦ ; then δ = (D, K) is a quaternion fusion packet. Further as μ(τb ) ∼ = E4 , μ(δ) = Z2 , so by 7.1.6 and 7.1.7, D is L2 [2m](1) − or L3 [m]. Further C = E(D) is the unique component of Fb containing z, so s ∈ S − NS (K) acts on C. Set J = K ∩ C and let s ∈ S − NS (K); then J s ≤ C centralizes J, a contradiction to the structure of D.  Lemma 9.5.5. (1) τa = (Fa , Ω) is a quaternion fusion packet with μ(τa ) ∼ = S4 . (2) One of the following holds: (i) Fa◦ = FTa (M ). ˇ 6. (ii) m = 8 and Fa◦ ∼ = AE + (iii) Fa◦ ∼ [m]. Ω = 6 Proof. Part (1) follows from 9.5.3.2 and 2.5.2. Replacing τ by τa , we may assume a ∈ Z(F) and set F + = F/a and τ + = (F + , Ω+ ). Then by 3.3.2, τ + is a quaternion fusion packet with μ(τ + ) ∼ = S4 . As μ(τ + ) ∼ = S4 , (F + )◦ is transitive on + Ω by 3.1.27 and 4.2.8.7. Then using 9.5.3.7 and observing that t+ = (V z)+ ∈ O + , τ + satisfies Hypothesis 9.3.1. Thus F + is described in Theorem 9.3.24. Then F appears in (2) by 5.1.21 and 3.3.16. 

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∗ Lemma 9.5.6. Let E = E(τ ) and DE = e∈E # D∗ (e). (1) Fe ≤ E for e ∈ E # . (2) Fa ≤ E. ∗ , there exists α ∈ homE (CS (s), S) with sα = z. (3) For each s ∈ DE Proof. Let g ∈ M with tg = z; then β = cg ∈ homE (CS (t), S) with tβ = z as NF (W ) ≤ E. By 3.4.3.4, Fz ≤ E, so Ft β ∗ ≤ Fz ≤ E. Then as β is an E-map, Ft = (Ft β ∗ )β −∗ ≤ E, establishing (1). By 1.3.2 in [Asc19], Fa = SFa◦ , NFa (Ta ) . Further W  NF (Ta ), so NFa (Ta ) ≤ NF (W ) ≤ E. By 9.3.20.1 applied in Fa /a and by 2.1.15, SFa◦ = SCFa◦ (z), NSFa◦ (W ) , so as Fz , NF (W ) ≤ E, (2) follows. Part (3) follows from an argument in 10.1.15. For s ∈ Z let B(s) = {α ∈ homE (CS (s), S) : sα = z}. If s ∈ D∗ (z) then by 9.1.10.3 there is γ ∈ AFz (s) with sγ = t. Then as Fz ≤ E, γβ ∈ B(s). Finally suppose r ∈ D ∗ (t) − ({z} ∪ D∗ (z)). Claim CS (r) ≤ CS (E). For if not, CS (r), COg (r) induces GL(E) on E, so r ∈ D∗ (z), contrary to the choice of r. As CS (E)  CS (E)M , by the claim, CS (r)β ≤ S, so β ∈ homE (CS (r), S). From the previous case there is δ ∈ B(rβ). Then βδ ∈ B(r), completing the proof of (3).  Lemma 9.5.7. Let s ∈ Z such that sFb ∩ E = ∅. Then (1) Either D ∗ (s) ∩ E = ∅ or s ∈ z Fb ∩ D o (z) and Fb◦ ∼ = L− 4 [m]. ∗ (2) Let D (s) ∩ E = ∅ and set U = CTb (s). Then either Fb ≤ E(τ ) or AutTb (U ) = AutS (U ). Proof. Assume D∗ (s) ∩ E = ∅ and let δ ∈ A(s). By 9.5.3.6.iii, s centralizes E. Thus [s, M ] ≤ CM (E) = O2 (M ). Therefore s induces an automorphism σ on M/W = μ ∼ = S4 centralizing μ/O2 (μ). Then as μ = Aut(μ) it follows that σ ∈ O2 (M/W ). Therefore s = cy for some y ∈ O2 (M ) and c ∈ CS (M/W ). Now c acts on each member of K M and y acts on some J ∈ K M , so s acts on V = J ∩ W . Set e = z(J). As D∗ (s) ∩ E = ∅, it follows that s ∈ Do (e), so s inverts V . Thus e ∈ [V, s]. But if s ∈ z Fb then s ∈ Tb so [V, s] ≤ Tb . On the other hand if s ∈ tFb then s = bu for some u ∈ Tb , so again [V, s] = [V, u] ≤ Tb . Hence e ∈ Tb ∩ E = z , so e = z. Therefore s ∈ Do (z), and by hypothesis, s is fused in Fb to z or t = bv. Hence there exists i ∈ z Fb ∪ v Fb inducing an outer automorphism on K. Then as, among the systems listed in 9.5.4.2, there exists such an involution in Tb inducing an outer automorphism on K iff Fb◦ ∼ = L− 4 [m], and in that case all such involutions are in z Fb , it follows that (1) holds. ∼ Now assume the setup of (2). By (1), Fb◦ ∼ = L− 4 [m], so from 9.4.16.7, U = E16 and from 9.4.18, U is in the set Z4 defined there. We may assume AutTb (U ) = AutS (U ), so by 9.4.22.1 and 9.4.28, Fb = CFb (z), AutFb (Ri ) : i = 1, 2 , where Ri ∈ R1 for i = 1, 2. However from 9.4.18 and 9.4.21, AutFb (Ri ) centralizes v or vz, and hence also centralizes t = bv or tz = bvz. Therefore AutFb (Ri ) ≤ AutE(τ ) (Ri ) by 9.5.6.1. Then as Fz ≤ E(τ ), Fb ≤ E(τ ), completing the proof of (2). 

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∗ Lemma 9.5.8. Let s ∈ Z. Then either s ∈ DE or the following hold: g Fb o (1) For some g ∈ M , s ∈ z ∩ D (z). Further either CS (s) ≤ CS (E) or we may take g = 1, and in either case γ = cg ∈ homE(τ ) (CS (s), S). (2) Fb◦ ∼ = L− 4 [m]. (3) There exists φ ∈ homFb (CS (sg ), S) with sg φ = z, and if AutTb (CTb (sg )) = AutS (CTb (sg )) then φ ∈ homE(τ ) (CS (sg ), S) and γφ ∈ homE(τ ) (CS (s), S). ∗ Proof. Assume s ∈ / DE and let α ∈ A(s) and F = Ea . By 9.5.3.6.iii, s ∗ / DE , f ∈ F − E. centralizes F . Then for some f ∈ F , f α centralizes v1 . As s ∈ If f = a then for δ ∈ A(aα), v1 δ ∈ D1 ∈ Δ, so either v1 δ ∈ W or zδ = z, and in either case zδ ∈ E. As ξ = αδ ∈ homFa (s, S) and sξ = zδ ∈ E, we have s ∈ z Fa . ∗ (cf. 9.3.14.1). This is a contradiction as z Fa ⊆ DE M Therefore f ∈ b , so there is g ∈ M with f g = b. Now f α = CF α (v1 ), so f is CS (s)-invariant. But if [CS (s), E] = 1 then b is the unique member of bM centralized by CS (s), so we may take g = 1. On the other hand if CS (s) ≤ CS (E) then CS (s)g ≤ S, as in the proof of 9.5.6.3. Therefore in either case, γ = cg ∈ homE (CS (s), S). Let β ∈ A(f α), ζ = γ −1 αβ, and r = sγ. Then ∗ / DE and ζ ∈ homFb (CS (s)γ, S) with v1 β ∈ W or zβ = z, so rζ ∈ E. Now as s ∈ ∗ g Eγ = E, r ∈ / DE , so (1) and (2) hold by 9.5.7.1 applied to r = s in the role of s. Let φ ∈ AFb (r); then φ satisfies the conditions of (3), using 9.5.7.2 to verify the second condition. 

Lemma 9.5.9. (1) E(τ ) = Fz , NF (W ) . (2) Set B(τ ) = E(τ ) unless Fb◦ ∼ = L− 4 [m] and AutS (CTb (s)) = AutTb (CTb (s)) Fb o for s ∈ z ∩ D (z), where we set B(τ ) = E(τ ), Fb . Then F = B(τ ). Proof. Part (1) follows from the definition of E(τ ) in Definition 3.4.2, and the fact that Ω = Ω(z). To establish (2), we argue as in the proof of 10.1.15. Set B = B(τ ). We first observe that: (a) For each s ∈ Z there exists α ∈ homB (CS (s), S) with sα = z. ∗ ∗ this is 9.5.6.3, while if s ∈ / DE it is 9.5.8.3. If s ∈ DE Now (2) follows from (a) and 2.1.8.  Notation 9.5.10. Let S0 = CS (O), R0 = OS0 , and Gz a model for NF (R0 ). Set Hz = W Gz , Lz = O 2 (Hz ), and Bz = bLz . If m > 8 set O = O(z) and for c ∈ {a, b} set Oc = CFc◦ (z)◦ . If m = 8 set O = FO (OO 2 (CLz (S0 )) and Oc = O 2 (CFc◦ (z))O, except when Fb◦ = O set Ob = O. Lemma 9.5.11. (1) [W, S0 ] ≤ CO(t) (O), CO(t) (O) is cyclic, and if m = 8 then CO(t) (O) = b . (2) If m > 8 then either O(z) = O, or OK ∼ = SL2 [m] ∗ = SL2 [m] and O(z) ∼ SL2 [m]. (3) If m = 8 then one of the following holds: (i) Hz = W . (ii) Lz centralizes S0 , and is of index 1 or 3 in SL2 (3) ∗ SL2 (3). (iii) O2 (Lz ) = O × Bz where Bz = bLz ∼ = E4 , and Lz = O2 (Lz )Xz for Xz ∈ Syl3 (Lz ). Moreover Lz /CLz (Bz ) ∼ = Z3 , Xz ∼ = E3e for some 1 ≤ e ≤ 3, and Z(Lz ) = z . (4) If m = 8 then O 2 (AutF (O)) = O 2 (AutHz (O)) and O 2 (CGz (S0 )) ≤ Lz . Proof. For g ∈ M set Vzg = (O ∩ W )g . From 5.7.4, W = Vz Vt , so [W, S0 ] = [Vt , S0 ]. Further S0 ≤ CS (v) = CS (t) by 9.5.3.5. Thus [Vt , S0 ] ≤ Vt ∩ S0 = CVt (O),

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while CO(t) (O) is cyclic by 5.7.5.3, with CO(t) (O) = b when m = 8. Thus (1) holds. Part (2) follows from 9.1.11.1. Thus it remains to prove (3) and (4), so we may assume m = 8 and set G+ z = Gz /O. We first prove (3). Observe that by 5.7.4.1, W + = w+ , where Z4 ∼ = w ∈ Δ with w2 = t. As b = vt, t+ = b+ . Set Bz = bLz . As b ∈ Z(S), Bz ≤ Z(R0 ) so O ≤ R0 ≤ CGz (Bz ). Set Hz! = Hz /CHz (Bz ). Then W ! = w! ∼ = Z2 , and as W is weakly closed in S, W ! is weakly closed in S ! , so by the Z ∗ -Theorem, L!z = O(L!z ). By (1), [S0 , W ] ≤ b . Suppose Lz centralizes b. Then as [S0 , W ] ≤ b , it follows that Lz centralizes S0 . Thus Hz ≤ CG (a), so from 9.5.5.2, (3i) or (3ii) + holds. Therefore we may assume Bz+ = [Bz+ , L+ z ] = b . ! + As w induces a transvection on Bz with center generated by b+ , and as L!z = O(L!z ), we conclude that Bz+ ∼ = E4 and Hz! = GL(Bz+ ). Indeed as [S0 , W ] = b , + ∼ + we have Hz = S4 with Bz = O2 (Hz+ ) and S0 = Bz CS0 (Hz ). Then w2 = t ∈ O2 (Hz ) ≤ OBz . Suppose Lz centralizes O. Then L = Lz w ∼ = Z2 × S4 has center generated by v = bt. Now by 9.5.3.3 there is α ∈ A(v) with vα = b, and we may assume v ∈ CM (K g ), where g ∈ M with t = z(K g ). Therefore b = vα ∈ O2 (L)α = z L ≤ Tb , contrary to 9.5.4.2. Thus Lz does not centralize O. Then as K2 ∈ K S , O = [O, Lz ] ≤ Lz , so O2 (Lz ) = OBz . Now as O ∼ = Q28 , Out(O) ∼ = O4+ (2) and then (3iii) holds, completing the proof of (3). ˜ z = Gz /z . For w ∈ W − OCS (O), w is nontrivial Let Σ = OutF (O) and G ˜ i for i = 1, 2, so w inverts O(Σ). Thus the first statement in (4) holds. Let on K X = O 2 (CGz (S0 )); as Out(O) is 2-nilpotent, X + is of odd order, so X = X0 Y where X0 is a Hall 2 -subgroup of X and Y = O ∩ X. As F ∗ (Gz ) = R0 , X0 is faithful on Y and then as w inverts O(Σ), X0+ = [X0+ , w] ≤ L+ z , so X ≤ Lz , completing the proof of (4).  Lemma 9.5.12. (1) O = Oc centralizes S0 for c ∈ {a, b}. ◦ ∼ + (2) If m > 8 then OK ∼ = SL2 [m] iff Fb◦ ∼ = P Sp4 [m] or L− 4 [m] iff Fa = Ω6 [m]. (3) O  Fz . Proof. By 2.6.12, S0 centralizes O(z). Thus if m > 8 then as a, b ≤ S0 , O = O iff Oa = O iff Ob = O, and then (1) holds in this case by 9.5.11.2. Then (1), 9.5.4, and 9.5.5 imply (2), while (3) holds as O(z)  Fz . Therefore we may assume that m = 8. Let c ∈ {a, b}. Then from 9.5.4 and 9.5.5, Oc centralizes S0 ∩ Tc , so Oc centralizes S0 . Therefore Oc ≤ O by 9.5.11.4. By construction, O centralizes S0 , so as c ∈ S0 we have O ≤ Fc . + ◦ ∼ 2 2 If Fa◦ ∼ = Ω+ 6 [m] or Fb = P Sp4 [m] or L4 [m] then O (AutOc (O)) = O (Aut(O)), so as Oc ≤ O, we have Oc = O. Suppose 9.5.5.2.i holds. Then FTa (M )  Fa , so NFa (O) ≤ NFa (Ta ), and hence as O 2 (AutF (Ta )) = 1, also O 2 (AutFa (O)) = 1, so O = Oa = O. 2 ◦ ˇ 6 or F ◦ ∼ Suppose Fa◦ = AE b = AE5 . Then O (AutFc (O)) acts on O2 (Fc ∩ O) 2 2 and hence O (AutFc (O))) = O (AutOc (O)). Thus again O = Oc . This leaves the case Fb◦ = O, where by definition O = Ob . This completes the proof of (1), so it remains to prove (3). By construction, O centralizes S0 and O is Sylow in O, so the central product O ∗ CF (O) is a subsystem of Fz by 2.3 in [Asc11]. Then as AutO (O)  AutF (O) by construction, we have O  Fz . This completes the proof of (3) and the lemma. 

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Lemma 9.5.13. If a ∈ Z(Fz ) then a ∈ Z(F). Proof. Assume a ∈ Z(Fz ). By 9.5.9.1, E = E(τ ) = Fz , NF (W ) , so as a ∈ Z(NF (W )), also a ∈ Z(E). Thus we may assume E = F, so by 9.5.9.2, Fb◦ ∼ = L− 4 [m]. Let s ∈ Dm (z) ∩ z Fb , Z = s, z, v , R = CS (Z), GR a model for NFb (R), GZ = NGR (Z), and HZ = v1GZ . By 9.4.21 there is h ∈ HZ with z h = s, and from the proof of 9.4.21, v0 = v1 v1s ∈ Z(HZ ). Now v0 = v or vz, and h centralizes bv0 = e = t or tz. By 9.5.6.1, Fe ≤ E, so as a ∈ Z(E) we have ah = a. However  a = bz, so ah = bz h = bs = bz = a, a contradiction. Lemma 9.5.14. Assume Fb◦ = O(z). Then (1) t ∈ Z(Fvz ). (2) CF (O) ≤ CFz (vz) ≤ CFz (a). (3) Assume a ∈ / Z(F). Then m = 8, E4 ∼ = Bz  Fb , and O 2 (AutF (O)) = O 2 (AutO (O)). Proof. By hypothesis, O(z)  Fb , so z ∈ Z(Fb ). Let g ∈ M interchange t and z. Then g interchanges at = vz and az = b, so (1) holds. By (1), CFz (vz) = CFz (vz, t ) ≤ CFz (a), and of course CF (O) ≤ CFz (vz), so (2) holds. Assume a ∈ / Z(F). Then a ∈ / Z(Fz ) by 9.5.13. Let F0 = O(z)CF (O). By 2.6.14, CF (O) = CF (O(z)) and F0  Fz . By (2), a is in the center of SF0 , so Fz = SF0 . But by 1.3.2 in [Asc19], Fz = SF0 , NF (OS0 ) . Recall that Gz is a model for NF (OS0 ). If AutF (O) is a 2-group then Gz = SCGz (O) and FS0 (CGz (O)) ≤ SF0 , so NF (OS0 ) ≤ SF0 , contradicting Fz = SF0 . It follows that m = 8. By 9.5.11.4, Gz = SHz CGz (O), so if Bz = b then Gz centralizes a, z = b, z , contradicting a ∈ / Z(Fz ). Then by 9.5.11.3, Bz ∼ = E4 . Finally by 9.5.12.1, O = Oa centralizes a. By 9.5.11.4, AutLz (O) = O 2 (AutF (O)), so if O 2 (AutF (O)) = O 2 (AutO (O)) then as O centralizes S0 and b ∈ S0 by 9.5.3.3, we have Lz = O 2 (CHb (a)CLz (O), so as CLz (O) ≤ CHz (a) by (2), and as CHz (Bz )  centralizes b, Hz centralizes a, z = b, z , contradicting Bz ∼ = E4 . Lemma 9.5.15. Assume Fb◦ = O(z) and a ∈ / Z(F). Let k = |AutO (O)|3 and T = Ta Bz . Then m = 8 and: (1) Either (a) k = 1, O = O, and Fa = FTa (M ), or ˇ 6. (b) k = 3, O is of index 3 in SL2 [8] ∗ SL2 [8], and Fa ∼ = AE (2) There exists X of order 3 in Hz such that COBz (X) = z and XS = SX. (3) Let A = tXO and τA = (NF (A), Ω). Then E8 ∼ = A  S, τA is a quaternion fusion packet, and NF (A)◦ = FT (T X), FT (T M ) ∼ = L3 (2)/E64 has Sylow group T. (4) CF (O) ≤ CFz (Bz ). (5) Fz = SCF (O), FS (Gz ) . (6) Bz  Fz . (7) FT (T M )  NF (W ). (8) T  SCF (O). (9) FT (T Lz )  Fz . (10) If k = 1 then F ◦ = NF (A)◦ ∼ = L3 (2)/E64 .

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∼ E4 and O 2 (AutF (O)) = O 2 (AutO (O)). In Proof. By 9.5.14.3, m = 8, Bz = 2 2 particular O (AutO (O)) = O (Aut(O)) so k = 1 or 3, while by 9.5.12.1, O = Oa , and then (1) follows from 9.5.5.2. As b ∈ Bz ∼ = S3 and then as b ∈ S0 we = E4 , it follows that AutHz (Bz ) ∼ conclude that case (iii) of 9.5.11.3 holds, so that P = O2 (Lz ) = OBz and hence CP (Xz ) = z . Set X0 = CXz (Bz ) and observe that X0 is faithful on O. As b ∈ Bz and a, z = b, z , we have X0 = CXz (a) = CXz (b). As X0 ≤ W CGz (a) , AutX0 (O) is induced in Oa , so |X0 | = k. Then as AutLz (Bz ) ∼ = S3 we have |Xz | = 3k. Hence if k = 1 then Xz ∼ = Z3 and we take X = Xz . As P X = Lz  Gz , SX = XS, so (2) holds when k = 1. If k = 3 then X0 is of order 3 and P X0  Gz , so as S interchanges K1 and K2 , CO (X0 ) = z . As O 2 (AutF (O) = O 2 (AutO (O), Xz is faithful on O. It follows that Xz has a unique subgroup X of order 3 with CP (X) = z , so again (2) holds. ˜ z = Gz /z . As C ˜ (X) = 1, t˜X ∼ Set G = E4 , so A ∼ = E8 , and as P X  Hz S P ˜ ˜ and t ∈ Z(S), A  S. Indeed AutSX (A) is the stabilizer in GL(A) of z, so either AutF (A) = AutSX (A) or AutF (A) = GL(A). Set Y = NF (A); in any event τA = (Y, Ω) is a quaternion fusion packet by 2.5.2. Suppose AutF (A) = GL(A). Then A# = z Y and FTa (M ) ≤ Y, so μ(τA ) ∼ = S4 . We conclude from Theorem 2 and 5.10.7.1 that one of the first three cases in Theorem 2 holds and Z(Y ◦ ) = 1. Then as μ(τA ) ∼ = S4 it follows from 6.2.8 that Y◦ ∼ = L3 (2)/E64 . Let T be Sylow in Y ◦ ; then |T | = 29 , so as |Ta | = 28 and Bz ≤ T with Bz ∩ Ta = b , |Ta Bz | = 29 , so that T = Ta Bz . As Y ◦ ∼ = L3 (2)/E64 , Y ◦ = FT (T X), FT (T M ) , so that (3) holds in this case. Thus to complete the proof of (3), we may assume AutF (A) = AutXS (A), and it remains to derive a contradiction. Let x ∈ X # , r = tx , and c = bx . Then A = Er and as t = vb, r = x v c = kc, where k = k1 k2 with ki ∈ K − W . Let g ∈ M have cycle (z, t) and w ∈ O g ∩ W of order 4. If r ∈ D∗ (t) then [A, O g ] ≤ A and [z, O g ] = t , / D∗ (t), so s = r g ∈ / D ∗ (z). As contradicting AutF (A) = AutFz (A). Therefore r ∈ w induces an outer automorphism on Ki for i = 1, 2, w inverts XP/P , so [c, w] = b and hence K gc = K g . Also K gk = K g , so r acts on K g and hence s acts on K, so s ∈ Do (z). Thus s induces outer automorphisms on Ki , so s inverts XP/P and hence [c, s] = b. Therefore |sOBz | = |OBz : COBz (s)| = 16. However A  S, so Ag  CS (E). Set θ = Ag − E; it follows that θ S = θ ∪ θ k is of order at most 8, contradicting sOBz ⊆ θ S . This contradiction completes the proof of (3). By 9.5.14.2, CF (O) ≤ CFz (a) = CFz (b), so CF (O) centralizes Bz = bGz , proving (4). By 1.3.2 in [Asc19], Fz = SCF (O), NF (R0 ) , so as NF (R0 ) = FS (Gz ), (5) holds. As Bz  Hz , (4) and (5) imply (6). By (1), Ta ∈ Syl2 (M ). Thus by 1.3.2 in [Asc19], NF (W ) = FS (SM ), NF (Ta ) , so as NF (Ta ) ≤ Fz , (7) follows from (6) and Theorem 1.5.2 in [Asc19]. Similarly by (4), CF (O) centralizes E, so as W  NF (E), CF (O) ≤ NF (W ). Then as T = CFT (T M ) (z), (8) follows from (7). Set F1 = FS (Gz ), E1 = FT (T Lz ), F2 = SCF (O), and E2 = T . Observe that Hypothesis 1.4.7 in [Asc19] is satisfied by this tuple, so (9) follows from 1.4.8 in [Asc19]. Assume k = 1. Then A  SHz CGz (P ) = Gz , while A  SCF (O) by (4). Therefore A  Fz by (5). Hence A − z = tFz = D∗ (z) by 9.1.10.3. Then as

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∗ AutF (A) = GL(A), A# = DE , so A  NF (W ). Therefore by 9.5.9, A  F, so (3) implies (10). 

/ Z(F), and |AutO (O)|3 = 3. Then Lemma 9.5.16. Assume Fb◦ = O(z), a ∈ (2). Sp F◦ ∼ = 6 Proof. Adopt the notation of 9.5.15 and set F1 = Fz , E1 = FT (T Lz ), F2 = NF (W ), E2 = FT (T M ), and X = E1 , E2 . We will show that X ∼ = Sp6 (2); in particular X is saturated. Then we verify that the tuple satisfies Hypothesis 1.4.7 in [Asc19]. Finally we appeal to 1.4.8 in [Asc19] to conclude that X  F, so it follows that F ◦ = X ∼ = Sp6 (2). The verification of Hypothesis 1.4.7 in [Asc19] has for the most part already been done. By 9.5.9, F = F1 , F2 . By 9.5.15.9, E1  F1 and by 9.5.15.7, E2  F2 . As z and W are invariant under AutF (T ), AutF (T ) = AutFi (T ) for i = 1, 2. Thus it remains to show that X ∼ = Sp6 (2). ˜ be the group Sp6 (2), S˜ ∈ Syl2 (G), ˜ and F˜ = F ˜ (G). ˜ Thus F˜ ∼ Let G = Sp6 (2) S 9 ˜ ˜ ≤ satisfies the hypothesis of the lemma with |S| = 2 = |T |. For example G ∼ ˆ ˆ ˜ G = Ω7 (3) contains a Sylow 2-subgroup of G, so by 2.5.2, G inherits a quaterˆ From section 5.3, the Lie packet of G ˆ satisfies Hynion fusion packet τ˜ from G. ˜ pothesis 9.5.1; then τ˜ inherits that hypothesis. Next G has a maximal parabolic subgroup Y˜ over S˜ such that Y˜ ∼ = L3 (2)/E64 . Let Y = NF (A)◦ ; from 9.5.15.3, ˜ = Ω, and Y = FT (Y˜ ). MoreY∼ = L3 (2)/E64 ∼ = FS˜ (Y˜ ), so we may take S˜ = T , Ω over from 9.5.15.3, Y = FT (T X), FT (T M ) . Further E1 = FT (T X), FT (T X0 ) , where X0 = CXz (Bz ) appears in the proof of 9.5.15. Thus X = Y, AutE0 (P0 ) , where E0 = FT (T X0 ) and P0 = O2 (E0 ). Hence it suffices to show that AutE0 (P0 ) = AutE˜0 (P˜0 ). Next T Lz = Lz NT Lz (Xz ) and NT Lz (Xz ) = Xz I, where I = z, u1 , u2 ∼ = E8 with z, u1 = CO2 (Y ) (X) with u1 ∈ z Y ∩ D m (z) and u2 is an involution in Y inverting X with K1u2 = K2 . For example Y splits over O2 (Y ), which is the core of the permutation module for Y /O2 (Y ), and hence m(CO2 (Y ) (u2 )) = 4. Therefore T Lz is determined up to isomorphism. Recall P = O2 (Lz ) and Bz = [Z(P ), X] is determined by FT (T X) with Xz = O(NT Lz (X)) and X0 = CXz (Bz ). Finally P0 = O2 (E0 ) = P u2 and O 2 (AutE0 (P0 )) = O 2 (CAut(P0 ) (Bz )), so indeed P˜0 = P0 and AutE0 (P0 ) = AutE˜0 (P˜0 ). This completes the proof.  Theorem 9.5.17. Assume Fb◦ = O(z). Then one of the following holds: ˇ 6 or Ω+ [m]. (1) a ∈ Z(F) and F ◦ = FTa (M ) or F ◦ ∼ = AE 6 (2) F ◦ ∼ = L3 (2)/E64 . (3) F ◦ ∼ = Sp6 (2). Proof. If a ∈ Z(F) then (1) holds by 9.5.5, so assume otherwise. Then by 9.5.15.1, k = |AutO (O)|3 = 1 or 3. Therefore (2) or (3) holds by 9.5.15.10 and 9.5.16.  Because of Theorem 9.5.17, during the remainder of the section we assume the following hypothesis: Hypothesis 9.5.18. Hypothesis 9.5.1 holds and Fb◦ = O(z).

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Lemma 9.5.19. Let g ∈ M interchange z and t. (1) v, z  S. (2) g fixes v and interchanges b and vz, and β = c∗g ∈ A(b, t ) with CS (b, t ) = CS (E) = CS (v, z ). (3) There exists a unique (solvable) component Lb,t of CFb◦ (t). Let Sb,t be Sylow in Lb,t and set Lvz,z = Lb,t β ∗ and Svz,z = Sb,t β. (4) Lb,t ∼ = L2 [m] if Fb◦ ∼ = AE5 or P Sp4 [m], and Lb,t ∼ = L2 [2m] if Fb◦ ∼ = L− 4 [m]. (5) Sb,t = (vz)Lb,t . (6) Lvz,z  CF (O) is S-invariant and Svz,z = bLvz,z . (7) If Fb◦ ∼ = P Sp4 [m] or L− 4 [m] then Fz = SOCF (O) and Lvz,z  Fz . Proof. Part (1) is an observation. As g fixes a = vzt and interchanges z and t, it fixes v, and then interchanges vz and vt = b. Also CS (b, t ) = CS (t) = CS (E) = CS (Eb ) = CS (v, z, b ) = CS (v, z ), establishing (2). Parts (3)-(5) follow from the local structure of X = Fb◦ , as we see in a moment. ∼ By 9.5.18 and 9.5.4, X is AE5 , P Sp4 [m], or L− 4 [m]. First, if X = AE5 then ∼ ∼ X = FTb (Hb ) where Hb is the split extension of Ub = E16 by Yb = A5 with Ub the core of the permutation module for Yb = Alt(I) and I = {1, . . . , 5}. We may take z = e1,2,3,4 and v = e1,2 . Then CHb (t) = CHb (v) = Ub CYb (v) ∼ = Z2 /(E4 × A4 ), so Xb = O 2 (CHb (v)) ∼ = A4 with vz = e3,4 ∈ O2 (Xb ) = vz, e3,5 . Here take Lb,t = FSb,t (Xb ) ∼ = L2 [8]. Thus (3)-(5) indeed hold in this case. Suppose next that X ∼ = Ωk [m], where k = 5 or (k, ) = (6, −1). Then X = ∼ FTb (Hb ) where Hb = Ω(U ) for some k-dimensional orthogonal space U over Fq for some suitable prime power q. We can take z to be an involution in Hb with [U, z] 4-dimensional of sign +1, while [U, v] is a 2-dimensional subspace of [U, z]. Then CHb (v) has a (solvable) component Xb ∼ = Ωk−2 (q) ∼ = L2 (q) or L2 (q 2 ) for k = 5 or 6, respectively. In this case Lb,t = FSb,t (Xb ), and again one can check (3)-(5). Next as vz ∈ O, CF (O) ≤ CFz (vz). Also Lb,t is a (solvable) component of CFb (t), and hence by (2) and 2.2 in [Asc10], L = Lvz,z = Lb,t β ∗ is a (solvable) component of CFz (vz). If m > 8 then L is a component, so by E-balance (cf. Theorem 7 in [Asc11]), and as CO(z) (vz) is a 2-group, L centralizes O(z), and in particular L ≤ CF (O) ≤ CFz (vz), so L  CF (O) as Lb,t  CF (b, t ) by (3). As S = OCS (vz), S acts on L, completing the proof of (6) when m > 8. / Svz,z , L centralizes Suppose m = 8. As v, z  O and L  CF (v, z ) with z ∈ O, completing the proof of (6). ∼ Suppose Fb◦ ∼ = P Sp4 [m] or L− 4 [m]. By 9.5.12, O = SL2 [m] ∗ SL2 [m] and O  Fz . Then by 2.6.14, Fz = SOCF (O), so L  Fz by (6), proving (7).  ∼ AE5 and let L be the subnormal subgroup of Gz Lemma 9.5.20. Assume Fb◦ = which is the model for L = Lvz,z . Then ˇ 6. (1) Fa◦ ∼ = AE (2) Lz = L × L0 where L ∼ = A4 , L0 = CLz (L) is of index 3 in SL2 (3)2 with O = O2 (L0 ) and FO (L0 ) = O, and Bz = O2 (L) = Svz,z . (3) Fz = SOCF (O) and L  Fz . (4) Fz = SL, CFz (b) .

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∼ AE5 , Ob is of index 3 in SL2 (3)2 . By 9.5.12.1, Ob = O = Oa , Proof. As Fb◦ = so Oa is of index 3 in SL2 (3)2 , and hence (1) follows from 9.5.5.2. By 9.5.19.4, L ∼ = L2 [m], so L ∼ = A4 as m = 8. By 9.5.19.6, O2 (L) = Svz,z = bL . Then as Bz = bLz  Gz is of order 2 or 4 by 9.5.11.3, it follows that E4 ∼ = O2 (L) = Bz . Let Xz ∈ Syl3 (Lz ). If x ∈ Xz# with Lx = L then by 2.2.2.1, 2 Lx = L × Lx × Lx , a contradiction as E4 ∼ = Bz  Gz . Therefore Lz acts on L. As [Bz , W ] = b by 9.5.11.1, L = [L, W ] ≤ Lz and then by 9.5.11.3 and 9.5.19.6, L = CLz (O) and Lz = L × L0 where L0 = CLz (L). Thus, using 9.5.11.4 and 9.5.12.1, O = FO (L0 ), so as O is of index 3 in SL2 (3)2 , the same holds for L0 . This completes the proof of (2). By 9.5.11.1, [W, S0 ] ≤ Bz , so as L0 centralizes Bz , it also centralizes S0 . Therefore by 2.3 in [Asc11], Fz contains a central product B = O ∗ CF (O). As L0  Gz , we conclude from 1.5.1 in [Asc19] that B  Fz . By 9.5.19.4, O 2 (AutF (O)) = AutLz (O), and by (2), AutLz (O) = AutL0 (O) = AutO (O), so Gz = SLz CGz (O) and hence NF (OS0 ) = SNB (OS0 ). Then Fz = SOCF (O) by 1.3.2 in [Asc19]. By 9.5.19.6, L = Lvz,z  CF (O) is S-invariant, while O centralizes L by (2). Therefore L  SOCF (O) = Fz , proving (3). As A4 ∼ = L  CF (O), O 2 (CF (O)) = L × O 2 (CF (P )), where P = OBz . Then  by (3), Fz = SL, SOO 2(CF (P )) = SL, CFz (b) , proving (4). Lemma 9.5.21. If Fb◦ ∼ = AE5 then F ◦ ∼ = AE7 . Proof. For c ∈ {a, b} set Uc = O2 (Fc◦ ), let Gc be a model for NFc (CS (Uc )), and let Hc  Gc be a model for Fc◦ . Then Hb is the split extension of Ub ∼ = E16 by ˇ 6 , so Ua ∼ A5 with Ub the A5 -module for Hb /Ub . By 9.5.20.1, Fa◦ ∼ = AE = E32 . By 9.5.12.1, O = Oc for c ∈ {a, b}. Now W O acts on a unique E8 -subgroup F of O, so F = O ∩ Ub = O ∩ Ua . Further Ua centralizes the hyperplane O ∩ Ub = O ∩ Ua of Ub , and that hyperplane is not the axis of a transvection in Aut(Hb ) on Ub , so Ua centralizes Ub . Set U = Ua Ub and let ρ = (NF (U ), Ω); by 2.5.2, ρ is a quaternion fusion packet. From the previous paragraph U is elementary abelian, so U = CUHb (Ub )  U Hb . Thus if Ub ≤ Ua then Hb acts on Ua = U and μ(ρ) = μ(τa ) ∼ = S4 , so by Theorem 2 ˇ 6 . Therefore F ◦ ≤ NF (U )◦ = Fa◦ , so Hb centralizes a, a and 5.1.21, NF (U )◦ ∼ = AE b contradiction as a = bz. Thus Ub ≤ Ua , so Ua ∩ Ub = O ∩ Ub and hence m(U ) = 6. As U  U Hb , Fb◦ ≤ NF (U ). As above, U = CUHa (Ua )  U Ha , so Fa◦ ≤ NF (U ). Thus μ(ρ) ∼ = E64 , and Fc◦ ≤ NF (U )◦ for c ∈ {a, b}, so we conclude from = S4 , U ∼ Theorem 2 and 5.1.21 that NF (U )◦ ∼ = AE7 . Thus to complete the proof it remains to show that U  F. By 9.5.20.4, Fz = SL, CFz (b) . As NF (U )◦ ∼ = AE7 , L ≤ FBz O (Lz ) ≤ NF (U )◦ . As CFz (b) centralizes a, b , it is contained in NF (U ). Therefore Fz ≤ NF (U ). By 9.5.9, F = Fz , NF (W ) , so it remains to show that NF (W ) ≤ NF (U ). But NF (W ) = SFS (M ), NFz (W ) and FTa (M ) ≤ NF (U )◦ , so the proof is complete.  Because of 9.5.21 and 9.5.4.2, during the remainder of the section we assume the following hypothesis: Hypothesis 9.5.22. Hypothesis 9.5.1 holds and Fb◦ ∼ = P Sp4 [m] or L− 4 [m]. Set L = Lvz,z , Tl = Svz,z , and T = Ta Tl .

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∼ Ω+ [m]. Lemma 9.5.23. (1) Fa◦ = 6 (2) Fz = SL, CFz (b) . (3) T  CNF (W ) (z) = CNF (W ) (b). (4) T O = CT Fa◦ (z)  CFa (z). (5) T OL  Fz . (6) FT (T M )  NF (W ). ∼ Proof. As Fb◦ is P Sp4 [m] or L− 4 [m], Ob = SL2 [m] ∗ SL2 [m], so by 9.5.12.1, ∼ Oa = SL2 [m] ∗ SL2 [m], and hence (1) follows from 9.5.5.2. By 9.5.19.7, L  Fz . Then O 2 (CF (O)) = L × O 2 (CF (P )), where P = OTl , while by 9.5.19.7, Fz = SOCF (O), so Fz = SL, SOO 2(CF (P )) = SL, CFz (b) , proving (2). Next FTa (M )  NF (W ), so a ∈ Z(NF (W )) by 9.5.3.2. Then as a = bz, CNF (W ) (z) = CNF (W ) (b). Also Ta = CM (z), and Tl = CL (a), so (3) holds, and (3) and Theorem 1.5.2 in [Asc19] imply (6). Similarly NFa (Ta ) ≤ NFz (W ) ≤ NF (T ) by (3), so (4) follows from Theorem 1.5.2 in [Asc19]. Next O  Fz by 9.5.12.3 and L  Fz by 9.5.19.7, so X = OL  Fz by Theorem 3 in [Asc11]. Suppose m > 8. Then b = Z(Tl ) so NFz (P ) ≤ CFa (z), and then as NO (O) = O, T  NFz (P ) by (4). Then (5) follows from Theorem 1.5.2 in[Asc19] in this case. So assume m = 8; we verify Hypothesis 1.4.7 in [Asc19] with F1 = SX , E1 = T X , F2 = CFz (b), and E2 = T O. By (2), Fz = F1 , F2 and by (4), E2  F2 . As T  S, E1  F1 by 1.3.4 in [Asc19]. Therefore Hypothesis 1.4.7 in [Asc19] is satisfied, so T X = E1 , E2  Fz by 1.4.8 in [Asc19], proving (5).  Lemma 9.5.24. (1) OL is transitive on the set I of involutions ij such that i ∈ O − z and j ∈ Tl . Moreover I ⊆ Z. (2) COg (O)  Tl is cyclic of order m/4. (3) Let r be an involution in Tl − b . Then r inverts COg (O). (4) Choose generators ki , xi , 1 ≤ i ≤ 3, for M as in 4.3.1 and 5.7.4, but with xi in the role of vi and [K1 , K2 ] = 1. Set x = x1 x2 x23 , k4 = k3k1 k2 , and s = k1 k2 r. Then COg (O) = x , K gr = K g , xr3 = x4 , s ∈ D ∗ (t) ∪ Do (t), s inverts x3 , and [x3 , r] = x−1 , so r x3 = rx. (5) We may choose k1 and k2 so that s ∈ D∗ (t), and k3 so that k3r = k4 . In particular M r is determined up to isomorphism. (6) Tb , Tl = Tb × Tl . (7) If Fb◦ ∼ = L− 4 [m] then Tl = y, r is dihedral of order m and we may choose y of order m/2 with y 2 = x, y centralizes OW , and k3y = k3 x3 . In particular M T is determined up to isomorphism. Moreover T = Tb × Tl . ∼ SL2 [m] ∗ SL2 [m], so O is transitive on involutions in Proof. By 9.5.22, O = O − z ; by 9.5.19.4, L is transitive on involutions in Tl ; and by parts (6) and (7) of 9.5.19, L centralizes O. This gives the transitivity in (1). As t = vb ∈ Z, the second statement in (1) also holds, completing the proof of (1). By 9.5.11.1, COg (O) is cyclic of order m/4. Let K g ∩ W = w . By 9.5.19.7, L  Fz , so Tl  S. By 9.5.11.1, [w, r] ∈ COg (O), so as b is the involution in COg (O), either K gr = K g or r centralizes w. In the latter case r0 = r or br centralizes K g , so r0 centralizes M = O, K g , so as b does not centralize M , neither does br0 . Then as br0 satisfies the hypothesis of r, we conclude that K gr0 b = K g , and then as b acts on K g , it follows that K gr = K g . Therefore |[w, r]| = m/4, so as |COg (O)| = m/4

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and Tl  S, we have x0 = [w, r] ∈ Tl a generator of COg (O). If m = 8 then x0 = b ∈ Z(Tl ) and (2) and (3) hold. If m > 8 then |x0 | > 2, so as Tl is dihedral, x0  Tl and r inverts x0 . This completes the proof of (2) and (3). It follows from (1)-(3) that the hypotheses of 5.7.5.5 are satisfied with τM , τ in the roles of τ , ρ, so (4) follows from parts (a) and (b) of 5.7.5.5.  o Let k = k1 k2 and k1 = x−1 1 k1 ; then k1 ∈ K1 − W and if s ∈ D (t) then as x1 g   induces an outer automorphism on K , s = k1 k2 r induces an inner automorphism on K g , so s ∈ D ∗ (t). Thus replacing k1 by k1 and appealing to (1), we may assume s ∈ D∗ (t). Now 5.7.5.5.c completes the proof of (5). ∼ ∼ Suppose Fb◦ ∼ = L− 4 [m]. By 9.5.19.4, L = L2 [2m] so Tl = Dm . Therefore Tl = y, r where y is of order m/2, and by (2) and (4) we may choose y so that y 2 = x. As Tl centralizes O, so does y. Applying (4) to r and yr, xr3 = x3 x−1 = xyr 3 , so y centralizes x3 , and hence also W = (W ∩ O)x3 . We have established the hypothesis of 5.7.5.5.d, so by that lemma we may choose y so that k3y = k3 x3 . This establishes the first sentence in (7). From 9.5.22, T = Ta Tl so T M = Tl M . As Tl = y, r with Tl /x ∼ = E4 , T M is determined by the action of r and y on M , together with the power maps y 2 = x and r 2 = 1 and the fact that r inverts y. The action of r is determined by (5), and as y centralizes OW with y 2 = x and k3y = k3 x3 , the action of y on M is determined. Thus T M is determined up to isomorphism, establishing the second sentence in (7). Assume (6). Then |Tb Tl | = |Tb ||Tl | = 2m2 · m = 2m3 . Similarly |T | = |Ta ||Tl |/|Ta ∩ Tl | = (m3 /2) · m/(m/4) = 2m3 . Finally Tb = Os1 , s2 where si ∈ D m (z) ∩ z Fb , so by 9.5.7.1, we may take si ∈ D ∗ (t). Then, choosing g as in 9.5.19, sgi ∈ D ∗ (z) ⊆ OTl ≤ T , so si ∈ T and hence Tb ≤ T . Therefore Tb Tl ≤ T with |Tb Tl | = |T |, so T = Tb × Tl by (6), completing the proof of (7). It remains to prove (6). By 9.5.19.7, Tl  S and by 9.5.3.4, Tb  S. Also Tl ∩ Tb ≤ CTb (O) = z and z ∈ / Tl , so Tb ∩ Tl = 1. Hence (6) holds.  Lemma 9.5.25. If Fb◦ ∼ = P Sp4 [m] then Tl ∼ = Dm/2 , |T : Ta | = 2, and T M is determined up to isomorphism. Proof. By 9.5.19.4, L ∼ = Dm/2 . By 9.5.24, x ∈ Tl is of order = L2 [m] so Tl ∼ m/4, so Tl = x, r and x = Tl ∩ Ta is of index 2 in Tl . Therefore T = Ta Tl is of order 2|Ta |, so T M = r M is determined by the action of r on M , and hence is determined up to isomorphism by 9.5.24.5.  ◦ ∼ Notation 9.5.26. By 9.5.22, Fb◦ is P Sp4 [m] or L− 4 [m]. If Fb = P Sp4 [m] set − − ◦ ∼ ˜ ˜ ˜ F = Ω7 [m], while if Fb = L4 [m] set F = Ω8 [m]. Let S be Sylow in F˜ and let ˜ Ω) ˜ be the Lie packet of F˜ . Let η˜ ∈ η(˜ ˜ = ˜ ˜ a model of τ˜ = (F, τ ), W η , and G ˜ ). NF˜ (W

˜∼ ˜ = T M , S˜ = T , Ω ˜ = Ω, and Lemma 9.5.27. (1) G = T M , so we may take G ˜ ) = FT (T M ). NF˜ (W ˜ (2) a = Z(G). Proof. By 9.5.24.7 and 9.5.25, T M is determined up to isomorphism by our ˜ = |T | by 5.2.7 and 5.3.5.4, we hypothesis, so as F˜ satisfies that hypothesis with |S| ˜∼ conclude that G T M , and then (1) follows. Then as Z(T M ) = a , (2) holds.  = Lemma 9.5.28. (1) T Fa◦ = F˜a .

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˜ (2) T O = T O. ˜ ˜ Proof. By 9.5.23.1, X = Fa◦ ∼ = Ω+ 6 [m]. Set Y = T X and Y = Fa . By 9.3.20 ˜ ) . By 9.5.27.1, N ˜ (W ˜) = and 2.1.15, Y = Yz , FT (T M ) and Y˜ = Y˜z , NF˜ (W F FT (T M ), so to prove (1) it suffices to show Yz = Y˜z . But Yz = T O, so (2) implies (1). Finally (2) follows from Theorem 2.7.3.  Lemma 9.5.29. If Fb◦ ∼ = P Sp4 [m] then F ◦ ∼ = Ω7 [m]. Proof. Let E1 = T OL, F1 = Fz , E2 = FT (T M ), and F2 = NF (W ). By 9.5.9, F = F1 , F2 and F˜ = F˜z , FT (T M ) . We claim that E1 = F˜z , so F˜ = E1 , E2  ˜ By 9.5.23.5, E1  F1 and by 9.5.23.6, E2  F2 . is saturated with F˜ = O 2 (F). Therefore Hypothesis 1.4.7 in [Asc19] is satisfied, so by 1.4.8 in [Asc19], F˜  F. Then F ◦ = F˜ ∼ = Ω7 [m], establishing the lemma. Thus it remains to prove the claim. ˜ Further E1 = T OL, so E1 = T O, T L . Thus it By 9.5.28.2, T O = T O. ˜ But L ∼ remains to show T L = T L. = L2 [m] ∼ = L˜ and by 9.5.24.4, r x3 = rx, so ˜ e , with T is transitive on the 4-subgroups of Tl , and hence on (T L)e and (T L) representative R = CT (U ) for U = b, r . From 9.5.24.4, R = OU j , where j = k3 k4 . Also O 2 (AutT L (R)) centralizes O, and we see in a moment that it also centralizes i = jr, so AutT L (R) = AutT L˜(R), completing the proof of the claim. To see that i is centralized, we show that i is the unique member of iU in ˜ First k3r = k4 , so r k3 = k−1 k4 r = tjr = ti Z and the unique member in Z. 3 ˜ and (br)k3 = bk3 r k3 = bt · ti = bi are in bF ∩ bF . Also b, j ∈ O g , so bj ∈ O g ˜ ˜ and hence bj ∈ bF ∩ bF . Finally K3i = K4 and we showed that ti ∈ bF ∩ bF , so ˜ i = t · ti ∈ Z ∩ Z.  It remains to treat the case Fb◦ ∼ = L− 4 [m], so we assume: Hypothesis 9.5.30. Hypothesis 9.5.1 holds and Fb◦ ∼ = L− 4 [m]. Set F1 = Fz , F2 = NF (W ), F3 = Fb , E1 = T OL, E2 = FT (T M ), and E3 = T Fb◦ . Set j = k3 k4 , i = jr, l = x3 y −1 , i2 = il−1 , and r2 = ry. Lemma 9.5.31. (1) F = Fi : 1 ≤ i ≤ 3 . (2) F˜ = F˜z , E2 . (3) Ei  Fi for 1 ≤ i ≤ 3. (4) T0 = i, l ∼ = Dm centralizes Tl with i ∈ Z ∩ Z˜ inverting l of order m/2. ˜ ˜ (5) bi2 and i2 r2 are in bF ∩ bF and i2 ∈ Z ∩ Z. Proof. Part (1) follows from 9.5.9. Similarly by 9.5.24.7, T = Tb × Tl , so as T is Sylow in F˜ by 9.5.27, applying 9.5.9 to F˜ we get F˜ = F˜z , NF˜ (W ) and then (2) follows from 9.5.27.1. Next E1  F1 by 9.5.23.5 and E2  F2 by 9.5.23.6. By 9.5.23.3, T  CNF (W ) (b), so as W  NFb (Tb ), E3  F3 by Theorem 1.5.2 in [Asc19] applied to Fb◦  Fb . This proves (3). As k3r = k4 , r centralizes j = k3 k4 , so i = jr is an involution centralizing r. Next −1 y = x3 y −2 y = x3 y −1 = l, lr = (x3 y −1 )r = x4 y = x3 x−1 3 x4 y = x3 x

so r centralizes l. Therefore l centralizes r, y = Tl .

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253

−1 Also k3y = k3 x3 , so y k3 = yx−1 . Then 3 =l

(y −1 )k4 = y rk3 = (y k3 )r = (l−1 )r = l−1 , r

so y k4 = l. Then k4 −1 y j = y k3 k4 = (l−1 )k4 = (yx−1 = lx−1 , 3 ) 3 =y

so j inverts y. Then as r inverts y, i centralizes y, so i centralizes Tl . Thus T0 = i, l centralizes Tl . Finally |l| = |y| = m/2 and we showed that j inverts y, so as j also inverts x3 , j inverts l. Hence as r centralizes l, i inverts l, so T0 ∼ = Dm . From the ˜ completing the proof of (4). proof of 9.5.29, i ∈ Z ∩ Z, ˜ As k3r = k4 , r k3 = tjr = ti. Thus r2k3 = r k3 y k3 = til−1 = ti2 ∈ bF ∩ bF , so as r2 k ˜ Also (br2 )k3 = bk3 r 3 = (bt)·(ti2 ) = bi2 , interchanges K3 and K4 , i2 = t·ti2 ∈ Z∩Z. 2 ˜ −1 −1 F F −1 ry = iyx r = ix−1 so bi2 ∈ b ∩ b . Finally i2 r2 = (il ) · ry = iyx−1 3 3 y 3 r = ˜ −1 F F g jrx3 r = jx4 ∈ b ∩ b as jx4 ∈ O is an involution. This completes the proof of (5) and the lemma.  ◦ ∼ − Lemma 9.5.32. If Fb◦ ∼ = L− 4 [m] then F = Ω8 [m].

Proof. We first observe that the tuple Fi , Ei , 1 ≤ i ≤ 3, satisfies Hypothesis 1.4.3 in [Asc19] by parts (1) and (3) of 9.5.31. Therefore by 1.4.4 in [Asc19], Y = Ei : 1 ≤ i ≤ 3 is an F-invariant subsystem of F on T , so in particular T is ˜ Thus Y = O 2 (Y) strongly closed in S with respect to F. We will show that Y = F. is saturated, and hence Y  F by 1.4.5 in [Asc19]. Then F ◦ = Y ∼ = Ω− 8 [m]. It ˜ remains to show Y = F . ˜ Further E1 = T OL, so We first show that E1 = F˜z . By 9.5.28.2, T O = T O. ˜ ˜ ˜ E1 = T O, T L , and similarly Fz = T O, T L , so to show E1 = F˜z it suffices to show ˜ From 9.5.31.4, T = OT0 × Tl . Let r1 = r, i1 = i, and Ui = ri , b that T L = T L. for i = 1, 2. As L ∼ = L2 [2m] and U1 and U2 are representatives for the orbits of Tl on its 4-subgroups, Ri = OT0 Ui , i = 1, 2, are representatives for the orbits of T on ˜ e , so it suffices to show AutT L (Ri ) = Aut ˜(Ri ) for i = 1, 2. Let (T L)e and (T L) TL ˜ i be model for NT L (Ui ), N ˜ (Ui ), respectively. Then Gi = Hi × Xi , where Gi , G TL Hi ∼ = S4 has Sylow 2-group NTl (Ui ) and Xi = CGi (Hi ) = OCT0 (Hi ). Moreover Φ(T0 ) = l2 = x23 x−1 = (x1 x2 )−1 ≤ O ≤ CT (Hi ), so Xi = Oie bi,e : e = 1, 2 . ˜ i . Thus it remains to show that i,e = ˜i,e for i, e ∈ Define similar notation in G ˜ 1 , so 1,1 = 0 = ˜1,1 . The {1, 2}. From the proof of 9.5.29, i centralizes H1 and H same proof together with 9.5.31.5 shows that 2,2 = 0 = ˜2,2 . We claim that i does not centralize L. Given the claim, 1,2 = 1 = ˜1,2 and similarly 2,1 = 1 = ˜2,1 , showing that E1 = F˜z . So assume i centralizes L; we produce a contradiction. By 9.5.31.4, i ∈ Z so there is α ∈ A(i) with iα = z. Let X = O 2 (CO (i)). As i K = K2 , X ∼ = L2 [m]. Then B = (X L)α∗ ≤ O 2 (Fz ) and as T is strongly closed in S with respect to F and E1  Fz , B ≤ O 2 (E1 ) = OL. Indeed from 9.5.24, bF ∩ OL is the set of involutions in O − z and Tl , so Lα∗ is contained in CO (zα) or L. As O 2 (CO (zα)) ∼ = L2 [m] it follows that Lα∗ = L, so in particular bα = b. Therefore (ib)α = zb = a, a contradiction as we saw during the proof of 9.5.29 that ib ∈ bF . This finally completes the proof that E1 = F˜z . As E1 = F˜z , E1 , E2 = F˜z , E2 = F˜ by 9.5.31.2. Thus F˜ ≤ Y, so to complete the proof it suffices to show that E3 ≤ F˜ . By 9.4.22 and 9.4.28, E3 = T O, AutE3 (Ri ) : i = 1, 2 , where Ri = CT (Zi ) and Zi = z, v, ti with ti ∈ Dm (z) ∩ z E3 . By 9.4.21, ui = v or vz is in the center of NE3 (Ri ), so NE3 (Ri )

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centralizes ei = bui = t or tz. However there is α ∈ AE2 (ei ) with ei α = z, so NE3 (Ri )α∗ ≤ CF3 (ei )α∗ ≤ Fz . However AutF3 (Ri ) = AutT (Ri )AutE3 (Ri ) , so as T is strongly closed in S with respect to F and E1  Fz , AutE3 (Ri )α∗ ≤ AutE1 (Ri α) ≤ AutF˜ (Ri α). Then as α is an F˜ -map, AutF3 (Ri ) ≤ AutF˜ (Ri ), so indeed E3 ≤ F˜ . This finally completes the proof of the lemma.  Collecting the results in this section, we have proved: Theorem 9.5.33. Assume Hypothesis 9.5.1. Then either ˇ 6 or Ω+ [m], or (1) a ∈ Z(F) and F ◦ = FS∩M (M ) or F ◦ ∼ = AE 6 ◦ (2) F is isomorphic to L3 (2)/E64 , Sp6 (2), AE7 , Ω7 [m], or Ω− 8 [m]. Proof. By Theorem 9.5.17 we may assume Fb◦ = O(z), so Hypothesis 9.5.18 holds. Then by 9.5.21 and 9.5.4.2 we may assume Fb◦ is P Sp4 [m] or L− 4 [m]; hence Hypothesis 9.5.22 is satisfied. Then 9.5.29 and 9.5.32 complete the proof.  Theorem 9.5.34. Assume τ = (F, Ω) is a quaternion fusion packet such that A(z) = ∅, Ω = Ω(z) is of order 2, and F ◦ is transitive on Ω. Then either (1) μ ∼ = E4 and F ◦ is AE5 , P Sp4 [m], or L− 4 [m], or + ˇ (2) μ ∼ = S4 and F ◦ is FS∩M (M ) for M ∈ M (τ ), AE6 , P Ω+ 6 [m], AE6 , Ω6 [m], − L3 (2)/E64 , Sp6 (2), AE7 , Ω7 [m], or Ω8 [m]. Proof. Let W ∈ W (τ ) and W  M ∈ M (τ ). Then τW = (FS∩M (M ), Ω) is a quaternion fusion packet with μ(τW ) = μ(τ ), satisfying Hypothesis 4.2.1. Adopt Notation 4.2.3. As A(τ ) = ∅, D is a set of 3-transpositions of μ = M/W by 4.2.10. As Ω = Ω(z) is of order 2, Vd = d⊥ is of order 2 for d ∈ D, so either D(D) is disconnected or D = Vd . In the first case μ ∼ = S4 by 4.2.13 and in the second, . In the latter case the theorem follows from 9.4.34, and in the former it μ∼ E = 4 follows from 9.4.35 and 9.5.33. 

CHAPTER 10

|Ω(z)| > 2 Let τ = (F, Ω) be a quaternion fusion packet and z ∈ ZS (τ ). Chapter 10 treats the case where F ◦ is transitive on Ω, |Ω(z)| > 2, and the Inductive Hypothesis holds. We begin in section 10.1 with the case μ ∼ = Weyl(D4 ); this is the case where all examples appear; moreover in this case we do not need to assume the Inductive Hypothesis. Then the general case is dealt with in section 10.2. As a corollary, in Theorem 10.2.14 we are then able to determine those τ where F ◦ is transitive on Ω and Ω = Ω(z) is of order at least 3, using an inductive argument to establish the Inductive Hypothesis in a minimal counter example. An important byproduct of 10.2.14 is that in each example τ , F = Fz , FS (SM ) for M ∈ M (τ ). 10.1. |Ω(z)| = 4 and μ isomorphic to Weyl(D4 ) In this section we assume the following hypothesis: Hypothesis 10.1.1. τ = (F, Ω) is a quaternion fusion packet such that (1) For z ∈ ZS , |Ω(z)| = 4. (2) μ = μ(τ ) ∼ = Weyl(D4 ). Notation 10.1.2. As μ ∼ = Weyl(D4 ), F is transitive on Ω by 4.2.8.7 and |Ω| = 4 by 4.2.8.6. Hence as |Ω(z)| = 4 for z ∈ ZS , we have Ω = Ω(z). Let {z} = ZS and {Ki : 1 ≤ i ≤ 4} = Ω(z). Set K = K1 , let η ∈ η(τ ), W = η , G a model for NF (CS (W )), and M = K G . Set O = O(z) and let v1 be a NS (K)-invariant subgroup of K of order 4, vi ∈ v1S ∩ Ki for 1 < i ≤ 4, and v = v1 · · · v4 . Set E = v, z and T = S ∩ M . Let Dm (z) consist of those t ∈ Z such that t is fixed point free on Ω, and let Do (z) consist of those t ∈ Z inducing an outer automorphism on some member of Ω. ¯ /Z(M ¯ , where M ¯ =ω Lemma 10.1.3. (1) M = M ¯ (D4 , m). (2) v ∈ D∗ (z), E  M S, E # = ZΔ , and AutM (E) = GL(E). (3) |W | = m4 /64 and M/W ∼ = μ. (4) OW is the kernel of the action of T on Ω and T /OW ∼ = E4 is regular on Ω. (5) |W : O ∩ W | = 2 and for each K ∈ Ω and w ∈ W − O, w interchanges the two classes of Q8 -subgroups of K. (6) Ω1 (W ) = v1 v2 , v1 v3 , v, z ∼ = E16 , (7) Dm (z) ∪ Do (z) ⊆ CS (E). (8) For s ∈ Z, we have s ∈ D∗ (z) iff s induces a nontrivial inner automorphism on some Ki iff s induces a nontrivial inner automorphism on each Ki . 255

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Proof. By parts (1) and (2) of Hypothesis 10.1.1 and 5.8.7.3, v ∈ z M and (1) holds; in particular v ∈ D∗ (z). By 4.3.8.4, μ = M/W , and by 10.1.1.2, μ ∼ = Weyl(D4 ), so M/O2 (M ) ∼ = S3 , and hence |z M | = 3. Then as v ∈ z M and [v1 , k1 ] = z for k1 ∈ K − W , it follows that E # = z M = ZΔ , so (2) holds. By 5.8.7 and ¯ )| = 4, |W ¯ | = (m/2)4 and ¯ = ω 4.3.7.2, in the universal group M ¯ (D4 , m), |Z(M ¯ /Z(M ¯ ) by (1), so (3) follows. Indeed Z(M ¯ ) = ¯ W =W z1 z¯2 , z¯1 z¯3 ∼ = E4 , so the ¯ ) = ¯ preimage of Ω1 (W ) is contained in Ω2 (W v1 , . . . , v¯4 , v¯0 , where v¯02 = v¯. Further from 5.8.7.6, v¯ = v¯1 · · · v¯4 , so Ω1 (W ) = v1 v2 , v1 v3 , v, z , so (6) follows. As OW acts on Ki ∩ W for each 1 ≤ i ≤ 4, OW acts on Ki . Also M ∗ = M/W ∼ = T ∗ /O ∗ ∼ = E4 . Further = Weyl(D4 ) with E16 = O ∗  T ∗ , so T /OW ∼ ∗ ∗ ⊥ dK∩W , so T /OW is regular on Ω, establishing (4). T /O is regular on Next O ∩ W = i (Ki ∩ W ) is of order (m/2)4 /8 = m4 /128 as Ki ∩ Kj = z for all i = j. Therefore |W : O ∩ W | = 2 by (3). Further for v ∈ x ∈ η, x induces an automorphism on K interchanging the two classes of Q8 -subgroups by 4.2.5, completing the proof of (5). Part (8) follows from 3.1.15. Suppose t ∈ Dm (z). Then relabeling if necessary, v1t = v2 z  and v3t = v4 z δ , so t centralizes v1 v2 , v3 v4 and hence also v. Similarly if s ∈ Z induces an outer automorphism on K1 then by (8) either s is outer on each Ki or we may assume s is outer on K2 and K3s = K4 . In either case s centralizes v, so each member of  Do (z) centralizes v. Thus (7) holds. Lemma 10.1.4. Let t ∈ D∗ (z) and I = {1, . . . , 4}. Then (1) t = k1 k2 k3 k4 c where ki ∈ Ki is of order 4 and c ∈ CS (O(z)) with c2 = 1. (2) There exists φ ∈ homFz (k1 , . . . , k4 , t , W ) and σ ∈ Sym(I) with ki φ = vσ(i) , cφ ∈ CS (O(z)), and tφ = v. (3) Fz is transitive on D∗ (z). (4) D∗ (z) ⊆ O and we may take c = 1. (5) D∗ (v) ∪ D ∗ (vz) ⊆ T . Proof. Part (1) follows as t induces a nontrivial inner automorphism on each Ki . Then the proof of 9.1.10.2 shows we may choose φ ∈ homFz (k1 , . . . , k4 , t , W ) with tφ ∈ ZΔ and ki φ = vσ(i) for some σ ∈ Sym(I). By 10.1.3.2, ZΔ = E # , so conjugating in O, we may take tφ = v. That is (2) holds. Then (2) implies (3). As v ∈ O and O(z)  Fz , D ∗ (z) ⊆ O by (3). Thus t ∈ O, so c = k1 · · · k4 t ∈ CO (O) = z , so replacing k1 by k1−1 if necessary, we may take c = 1, completing the proof of (4). Similarly D∗ (v) ⊆ O g for g ∈ M with z g = v, so D ∗ (v) ⊆ COg (E) =  O g ∩ O2 (M ) ≤ T , establishing (5).  Lemma 10.1.5. Let w = v3 v4 and Q = g∈M O g . Then (1) τw = (Fw , Ωw ) is a quaternion fusion packet, where Ωw = {K1 , K2 }. (2) ηw = (K ∩W )CM (w) ∈ η(τw ), and setting Ww = ηw and Mw = K CM (w) , μw = Mw /Ww = μ(τw ) ∼ = S4 with w = Z(Mw ). ˇ 6 or Ω+ [m]. (3) Fw◦ = FS∩Mw (Mw ) or Fw◦ ∼ = AE 6 g (4) There is g ∈ CM (w) with z = v and w ∈ K3g K4g . (5) Ω1 (W ) ≤ Q. Proof. Part (1) follows from 2.5.2. Then (2) follows from 5.8.7, while (3) follows from (1), (2), and Theorem 9.5.33.

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By (2), AutMw (E) = GL(E), so there exists g ∈ Mw with z g = v. As w ∈ K3 K4 , (4) follows. As v ∈ O and AutM (E) = GL(E), we have E ≤ Q. By (4), w ∈ Q and then by symmetry among the vi and 10.1.3.6, (5) follows.  Lemma 10.1.6. (1) Dm (z) ⊆ D∗ (v) ∪ D ∗ (vz). (2) Dm (z) ⊆ T . Proof. Let t ∈ Dm (z) and w = v3 v4 . Without loss t acts as (K1 , K3 )(K2 , K4 ) on Ω, so [w, t] = v or vz, say the former. Choose g as in 10.1.5.4 and let Ji = Kig . Then t acts on O g by 10.1.3.7, so as w ∈ J3 J4 and [w, t] = v = [wt , t], t acts on each Ji , and then by 10.1.3.8, either t ∈ D ∗ (v) or t induces an outer automorphism on each Ji . But in the latter case, COg (t) = v1g vig , v : 1 < i ≤ 4 , so COg (t) = Ω1 (W ) by 10.1.3.6 and symmetry between z and v. Then t centralizes v1 v2 by 10.1.3.6, a contradiction. Therefore t ∈ D∗ (v). This proves (1), and (1) and 10.1.4.5 imply (2).  Lemma 10.1.7. Write Da (z) for the set of t ∈ Do (z) such that t ∈ T acts on each Ki and inverts W and set  D∗ (e). D ∗ (E) = e∈E #

Then (1) If t ∈ Do (z) is in T then t ∈ Da (z). (2) We have  D∗ (e). T ∩Z = e∈E # ∗

(3) If t ∈ D (E) then t = x1 x2 x3 x4 for some xi ∈ Ki − W . Proof. Suppose t ∈ Do (z) is in T . By 10.1.3.4, t ∈ OW and t acts on each member of Ω. Then as t induces an outer automorphism on some member of Ω, t ∈ / D ∗ (z) by 10.1.3.8, so by another application of 10.1.4.8, t induces an outer automorphism on each Ki , and hence t inverts O ∩ W . Therefore t ∈ kW where k = k1 · · · k4 with ki ∈ Ki − W . Now k inverts W by 4.3.5.2, establishing (1) and showing Da (z) ⊆ Z ∩ kW . Set  U= D∗ (e), e∈E #

and assume t ∈ T ∩ Z but t ∈ / U. Then by 10.1.6.1, t ∈ / Dm (z) ∪ O, so t ∈ Do (z) a and hence t ∈ D (z) by (1). Define Q as in 10.1.5 and set P = W ∩ Q, U = k W , and M + = M/P . Let X ∈ Syl3 (M ). Then X acts on U + and by 5.8.7.7, W + = [U + , X] ∼ = E4 so U + = CU + (X) × W + with |CU + (X)| = 2. We may take k1+ to invert X + , so CU + (X) ≤ CU + (k1 ) = k+ , (K ∩ W )+ ≤ O + , so each orbit of X on U + contains a member of U ∩ O, and in particular the preimage U0 in U of CU + (X) is contained in Q. Thus Z ∩ U0 ⊆ D ∗ (E) ∪ E # and Da (z) ⊆ Z ∩ kW ⊆ U, establishing (2). Suppose t ∈ D∗ (E). By 10.1.4, t = x1 x2 x3 x4 with xi ∈ Ki of order 4. As W ∩ Z = E # by 10.1.3.2, we have t ∈ / W . As t ∈ D∗ (E), we have t ∈ Q ≤ QW = k W , so t = kw for some w ∈ W . Then w = kt ∈ W ∩ O, so as ki ∈ Ki − W , also  xi ∈ Ki − W , proving (3).

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Lemma 10.1.8. (1) O  NFz (E). (2) FT (M )  NF (E) = NF (W ). (3) T  NFz (E). (4) T  SCF (O). (5) Either O  Fz or m > 8 and OKi ∼ = SL2 [m] for each 1 ≤ i ≤ 4. Proof. As v = v1 · · · v4 with vi  Ki of order 4, NO(z) (E) = O. So as O(z)  Fz , (1) follows from 8.23.2 in [Asc11]. Let Y = NF (E). By (1) and 2.2 in [Asc10], O  CY (z). Thus CO (E)  CY (E) = X by 8.23.2 in [Asc11], so CO (E) ≤ O2 (X ). But, using 10.1.3.5 for example, W = (O ∩ W )M , so W ≤ O2 (X ) ≤ O2 (Y). Then as W is weakly closed in S with respect to F, W  Y. On the other hand by 3.1.11.7 and 10.1.3.2, E # = Z ∩ W , so E  NF (W ), and hence NF (W ) = Y. Moreover by 3.3.10.2, M = [Ω]Y is constrained, and by 3.3.10.3, M is a model for M, so FT (M ) = M  Y, completing the proof of (2). Further T = CM (z)  CY (z) = NFz (E), so (3) holds. Set B = SCF (O). By 10.1.3.2, B ≤ NFz (E) = CY (z), while by (3), T  CY (z), so T  B, establishing (4). If m = 8 then O = O(z)  Fz , so (5) holds in this case. Thus we may assume m > 8. By 10.1.3.5, w ∈ W −O induces an outer automorphism on K interchanging the two classes of Q8 -subgroups of K, so if OK = K then OK ∼ = SL2 [m] by 2.5.7. Then OKi ∼ = SL2 [m[ for each 1 ≤ i ≤ 4 as T is transitive on Ω by 10.1.3.4. Similarly  if OK = K then O = O(z)  Fz , completing the proof of (5). Lemma 10.1.9. There exists a normal subsystem O of Fz with Sylow group O such that T O  Fz , Σ = O2 (

4 

NAutF (O) (Ki )) = O 2 (AutO (O)),

i=1

and one of the following holds: (1) O ∼ = SL2 [m]4 . (2) O = O. (3) m = 8 and O is of index 3a in SL2 [m]4 for some 1 ≤ a ≤ 3. Proof. Suppose first that m > 8 and set O = O(z). From 10.1.8.5, either (1) or (2) holds. Further Aut(O) is a 2-group, so O 2 (AutF (O)) = O 2 (Aut(O)) = 1. Thus it remains to show T O  Fz . Applying Theorem 1.5.2 in [Asc19] to O in the role of “F0 ” and T in the role of “E”, it suffices to show T NO (O)  NF (O). As m > 8, NO (O) = O and E  NF (O), completing the proof in this case. So we may assume m = 8. Let S0 = CS (O), R = OS0 , G0 a model for NF (R), H = T G0 , B = CG0 (O), and G∗0 = G0 /B. Thus G∗ = AutF (O). By 10.1.8.4, T  SB, so [T, B] ≤ CT (O) = z , and hence L = O 2 (H) centralizes B. Let G1 be the kernel of the action of G0 on Ω and G2 the preimage in G of O2 (Sym(Ω)). Then G1 T = G2  G0 by 10.1.3.4. By a Frattini argument, G0 = G2 NG0 (T S0 ). Further as E = Z2 (T S0 )∩O, NG0 (T S0 ) ≤ NG0 (E), so T  NG0 (T S0 ) by 10.1.8.3. Therefore T G0 = T G2 = T G1 , so H = T G0 = T G1 ≤ G2 . Then as L centralizes B, L is of index 3a in SL2 (3)4 for some 0 ≤ a ≤ 4. Further G1 ≤ SBL so T G1 = T SBL = T L , so H = T G1 = T L = T L. Let w ∈ W − O; by 10.1.3.5, w induces an outer automorphism on Ki for 1 ≤ i ≤ 4. Therefore AutL (O) = Σ. Set O = FO (L), F1 = SCF (O), F2 = NFz (S0 ), E1 = T , and E2 = T O. Observe that this 4-tuple satisfies Hypothesis 1.4.7 in [Asc19]. For example Fz = F1 , F2

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by 1.5.2 in [Asc19], E1  F1 by 10.1.8.4, and E2  F2 as T L  G. Therefore  T O = E1 , E2  Fz by 1.4.8 in [Asc19], completing the proof. Notation 10.1.10. Write O for the normal subsystem of Fz appearing in 10.1.9. If m = 8 then T O is constrained, so it possess a model Hz , with Lz = O 2 (Hz ) a model for O. Moreover Lz = OXz where O ∼ = Q48 and Xz is an elementary abelian 3-group; write e(O) for m3 (Xz ). Lemma 10.1.11. (1) Fz = SO, NFz (E) . (2) D∗ (z) = v O . Proof. By 10.1.9, T O  Fz , so by 1.3.2 in [Asc19], Fz = SO, NFz (T ) . As W is weakly closed in S and W ≤ T , NFz (T ) ≤ NFz (W ), and then 10.1.8.2 completes the proof of (1). By 10.1.4.3, D∗ (z) = v Fz . Further if α ∈ homFz (v, O) then as T O  Fz , α = ϕφ for some ϕ ∈ AutFz (T ) and φ ∈ homT O (vϕ, T ). As AutF (T ) acts on E, vα = vφ or vφz. Similarly as O  T O, vφ = vψ for some ψ ∈ homO (v, O), so (2) holds.  Lemma 10.1.12. Let t ∈ Z and α ∈ A(t). Assume that D∗ (zα) ∩ D∗ (z) = ∅. Then t ∈ T . Proof. Assume x ∈ D∗ (z) ∩ D∗ (zα), and let γ ∈ A(zα) with zξ = z, where ξ = αγ. Then ξ is a Fz -map and xγ ∈ D∗ (zγ) ∩ D ∗ (zξ) = D∗ (tξ) ∩ D∗ (z), so by 10.1.4.3 there is δ ∈ AFz (xγ) with xγδ = v. Then tξδ ∈ D∗ (v) ⊆ T by 10.1.4.5, so as ξδ is a Fz -map, and as T is strongly closed in S with respect to Fz by 10.1.9. we conclude that t ∈ T .  Lemma 10.1.13. If t ∈ Do (z) then t ∈ D∗ (u) for some u ∈ {v, vz}. Proof. Assume otherwise; conjugating in Fz , we may assume t ∈ Fzf . Let α ∈ A(t) and St = CS (t). If t ∈ T then the lemma follows from 10.1.7, so t ∈ / T . Hence by 10.1.12, / T by 10.1.7.2, and neither vα nor (vz)α is in D ∗ (z). D∗ (z) ∩ D∗ (zα) = ∅, so zα ∈ Therefore zα ∈ Do (z) by 10.1.6.2. As s = zα ∈ Do (z), we may assume s acts on K1 . If u = vα acts on K1 then as neither u nor us is in D∗ (z), both invert v1 , so s = u(us) centralizes v1 , a contradiction. Thus we may take (K1 , K2 ) to be a cycle of u. Then (K3 , K4 ) is a cycle in u or su, and we may assume the former. Hence u ∈ Dm (z), so there is y ∈ {v, vz} with y ∈ D∗ (u) by 10.1.6.1. As t ∈ Fzf it follows from 2.2 in [Asc10] that s ∈ Fzf and St α = CS (s). By 10.1.3.7, y ∈ St α. Let β ∈ A(s) and ξ = αβ. Then ξ ∈ homFz (St , S) with tξ ∈ Fzf . As vξ ∈ D∗ (z), by 10.1.4.3 there is γ ∈ AFz (vξ) with vξγ = v. Set ζ = ξγ, t¯ = tζ, and y¯ = yβγ. Let g ∈ M with v g = z. As u ∈ Dm (z) and y ∈ E, u centralizes y by 10.1.3.7, so y¯ = yβγ centralizes uβγ = v, and hence y¯ ∈ CS (E) ≤ O2 (M S). Similarly u centralizes z so t¯ = zβγ centralizes uβγ = v, so t¯ ∈ O2 (M S). As y ∈ D∗ (z) we have y¯ ∈ D∗ (t¯), and hence y¯g ∈ D∗ (t¯g ). As y ∈ D∗ (u), we have y¯ ∈ D∗ (v), so y g ) with y¯g δ = v. Then as y¯g ∈ D∗ (z), and then by 10.1.4.3 there is δ ∈ AFz (¯ g ∗ ¯g g y¯ ∈ D (t ), we have t¯ δ ∈ T by 10.1.4.5, while by 10.1.9, T is strongly closed in S with respect to Fz , so t¯g ∈ T . Then as T is Sylow in M , t¯ ∈ T . As t¯ = tζ and ζ is an Fz -map, also t ∈ T , contrary to an earlier reduction. This completes the proof. 

10. |Ω(z)| > 2

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Lemma 10.1.14. (1) Z ⊆ T . (2) We have Z=



D∗ (e).

e∈E #

(3) Do (z) = Da (z). Proof. Let t ∈ Z − O. Then t ∈ Dm (z) ∪ Do (z), so t ∈ T by 10.1.6.2, 10.1.13, and 10.1.4.5. Thus (1) holds. Then (1) and 10.1.7.2 imply (2), while 10.1.13 implies (3).  Lemma 10.1.15. Let E = E(τ ). (1) For each s ∈ Z, there exists α ∈ homE (CS (s), S) with sα = z. (2) F = E(τ ) = Fz , FS (SM ) . Proof. We first prove (1). Let s ∈ Z and B(s) = {α ∈ homE (CS (s), S) : sα = z}. Let g ∈ M with v g = z. Then CS (v) = CS (E)  M S, so β = cg ∈ B(v). Next let s ∈ D∗ (z). By 10.1.4.3 there is γ ∈ AFz (s) with sγ = v. Then γβ ∈ B(s). Finally let s ∈ D ∗ (v) − O. Claim CS (s) ≤ CS (E). For if not, CS (s), COg−1 (s) induces GL(E) on E, so s ∈ O, contrary to the choice of s. By the claim, CS (s)β ≤ S with sβ ∈ D∗ (z). By the previous case there is δ ∈ B(sβ). Then βδ ∈ B(s). Now 10.1.14.2 completes the proof of (1). Next by definition of E(τ ) in 3.4.2, E = SO(z), NF (O), NF (W ) . Observe that Fz ≥ SO(z), NF (O) and by 10.1.8.2, FT (M )  NF (W ), so by 1.3.2 in [Asc19], NF (W ) = FS (SM ), NF (T ) . Finally as z = Z(T ), NF (T ) ≤ Fz , so E = Fz , FS (SM ) , establishing the second equality in (2). Also F = E by (1) and 2.1.8, completing the proof of (2).  Lemma 10.1.16. Assume t0 ∈ D ∗ (E) and set E0 = Et0 , Ω0 = {NKi (E0 ) : 1 ≤ i ≤ 4}, and τ0 = (NF (E0 ), Ω0 ). Then τ0 is a quaternion fusion packet and NF (E0 )◦ ∼ = L3 (2)/23+6 . Proof. Observe that E0 ∼ = E8 . From 10.1.7.3, t0 = k1 · · · k4 where ki ∈ Ki − W . It follows that NKi (E0 ) ∼ = Q8 , so NO (E0 ) ∼ = Q48 . Then τ0 is a quaternion fusion packet by 2.5.2. Next Aut O (E0 ) is the group of transvections with center z, so t0 z ∈ Z. Further E0 ≤ Q = g∈M O g , so for each e ∈ E # , E0 − e ⊆ D∗ (e). Then for each distinct a, b ∈ E0 − E, there is e = ab ∈ a, b ∩ E # , so as e ∈ D∗ (a), also b = ae ∈ D∗ (a). Hence for each a ∈ E0# , E0 − a ⊆ D ∗ (a). Thus for α ∈ A(a), Qa = NO (E0 α) ∼ = Q48 and AutQa (E0 α) is the group of transvections on E0 α with center z. Hence Aa = AutQa (E0 α)α−∗ is the subgroup of A = AutF (E0 ) of all transvections with center a. Therefore GL(E0 ) = Aa : a ∈ E0# . As Ω0 = Ω0 (z) is of order 4, it follows from Theorem 2 and 5.10.7.1 that NF (E0 )◦ ∼ = L3 (2)/23+6 .  Lemma 10.1.17. If O = O then F ◦ = FT (M ). Proof. Assume O = O. Then OT = T is normal in Fz by 10.1.9, so W  Fz . Then by 10.1.15.2, W  Fz , FS (SM ) = F, so FT (M )  F by 10.1.8.2, the lemma follows.  Lemma 10.1.18. If m = 8 and e(O) = 1 then F ◦ ∼ = L3 (2)/23+6 . Proof. Recall Notation 10.1.10, and assume m = 8 and e(O) = 1, and set ˜ z = Hz /z . Then X = Xz is of order 3, and as T is transitive on Ω, C ˜ (X) = 1. H O

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261

∼ E4 ; let E0 be the preimage of E ˜0 = ˜ ˜0 in O and t ∈ E0 − E. By Therefore E vX = 10.1.11.2, E0 = v O = v Fz , so E0  Fz . Suppose t ∈ D∗ (E) and let F0 = NF (E0 )◦ ; by 10.1.16, F0 ∼ = L3 (2)/23+6 , so F0 has a model G0 , and M0 = NG0 (E) ∼ = M , so E0  M . Thus E0  Fz , FS (SM ) = F by 10.1.15.2, so F0 = F ◦ . Thus the lemma holds in this case, so we may assume t∈ / D∗ (E) and it remains to derive a contradiction. ˜ i , X] = K ˜ i for each i, t = k1 · · · k4 with ki ∈ Ki − W for each i. Then As [K from the proof of 10.1.7, t inverts W , so t acts on each member of K G . Therefore / D∗ (e) for some e ∈ E # , so by 10.1.7.1, s = tg ∈ D a (z) for some as t ∈ / D∗ (E), t ∈ g ∈ M −O. Thus U = v1 , . . . , v4 = [O, s] is inverted by s and Ω1 (W ) = CO (s) is of order 16, so as |U | = |O|/16 it follows that sU = sO ⊆ Z. Then sΩ1 (W ) ⊆ O g ∩ Z by 10.1.5.5, a contradiction as O ∩ Z = E0# is of order 7 and |Ω1 (W )| = 16. This contradiction completes the proof.  Lemma 10.1.19. If m = 8 and e(O) = 2 then F ◦ ∼ = AE8 . ˜ = H/z . Then Proof. Assume m = 8 and e(O) = 2, and set H = Hz and H ∼ X = Xz = E9 , and by 10.1.3.4, T is transitive on Ω with W O the kernel of the action and T /OW ∼ = E4 . Therefore NT (X) is transitive on Ω, so CO˜ (X) = 1 and ˜ is self centralizing in T˜. We see below that T /O ∼ O = E8 and T splits over O. Then ˜ = ˜ ˜ t˜, r˜ , where A˜ = ˜ ˜ Let NH˜ (X) s × X s, t˜, r˜ ∼ E and s˜ is fixed point free on Ω. = 8 Y be the set of hyperplanes Y of X such that CO˜ (Y ) = 1. Then  ˜= O CO˜ (Y ), Y ∈Y

so as T is transitive on Ω it follows that we may chose notation so that X = X1 ×X2 ˜ with [O, X1 ] = K1 K2 = CO (X2 ) and [O, X2 ] = K3 K4 = CO (X1 ), r˜ inverts X, t s ˜ X1 = X2 , K1 = K2 , and r acts on K1 . As NT˜ (X) is abelian and transitive on ˜ a ˜ ˜] = CO˜ (˜ a), so O Ω, r acts on each member of Ω. Then for each a ˜ ∈ A˜# , [O, ˜ is transitive in the involutions in a ˜[O, a ˜], so in particular if a[O, a] contains an involution then a is an involution. On the other hand recall that T splits over O, so indeed a[O, a] contains an involution. In particular we may choose s to be an involution, so U0 = [O, s] = CO (s) ∼ = E32 and then U = U0 s ∼ = E64 . As ˜ s˜ ∈ Z(NH˜ (X)), H = ONH (X) acts on U . Indeed U0 = v H = D∗ (z) by 10.1.11.2 and U = CT (U0 ), so U  Fz . We see in a moment that T has four E64 -subgroups, and all are normal in M . Therefore U  SM . Then by 10.1.15.2, U  Fz , FS (SM ) = F. Now as U ∼ = E64 and μ ∼ = AE8 . = Weyl(D4 ), we conclude from Theorem 2 that F ◦ ∼ ¯ = AE8 satisfies It remains to check our two claims; to do so we observe that G our hypothesis, so as M is determined up to isomorphism by 4.3.8 and 10.1.3.1, we ¯ = O2 (G) ¯ is the natural module for the complement may take M = NG¯ (E). Now U ¯ ¯ , where NY¯ (E) is the stabilizer Y = Alt(I), where I = {1, . . . , 8} and M = NY¯ (E)U of the partition Λ = {{i, i + 1} : i ∈ {1, 3, 5, 7}} of I. We may take z = e1,2,3,4 and then ¯ = ei,j : i, j ∈ {1, 2, 3, 4} or i, j ∈ {5, 6, 7, 8} O∩U is a hyperplane of U and O = (O∩U )(1, 2)(3, 4), (1, 3)(2, 4), ((5, 6)(7, 8), (5, 7)(6, 8) . Then ¯ s, t¯, r¯ is a complement to O in T isomorphic to E8 , where s¯ = e4,8 , t¯ = (1, 5)(2, 6)(3, 7)(4, 8), and r¯ = (1, 2)(5, 6).

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Next the E64 subgroups of T are U , together with the subgroups U (A) = ACU (A) for A ≤ T0 = T ∩ Y¯ such that A is an FF-offender on U , in the sense of section B of [AS04]. By B.3.2.6 in [AS04] there are exactly three choices for A: A(1) = (1, 2)(3, 4), (1, 2)(5, 6), (1, 2)(7, 8) and the two E8 -subgroups A(2) and A(3) of T0 which are regular on I. Regard Y¯ as L4 (2) with Borel subgroup T0 . Now NY¯ (A(1)) = NY¯ (E) is the middle node parabolic, while NY¯ (A(i)), i = 2, 3 are the maximal parabolics with Levi factor L3 (2). As all three parabolics contains the middle node parabolic, U (A(i)) is normal in M for each 1 ≤ i ≤ 3, and of course ¯ is normal in M . This completes the proof of the lemma. U  Lemma 10.1.20. (1) If m = 8 and e(O) = 3 then F ◦ ∼ = Ω+ 8 (2). + 4 ◦ (2) If O ∼ = P Ω8 [m]. = SL2 [m] then F ∼ ˜ ˜ be the group Ω+ (2), S˜ ∈ Syl2 (G), Proof. Under the hypothesis of (1), let G 8 ˜ Ω) ˜ the quaternion fusion packet of G ˜ inherited from the embedding of and τ˜ = (F, ˜ in P Ω+ (3). On the other hand under the hypothesis of (2) let F˜ = P Ω+ [m] and G 8 8 ˜ its Lie packet. In either case as M is determined up to isomorphism τ˜ = (F˜ , Ω) from the proof 4.3.8 and 10.1.3.1, and as τ˜ satisfies the corresponding hypothesis ˜ = |T | and F ˜ (M ˜ ) = N ˜ (E), ˜ we may take S˜ = T , M ˜ = M , and on τ with |S| S F ˜ Ω = Ω. Set Y = T O, FT (M ) ≤ F. We first claim that Y = F˜ . By 10.1.15.3 applied ˜ F˜ = T O, ˜ FT (M ) , so it suffices to show that T O ˜ = T O. to F, Suppose the claim holds. The second step in the proof is to set F1 = Fz , F2 = FS (SM ), E1 = T O, and E2 = FT (M ). We verify that this tuple satisfies Hypothesis 1.4.7 in [Asc19] and then appeal to 1.4.8 in [Asc19] to conclude that Y  F. Then as F ◦ = Y, proving the lemma. The verification of Hypothesis 1.4.7 in [Asc19] is not difficult. By 10.1.15.2, F = F1 , F2 . By 10.1.9, E1  F1 and by 10.1.8.2, E2  F2 . As Y ∼ = F˜ , Y is 2 saturated and Y = O (Y). Thus Hypothesis 1.4.7 in [Asc19] is satisfied. ˜ = T O, which we do separately in each of the two It remains to show T O ˜ = H/z . cases. Assume first that m = 8 and e(O) = 3, and set H = Hz and H ∼ ˜ ˜ Then X = Xz = E27 . Now H is contained in the stabilizer A in Out(O) of Ω ∼ and A = S4 wr S4 ; that is A = (A1 × · · · × A4 )B where B = Sym(I) permutes {Ai : i ∈ I} via (a1 · · · a4 )b = a1b · · · a4b for b ∈ B, I = {1, . . . , 4}, and Ai ∼ = S4 . Set A+ = A/O2 (A), so that A+ ≤ Out(O) and A+ ∼ = S3 wr S4 . We saw during the proof of 10.1.19 that T /O ∼ = E8 and T splits over O. Then ∼ E8 = T + ≤ A+ and E27 ∼ = E81 . Further the kernel T0+ of = X + ≤ O(A+ ) ∼ + + the action of T + on Ω, and hence also on the A+ i , is of order 2, with T /T0 + + regular on the Ai . It follows that T has four 1-dimensional weight spaces on O(A+ ). Moreover NA+ (T + ) is transitive on these weight spaces, so the four T + invariant hyperplanes of O(A+ ) are conjugate under NA+ (T + ). Hence, conjugating ˜ to be H, so that T O ˜ = T O. in Aut(H), we may indeed choose the model for T O (cf. 1.6.7 in [Asc19]) This completes the proof of (1). Next assume O ∼ = SL2 [m]4 ; that is O = O1 ∗ · · · ∗ O4 is a central product where ˜ by 2.7.3, completing the proof Oi ∼ = SL2 [m] with Ki Sylow in Oi . Hence T O = T O of (2) and the lemma.  Theorem 10.1.21. Assume Hypothesis 10.1.1. Then one of the following holds: (1) O = O and F ◦ = FT (M ).

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(2) (3) (4) (5)

263

∼ L3 (2)/23+6 . m = 8, e(O) = 1, and F ◦ = ◦ ∼ m = 8, e(O) = 2, and F = AE8 . m = 8, e(O) = 3, and F ◦ ∼ = Ω+ 8 (2). + 4 ◦ ∼ P Ω [m] and F [m]. O∼ SL = = 2 8

Proof. Recall the normal subsystem O of Fz is defined in 10.1.9, and by 10.1.9 one of the three cases in that lemma holds. If O = O then (1) holds by 10.1.17, while if O ∼ = SL2 [m]4 then (5) holds by 10.1.20.2. Therefore by 10.1.9 we may assume that m = 8 and, as in Notation 10.1.10, e(O) = m3 (Xz ) is 1, 2, or 3. Then 10.1.18, 10.1.19, and 10.1.20.1 say that (2), (3), or (4) holds, completing the proof of the theorem. 

10.2. |Ω(z)| large In this section we assume the following hypothesis: Hypothesis 10.2.1. τ = (F, Ω) is a quaternion fusion packet such that (1) For z ∈ ZS , |Ω(z)| > 2. (2) F ◦ is transitive on Ω. (3) The Inductive Hypothesis holds. Notation 10.2.2. Let z ∈ ZS with z ∈ F f and set {Ki : 1 ≤ i ≤ k} = Ω(z). Set K = K1 , let η ∈ η(τ ), W = η , G a model for NF (CS (W )), M = K G , and μ = μ(τ ). Set O = O(z) and ZS = {z1 , . . . , zr } with z = z1 . Lemma 10.2.3. Assume ρ = (Y, Γ) is a proper subpacket of τ such that Γ(z) = Ω(z) ∩ Γ = {K1 , . . . , Kkρ } with kρ > 2. Then (1) Theorem 4 holds in ρ. (2) Suppose ξ = (C, Σ) is a coconnected component of ρ◦ such that K ∈ Σ. Let θ ∈ η(ξ) and U = θ . Then either (a) ξ = (OK , K), and (OKi , Ki ) is a coconnected component of ρ◦ for each 1 ≤ i ≤ kρ , or (b) kρ = 4, Σ = Γ(z), μ(ξ) ∼ = Weyl(D4 ), and either U  C and Z(C) = 1 or + ∼ C = L3 (2)/23+6 , AE8 , Ω+ 8 (2), or P Ω8 [m]. Proof. Part (1) follows from 10.2.1.3 and 6.6.8. / Σ for some i. Then Assume the setup of (2), and suppose first that Ki ∈ there is a coconnected component ξ  = (C  , Σ ) of ρ such that Ki ∈ Σ . Then C  centralizes C, so z centralizes C, and hence ξ = (OK , K). Hence (2a) holds in this case, so we may assume Γ(z) ⊆ Σ. Then by induction on the order of τ , kρ = 4, Σ = Γ(z), and μ(ξ) ∼  = Weyl(D4 ). Then (2b) holds by Theorem 10.1.21. Lemma 10.2.4. Set τW = (FS∩M (M ), Ω). Then either μ is abelian and τW has rk = |Ω| coconnected components (J, {J}) for J ∈ Ω, or the following hold: (1) k = 4. (2) M = M1 × · · · × Mr with τi = (FSi (Mi ), Ω(zi )) a coconnected component of τW , for Si = S ∩ Mi and r = |Ω|/4. (3) μ = μ1 × · · · × μr with μi = μ(τi ) ∼ = Weyl(D4 ).

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Proof. By 6.2.6, G is transitive on Ω. Replacing F by NF (CS (W )), we ˆ × μ1 × may assume F = FS (G). By 3.2.2.1, A(τ ) = ∅. Then by 3.1.25, μ = μ · · · × μr , where μ ˆ is described in 3.1.24 and 3.1.25, and each μi is generated by 3-transpositions. As G is transitive on Ω and k > 2, we conclude from 6.1.6.2 that either μ = μ ˆ, so that μ is abelian, or μ = μ1 × · · · μr with G transitive on the μi , and we may assume the latter. From 4.1.4, each μi is generated by a conjugacy class Di of 3-transposition, and μi is nonabelian. Then by 4.2.10.2, M = M1 · · · Mr is a central product, where τi = (FS∩Mi (Mi ), Ωi ) is a coconnected component of τ . By 10.2.3.2, Ωi = Ω(zi ) for some zi ∈ ZS , μi ∼ = Weyl(D4 ) and Z(Mi ) = 1. Thus k = 4 and as Z(Mi ) = 1, the central product is direct. The proof is complete.  Lemma 10.2.5. One of the following holds: (1) Either μ is abelian or r = 1. Further {τJ = (OJ , J) : J ∈ Ω} is the set of coconnected components of τz◦ , or (2) r > 1, k = 4, μ is nonabelian, and {τKi : 1 ≤ i ≤ 4} ∪ {τj = (Cj , Ω(zj ) : 2 ≤ j ≤ r} is the set of coconnected components of τz◦ , with μ(τj ) ∼ = Weyl(D4 ) and FSj (Mj ) ≤ Cj for 1 < j ≤ r. Proof. See the proof of 9.3.10. We apply 10.2.3 to τz , zj in the role of ρ, z. By 10.2.3.1, τz◦ has coconnected components ξl = (Fl , Γl ), 1 ≤ l ≤ a. We may take Fi = OKi and Γi = {Ki } for 1 ≤ i ≤ k. By 10.2.3.2, for l > k, either ξl = (OJ , J) for some J ∈ Ω − Ω(z), or kξl = 4 = k, Γl = Ω(zi(l) ) for some i(j) > 1, and μ(ξl ) ∼ = Weyl(D4 ). If r = 1 then a = k and (1) holds, so we may assume r > 1. Suppose μ is nonabelian; then by 10.2.4, k = 4 and M and μ are described in 10.2.4. Then, arguing as in the proof of 9.3.10, for each i > 1, FSi (Mi ) ≤ Cj(i) for some j = j(i) > k with Weyl(D4 ) ∼ = μi ≤ μ(ξj ), so as μ(ξj ) ∼ = Weyl(D4 ), we conclude that μi = μ(ξj ). Therefore the map i → i(j) is a bijection of {2, . . . , r} with {5, . . . , a}, so (2) holds in this case. So assume that μ is abelian. Then as μ(τz ) ≤ μ, μ(τz ) is abelian, so that (1) holds in this case.  Lemma 10.2.6. Let u ∈ O − z be an involution and assume either (a) r > 1 and u ∈ / Z, or (b) μ is abelian. Then uFz ∩ O ∩ F f = ∅. Proof. If (b) holds then the argument establishing 9.4.4.1 works, using the hypothesis that μ is abelian in place of the appeal to 9.4.1.3. So assume (a) holds and μ is nonabelian, and let α ∈ A(u) and Σ = Ω − Ω(z); we can still argue as in the proof of 9.4.4.1. First u = u1 · · · ul with l > 0 even and ui ∈ Ki of order 4. Then ui α ∈ Di ∈ Δ(zα) centralizes Ω − {K1 , . . . , Kl }α, Sj α for j > 1, and v = uα, so Di ∈ Ω(zα) ∪ η(τ1 )α. Therefore either ZS α = ZS or l = 4. In the first case v ∈ / Z, so zα is the unique member of ZS such that v does not centralizes Ω(zα), and hence CS (v) ≤ CS (zα). Similarly in the second case zα ∈ ZΔ(τ1 )α , and v ∈ Q = D1 · · · D4 ≤ W1 α, where W1 = η1 for η1 ∈ η(τ1 ). However as v = u1 α · · · u4 α with ui of order 4, it follows from 10.1.3.2 that v ∈ Z, contrary to hypothesis. Therefore CS (v) ≤ CS (zα). Let β ∈ A(zα) with zζ = z, where ζ = αβ. As  O(z)  Fz , vβ = vζ ∈ O. As CS (v) ≤ CS (zα), vζ ∈ F f .

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Lemma 10.2.7. Let u ∈ O − z be an involution such that u ∈ F f . (1) For each coconnected subcomponent ρ = (Y, Γ) of τu◦ such that Γ∩Ω(zj ) = ∅ for some j > 1, ρ is a coconnected component of τz◦ . (2) If r > 1 then each coconnected subcomponent of τu◦ is a component of τz◦ , and if u centralizes Ki then τKi is a component of τu◦ . (3) Assume r = 1, μ is abelian, and CΩ(z) (u) = {K1 , . . . , Kl } with l > 0. Then for each 1 ≤ i ≤ l, (OKi , Ki ) is a coconnected component of τu◦ . Proof. Part (1) can be proved using the argument which establishes 9.4.4.2. Here we use 10.2.3.2 and the fact that if Y is one of the systems listed there then Sj is Sylow in Y and CSj (O(zj ))= zj . Assume r > 1 and let B2 = j>1 Cj as in 10.2.5. By (1), B2  Fu . Let T2 be Sylow in B2 . As in 8.1.4.1, T2 ∈ F f . Further for each Ki centralizing u, τKi is a coconnected component of CFu (T2 ), so τKi is also a component of Fu . Hence (2) holds. Assume the setup of (3) and let ρi = (Yi , Γi ) be the coconnected component of τu containing Ki . If l > 2 then applying 10.2.3 to τu in the role of ρ, and using the assumption that μ is abelian, (3) holds. If l = 2 and (3) fails then as μ is abelian, Yi ∼ = AE5 , P Sp4 [m], or L− 4 [m] by Theorem 9.4.33. In particular for Ti Sylow in Yi , CTi (OYi (z)) = z . Similarly if l = 1 then by Theorem 7.1.7, Yi ∼ = L2 [2m](1) or − L3 [m], so again CTi (OYi (z)) = z . Using these facts, the proof of 9.4.4.2 shows  that z centralizes Yi , a contradiction. Lemma 10.2.8. If r > 1 then μ is abelian. Proof. Assume r > 1 and μ is nonabelian. We check that Hypotheses 8.1.12 and 8.1.14 are satisfied. Then we appeal to 8.1.15 to see that F ◦ = C1 ∗ · · · ∗ Cr , contradicting F = F ◦ transitive on Ω. Condition 8.1.12.1 follows from 10.2.4. Condition 8.1.12.2 follows from 10.2.5. From Theorem 10.1.21 and its proof, Sj is Sylow in Cj and by 10.1.15.2, Cj = Sj Oj , FSj (Mj ) . Thus it remains to show that Hypothesis 8.1.14 holds. Here C1 = S1 O1 , FS1 (M1 ) is described in 10.2.3.2. Let η1 ∈ η(τ1 ) and W1 = η1 . Let u be an involution in S1 . We claim that u is fused into O under M1 . First all involutions in S1 /W1 are fused into OW1 under M1 . Further from 10.1.3.5, |OW1 : O| = 2 while (cf. 10.1.7.1) each involution in OW1 − W1 is conjugate under O to a member of O g for g ∈ M1 − O. Finally by 10.1.5.5, each involution in W1 is in O, completing the proof of the claim. By the claim and 10.2.6, we may choose u ∈ O ∩ F f , and then 8.1.14 holds by 10.2.7.2. This completes the proof of the lemma.  Lemma 10.2.9. z ∈ / Z(F). Proof. If z ∈ Z(F) then by 10.2.1.2, F ◦ = O(z), contradicting F ◦ transitive on Ω.  Lemma 10.2.10. r = 1. Proof. Assume r > 1; then μ is abelian   by 10.2.8. Set B1 = O(z), Γ1 = Ω(z), B2 = j>1 O(zi ), Γ2 = Ω − Γ1 , and T2 = j>1 O(zj ). We check that Hypothesis 8.1.2 is satisfied. Then we appeal to 8.1.6.2 to see that F ◦ = B1 ∗ B2 , contradicting F = F ◦ transitive on Ω. Hypothesis 8.1.1 holds by 10.2.1.3 and 6.6.8.

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Condition 8.1.2.1 follows from 10.2.5. Suppose x is an involution in O with x ∈ Of . By 10.2.6 we may assume x ∈ F f , and then it follows from 10.2.7.2 that all coconnected components of Fx are of the form τJ for some J ∈ Ω, and then in particular that B2  Fx . That is 8.1.2.2 holds. Condition 8.1.2.3 follows from 10.2.9, while 8.1.2.4 follows from the transitivity of F on Ω and 8.1.2.5 follows from the definition of Bi . This completes the proof of the lemma.  Lemma 10.2.11. μ is nonabelian. Proof. Assume μ is abelian. We begin a series of reductions. Set F˜z = Fz /z . Write U for the set of involutions in O −z . Observe that for u ∈ U, u = u1 · · · ul(u) for some even integer l(u), for some ui ∈ Kσ(i) of order 4, and ˜ i is ˜= K for some injection σ : {1, . . . , l(u)} → I = {1, . . . , k}. Moreover as O i∈I a direct product, σ is uniquely determined and the elements ui are determined up to multiplication by z. Further as Fz permutes the subsystems OKi , we have: (a) For u ∈ U and φ ∈ homFz (u, S), uφ ∈ O and l(uφ) = l(u). (b) For each u ∈ U and δ ∈ A(u), uδ ∈ O and δ ∈ homFz (CS (u), S). This follows from the proof of 9.4.4.1, using the fact that r = 1 by 10.2.10 and μ is abelian by hypothesis. As a consequence of (b): (c) For each u ∈ U, uF ∩ O = uFz . (d) If u ∈ U with l(u) < k then Fu ≤ Fz . For let δ ∈ A(u); by (b), v = uδ ∈ O and δ ∈ homFz (u, S), and by (a), l = l(u) = l(v), so we may assume CΩ (v) = {K1 , . . . , Ka }, where a = k − l > 0. By 10.2.7.3, {τKi : 1 ≤ i ≤ a} is the set of coconnected components of Fv , so Fv ≤ Fz . Then as δ is an Fz -map and Fu δ ∗ ≤ Fv ≤ Fz , (d) holds. Write Dm (z) for the set of t ∈ Z which are fixed point free on Ω, and Do (z) for the set of t ∈ Z − {z} which act on some member of Ω. As r = 1 and μ is abelian, D∗ (z) = ∅, so (e) If t ∈ Do (z) acts on K then t induces an outer automorphism on K with Kt ∼ = SD2m . By 10.2.9 there is t ∈ Z − {z}. Let α ∈ A(t). (f) There exists u ∈ CU (t) with l(u) = 2. If t acts on K1 and K2 then by (e), t inverts vi ∈ Ki of order 4, so v1 v2 ∈ CU (t). If K1t = K2 then v1 v1t ∈ CU (t). By (f) the set U(t) of u in CU (t) with l(u) < k is nonempty. Pick u ∈ U(t) and set B = t, u , s = zα, and v = uα. Let γ ∈ A(u) and β ∈ A(v) with vβ = uγ. Then ξ = αβγ −1 ∈ homFu (CS (B), S), so ξ is an Fz -map by (d). Similarly γ is an Fz -map by (b), so αβ = ξγ ∈ homFz (CS (B), S). In particular zαβ = z ∈ Z(CS (uα)β), so s = zα ∈ Z(CS (v)). Therefore (g) CS (v) ≤ CS (s). (h) If v acts on K then CK (v) = z and Kv ∼ = SD2m . For if not, z < CK (v), so z < CK (s) by (g), contrary to (e). (i) v acts on each member of Ω. ˜ ∼ For assume K v = K2 . Then X = {x1 xv1 : x1 ∈ K} ∼ =K = Dm/2 centralizes v, so X ≤ CKK2 (s) by (g). Hence s and v act on {K1 , K2 }, so some r ∈ {s, sv} acts ˜K ˜ 2 , contrary to (e) on K1 and K2 . Further r ∈ s, v ≤ CS (X), so r centralizes K and (h) and the fact that rα−1 = z or zu ∈ U(t). (j) s ∈ Dm (z).

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Suppose s acts on K and let V be a NS (K)-invariant cyclic subgroup of K of index 2. By (e) and (h), s and v invert V , so sv centralizes V . As zu ∈ U(t), this contradicts (h). By (i) and (j) we may take K1s = K2 and v acts on K1 and K2 . Hence sv K1 = K2 , contrary to (i) and the fact that zu ∈ U(t). This contradiction completes the proof of the lemma.  Lemma 10.2.12. k = 4 and μ ∼ = Weyl(D4 ). Proof. By 10.2.10, r = 1. Therefore by 10.2.4 either μ is abelian or the lemma holds, and in the former case 10.2.11 supplies a contradiction.  Theorem 10.2.13. Assume Hypothesis 10.2.1. Then Ω = Ω(z) is of order 4 for z ∈ ZS , μ(τ ) ∼ = Weyl(D4 ), and taking W ∈ W (τ ), either W  F or + F◦ ∼ = L3 (2)/23+6 , AE8 , Ω+ 8 (2), or P Ω8 [m]. ∼ Weyl(D4 ). Proof. By 10.2.10, Ω = Ω(z), and by 10.2.12, |Ω(z)| = 4 and μ = Therefore Hypothesis 10.1.1 is satisfied, so Theorem 10.1.21 completes the proof.  Theorem 10.2.14. Assume τ = (F, Ω) is a quaternion fusion packet such that F ◦ is transitive on Ω and for z ∈ ZS (τ ) we have Ω = Ω(z) is of order at least 3. Then (1) μ(τ ) ∼ = Weyl(D4 ). (2) Either F ◦ = FS∩M (M ) for M ∈ M (τ ) or F ◦ is isomorphic to L3 (2)/23+6 , + AE8 , Ω+ 8 (2), or P Ω8 [m]. (3) F = Fz , FS (SM ) for M ∈ M (τ ). Proof. Assume otherwise and pick τ to be a counter example of minimal order. We show that Hypothesis 10.2.1 is satisfied and appeal to Theorem 10.2.13 and lemma 10.1.15 to obtain a contradiction. By hypothesis, conditions (1) and (2) of 10.2.1 are satisfied, so it remains to consider a proper subpacket ρ = (Y, Γ) of τ satisfying the hypothesis of Theorem 1 with Y ≤ F and Γ ⊆ Ω, and to show ρ satisfies one of the conclusions of Theorem 1. As Ω = Ω(z) also Γ = Γ(z). Set k = |Γ|. If k > 2 then by minimality of τ , ρ satisfies one of the conclusions of our theorem, and hence one of the conclusions of Theorem 1. If k = 1 then ρ is a conclusion to Theorem 1 by Theorem 5. Finally if k = 2 then ρ is a conclusion to Theorem 1 by 9.5.24. This completes the proof. 

CHAPTER 11

Some results on generation Let τ = (F, Ω) be a quaternion fusion packet with Sylow group S. In Chapter 11 we consider the case where τ satisfies Hypothesis 11.2.1. In particular F is transitive on Ω and for z ∈ ZS (τ ), we have |Ω(z)| ≤ 2; indeed in Chapter 11, usually |Ω(z)| = 2. Our objective is to prove results on the generation of F; that is to say we wish to show that F = B, where B is the subsystem of F generated by some nice collection of subsystems of F. For example in section 11.3, B = B(τ ), where B(τ ) is generated by E(τ ) and the centralizers Fx of fully centralized involutions x in a certain set W(τ ) of involutions x with W(τ ) ⊆ W ∈ W (τ ) when |Ω(z)| = 2 and W(τ ) ⊆ WS when |Ω(z)| = 1. The major results in the chapter of this sort are Theorems 11.3.10 and 11.3.11; the heart of the proof of these theorems is the difficult analysis in section 11.2. Theorem 11.3.11 is refined in Theorem 11.5.3 in the case where F is an extension of the 2-fusion system of an orthogonal group. Similar refinements for all the classical groups appear later in the paper. ˇ n . Section Section 11.6 considers generation in the case where F ◦ is AEn or AE 11.4 contains some results on essential subgroups, used in the proof of 11.5.3 and later results of a similar flavor. Section 11.1 begins the general study of the case |Ω(z)| = 2 and μ ∼ = Weyl(Dn ) with n ≥ 4. 11.1. |Ω(z)| = 2, μ isomorphic to Weyl(Dn ), n ≥ 4 In this section we assume the following hypothesis: Hypothesis 11.1.1. τ = (F, Ω) is a quaternion fusion packet such that (1) For z ∈ ZS , |Ω(z)| = 2. (2) F ◦ is transitive on Ω. (3) μ ∼ = Weyl(Dn ) for some n ≥ 4. (4) The Inductive Hypothesis holds in τ . Notation 11.1.2. Pick z ∈ ZS ∩ F f and write {K, K  } for Ω(z). Indeed for J ∈ Ω, write J  for the second member of Ω(z(J)). By 11.1.1.3 and 3.3.14 there is a unique η ∈ η(τ ). Set W = η , let GW be a model for NF (CS (W )), Λ = K GW , M = Λ , TM = S ∩ M , D = Dη , and μ = μ(τ ). Let I = {1, . . . , n}, WΔ = ZΔ , and nS = |ZS |; thus n = 2nS or 2nS + 1 by ¯ =ω 4.2.8.6. By 5.8.5 there is a surjective homomorphism π : M ¯ (Dn , m) → M ¯ ). As |Ω(z)| = 2, u ¯ ∈ ker(π) by 5.8.5. with ker(π) ≤ CW ¯ (M ¯ is generated by elements k¯i , v¯i , i ∈ I, with K ¯ i = k¯i , v¯i quaterFrom 4.3.1, M ¯ i π, zi = z¯i π, etc, nion of order m with involution z¯i . As in 5.8.5 we set Ki = K except we write xi for v¯i π. Then {Ki : i ∈ I} ⊆ Λ and we set Vi = Ki ∩ W and 269

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¯ and z¯ = z(K ¯  ). Observe that ki ∈ Ki − Vi . Indeed u ¯ = z¯z¯ , where z¯ = z(K),    Kn−1 = Kn . Pick vi , vi of order 4 in Vi , Vi , respectively, and set v(i) = vi vi and WI = v(1)M . Let di ∈ D be the image ki∗ of ki in M ∗ = M/W , and write di for ki∗ , where ˜ = Sym(I), and ki ∈ Ki − Vi . Set μ0 = (d1 d1 )μ ; thus (cf. 5.8.3.2) μ/μ0 = μ ˜ we may choose notation so that di = (i, i + 1) for 1 ≤ i < n, and T˜M contains ˜ is n − 1, n − 2 for n even, odd, respectively. {d˜1 , d˜3 , . . . , d˜n˜ }, where n Notice that we write WI for the module W1 of 5.8.6 and WΔ for the module ¯ ∈ ker(π), it follows from 5.8.3 and 5.8.6 that μ0 centralizes W0 of 5.8.3. As u WI , and by 5.8.6 and its proof, WI is the n-dimensional permutation module for μ ˜ = Sym(I), or its image modulo its center eI , with v(i) of weight 1 in that module. Thus we can view WI as the power set of I with the group operation symmetric difference. For θ ⊆ I, write eθ for θ regarded as a member of WI . By 5.8.3, WΔ is the core of WI . For w ∈ WI set Λw = {J ∈ Λ : J ≤ CM (w)}, Mw = Λw , ηw = {J ∩ W : J ∈ Λw }, and Ww = ηw . Lemma 11.1.3. (1) zi = ei,i+1 for 1 ≤ i < n. (2) v(n − 1) = en . (3) Ω = {K1 , K1 , K3 , K3 , . . . , Kn˜ , Kn˜ }. (4) NF (W ) controls fusion in CS (W ) with AutF (CS (W )) = AutGW (CS (W )). Further WI and WΔ are AutF (CS (W ))-invariant. (5) AutF (WI ) = AutGW (WI ) = AutM (WI ), so AutS (WI ) = AutTM (WI ). (6) ZΔ , v(n − 1)F ∩ W is the set of vectors in WI of weight 2, 1, respectively. (7) For x ∈ WI , xF ∩ F f ⊆ WI and there exists g ∈ M such that c∗g ∈ A(x).  (8) If x ∈ WΔ ∩ F f then x = s∈σ s for some σ ⊆ ZS . ˜ x∗ = C ˜ (x) and d˜ ∈ C ˜ (x) iff d˜ acts on θ. (9) For each x = eθ ∈ WI , M D D Proof. Recall from 3.1.20.3 that [WΔ , di ] = zi , so (1) follows as d˜i = (i, i+1) and WI is the permutation module WI or its image WI /eI . Let w = v(n − 1). By 5.8.6, Cμ˜ (w) = d˜i : 1 ≤ i ≤ n − 2 , so Cμ˜ (w) = μ ˜n is ˜, the stabilizer in μ ˜ = Sym(I) of en . Thus as WI is the permutation module for μ w = en or en eI , and if eI = 1 then en = en eI . If eI = 1 and n is odd then en eI is even, so w = en as w ∈ WI − WΔ and WΔ is the core of WI . Finally if n is even and eI = 1 then we choose notation so that w = en . Thus (2) holds. Let J ∈ Λ. Then J = (J ∩ W )j with j ∗ = dJ∩W ∈ D. Hence J ≤ TM iff ˜ ∩ T˜M = ˜j ∈ T˜M iff J  ≤ TM . Finally from the notation established in 11.1.2, D d˜1 , d˜3 , . . . , , so (3) follows. Further for x ∈ WI , J ∈ Λx iff d˜J∩W = ˜j centralizes x, so (9) holds. The first statement in (4) follows as W is weakly closed in S with respect to F, and then the second follows from the first. By construction M  GW and Λ is GW -invariant. By 4.2.8, the map ζ : J → z(J) is a surjection of Λ onto ZΔ , so ZΔ is the set of elements of WI of weight 2 by (1) and WΔ = ZΔ  G. Similarly V = v(n − 1)M is the set of vectors of weight 1 by (2) and consists of the involutions in JJ  ∩ W − z(J) for J ∈ Λ, so V is GW -invariant, and hence WI = V  GW . This proves (4) and (6). ˜ = Sym(I), so EndF2 μ (WI /eI ) = Next WΔ /eI is the natural module for μ F2 , and hence AutF (WI ) = AutM (WI ). This proves (5).

11.1. |Ω(z)| = 2, μ ISOMORPHIC TO Weyl(Dn ), n ≥ 4

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Let x ∈ WI , and α ∈ A(x). As W ≤ CS (x) is weakly closed, W α = W , so xα ∈ WI by (4). Further by Sylow’s Theorem there is g ∈ M with CS (xg ) ∈ Syl2 (CSM (x)), so c∗g ∈ A(x), proving (7). Choose x as in (8). Then x is of even weight 2k so there is h ∈ M with xh = y = z1 z3 · · · z2k−1 . Thus y ∈ ZS by (3), while by (7) there is g ∈ G such that c∗g ∈ A(y) with y g = x. Then O(τ ) ≤ CS (y), so O(τ )g = O(τ ) and (8) follows.  Lemma 11.1.4. Let 1 = x = eθ ∈ WI with x ∈ F f , and set k = |θ|, Ωx = Λx ∩Ω and τx = (Fx , Ωx ). Set θ1 = θ, θ2 = I − θ, Λi = {J ∈ Λ : M ov(dJ∩W ) ⊆ θi }, Ωi = Ω ∩ Λi , Mi = Λi , ηi = {J ∩ W : J ∈ Λi }, and Wi = ηi . (1) τx is a quaternion fusion packet. . (2) ηx ∈ η(τx ),Mx is a model for [Ωx ]NFx (Wx ) , and μ(τx ) ∼ = Mx /Wx (3) Λx = Λ1 Λ2 , Mx = M1 ∗ M2 is a central product, ηx = η1 η2 , and τx◦ = τ1 ∗ τ2 with τi = (Fi , Ωi ) a quaternion fusion packet, ηi ∈ η(τi ), and μ(τx ) = μ(τ1 ) × μ(τ2 ), where μ(τi ) ∼ = Mi /Wi . (4) x ∈ WΔ iff k is even, in which case Ωx = Ω. (5) M1 ∼ = ω(Dk , m), μ1 ∼ = Weyl(Dk ), x ∈ Z(M1 ), M2 ∼ = ω(Dn−k , m), μ2 ∼ = Weyl(Dn−k ), and if M2 = 1 then eI x ∈ Z(M2 ). Proof. As x ∈ F f , Fx is a saturated fusion system. By definition, Λx consists of those J ∈ Λ centralizing x, so Ωx consists of those members of Ω centralizing x. Then (1) follows from 2.5.2. Next for J ∈ Λx , J is conjugate in CM (x) to a member of Ωx by Sylow’s Theorem, so as J ∩ W ∈ ηx , ηx ⊆ ν ∈ η(τx ). Conversely if V ∈ ν then V ≤ X ∈ Δ, and either X ∈ Ωx or V = X ∈ η. In the former case as X ∩W ∈ ν, V = X ∩W ∈ η. Thus ν = ηx , proving (2). By 11.1.3.9, Λx is the disjoint union of Λ1 and Λ2 , and hence ηx is the disjoint union of η1 and η2 . If Ji ∈ Λi then [d˜J1 ∩W , d˜J2 ∩W ] = 1, so [J1 , J2 ] = 1 by 3.1.22. Hence [M1 , M2 ] = 1, so Mx = M1 ∗ M2 is a central product. Let k1 = k and k2 = n − k. As WΔ is the core of the permutation module WI , x ∈ WΔ iff k is even. Moreover in that event as x ∈ F f , x ∈ ZS by 11.1.3.8, so Ωx = Ω. Hence (4) holds. Further we may choose notation so that θ = {1, . . . , k},  and hence (using the fact that x ∈ Z(M1 ) and so M1 = K1 , . . . , Kk−1 , Kk−1 eI x ∈ Z(M2 ) if M2 = 1) we have Mi ∼ = ω(Dki , m) and μi = Mi /Wi ∼ = Weyl(Dki ) from 5.8.6. Thus (5) holds. It remains to prove (3). By 11.1.1.4 and 6.6.6, τx◦ = ρ1 ∗ · · · ∗ ρr where ρi = (Yi , Γi ), 1 ≤ i ≤ r are the coconnected components of τx◦ . Note that if K ∈ Γi then by 3.1.6, either K  ∈ Γi or ρi = (OK , {K}). Hence in the former case for each z ∈ ZS (ρi ) we have Γi (z) = Ω(z) of order 2. Then as ρi satisfies Theorem 1, we conclude that μ(ρi ) ∼ = Weyl(Dl ) for some l ≥ 2. Suppose kj > 2 for some j ∈ {1, 2}. Then Mj is transitive on Λj so Ωj ⊆ Γi for some i, and ηj ⊆ η(ρi ) ⊆ η(τx ) = η1 ∪ η2 , so μ(ρi ) = μj × μ3−j for some μ3−j ≤ μ3−j . But we saw that μ(ρi ) ∼ = Weyl(Dl ), so we conclude that μ(ρi ) = μj and hence Γi = Ωj . Therefore (3) holds in this case with τj = ρi and τ3−j the produce of the remaining ρl . Therefore we may assume that kj ≤ 2 for j = 1, 2, so as k1 + k2 = n ≥ 4, we have n = 4 and k1 = k2 = 2. Therefore μx = μ1 × μ2 with μi ∼ = E4 . Hence for

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each i, μ(ρi ) is abelian, and we saw that either Γi = Ω(z) is of order 2 for some  z ∈ ZS (ρi ) or ρi = (OKi , {Ki }). We conclude that (3) holds. Lemma 11.1.5. (1) Fz◦ = O(z) ∗ C where ρ = (C, Ω − Ω(z)) is a quaternion fusion packet with C = C ◦ and μ(ρ) ∼ = Weyl(Dn−2 ). (2) Define M2 as in 11.1.4. Then Z(M2 ) = eI z is of order 2. (3) One of the following holds: (i) C = ω(Dn−2 , m) with n ≥ 5. ˇ 2n−4 with n ≥ 5 or AE2n−3 . (ii) C ∼ = AE − ∼ (iii) C = Ω+ 2n−4 [m] with n ≥ 5, Ω2n−3 [m], or Ω2n−2 [m]. (iv) n = 4 and C = O(zeI ). (v) n = 5 and C ∼ = L3 (2)/E64 or Sp6 (2). ˇ 3 (2)/23+6 or Ω ˇ + (2). (vi) n = 6 and C ∼ =L 8 Proof. As z ∈ ZS , z = eθ with |θ| = 2 by 11.1.3.1. Therefore by 11.1.4, τz = (Fz , Ω) is a quaternion fusion packet and τz◦ = τ1 ∗ τ2 with τi = (Fi , Ωi ) and μi = μ(τi ) satisfying μ1 ∼ = Weyl(D2 ) and μ2 ∼ = Weyl(Dn−2 ). From 2.6.11, O(z)  Fz , so F1 = O(z). Write C for F2 ; then (1) holds. Similarly (2) follows from 11.1.4.5. If n = 4 then μ(ρ) ∼ = E4 by (1), so, using 11.1.1.1, (3) follows from 9.4.33. If n = 5 then μ(ρ) ∼ = S4 by (1), so as |Z(M2 )| = 2 by (2), (3) holds by Theorem 9.5.33. Similarly if n = 6 then μ(ρ) ∼ = Weyl(D4 ), so (3) holds by Theorem 12.1.25, applied in an inductive context. Finally if n ≥ 7 then n − 2 ≥ 5, so (3) holds by induction on n (cf. Theorem 12.2.32).  Lemma 11.1.6. Suppose c = eI = 1. Then: (1) τc = (Fc , Ω) is a quaternion fusion packet with μ(τc ) ∼ = Weyl(Dn ). (2) If n > 4 and c ∈ Z(F) assume the Extended Inductive Hypothesis. Then one of the following holds: (i) Fc◦ = FTM (M ). ˇ 2n . (ii) Fc◦ ∼ = AE ◦ ∼ + (iii) Fc = Ω2n [m]. ˇ 3 (2)/E23+6 or Ω ˇ + (2). (iv) n = 4 and Fc ∼ =L 8 Proof. Part (1) follows from 11.1.4. Set Fc+ = Fc /c ; by 3.3.2, τc+ is a quaternion fusion packet with μ(τc+ ) ∼ = μ(τc ). If n = 4 then Fc+ is described in 10.1.21, so that (2) holds, using 5.1.21.5, 5.10.7.1, 5.10.12, and 3.3.16. Thus we may take n ≥ 5. If c ∈ / Z(F) then Fc◦ satisfies one of the conclusions of Theorem 1 by 11.1.1.4, so, inspecting the lists of conclusions in Theorems 1 and 2, (2) holds in this case, again using 5.1.21.5 and 3.3.16. Thus we may assume c ∈ Z(F) and the Extended Inductive Hypothesis holds. Hence by 6.6.9, τ + satisfies Hypothesis 11.1.1, so by induction on the order of τ , F + is described in Theorem 12.2.32 in the case c = 1, and then, arguing as above and as c ∈ Z(F), F appears in (2). Formally this approach amounts (for fixed n) to treating the case c = 1 in Theorem 12.2.32 before proving 11.1.6 when c = 1.  In the next lemma we appeal to Theorems 12.1.25 and 12.2.32 at n−1. Formally this amounts to treating the n − 1 case to its end, before going on to treat the n case.

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Lemma 11.1.7. Let b ∈ WI ∩ F f be of weight 1 and a ∈ {b, bc}. Then (1) τa = (Fa , Ωa ) is a quaternion fusion packet with μ(τa ) ∼ = W eyl(Dn−1 ). (2) If n is odd then a ∈ Z(S) so Ωa = Ω. (3) One of the following holds: (i) Fa◦ ∼ = ω(Dn−1 , m). ˇ 2n−2 with bc ∈ Z(Fa◦ ). (ii) Fa◦ ∼ = AE (iii) a = b and Fb◦ ∼ = AE2n−1 . + Ω [m] and bc ∈ Z(Fa◦ ). (iv) Fa◦ ∼ = 2n−2 ◦ ∼ (v) a = b and Fb = Ω2n−1 [m] or Ω− 2n [m]. (vi) n = 4, a = b, and Fa◦ ∼ = L3 (2)/E64 or Sp6 (2). ˇ 3 (2)/23+6 or Ω ˇ + (2), and bc ∈ Z(Fa◦ ). (vii) n = 5, Fa◦ ∼ =L 8 Proof. Part (1) follows from 11.1.4 and (2) follows from 11.1.3.5. From 11.1.4, Mb = Mbc , and from 11.1.4.5 we have bc ∈ Z(Ma ), so if Z(Fa◦ ) = Z(Ma ) then bc ∈ Z(Fa◦ ), while if Z(Fa ) = 1 then a = bc so a = b. Suppose n = 4. Then (3) follows from Theorem 9.5.33. Suppose n = 5. Then (3) follows from Theorem 12.1.25. Finally assume n > 5; then n − 1 ≥ 5. Now Fa is proper in F, so by 6.6.12, τa satisfies the Extended Inductive Hypothesis. Then (3) holds by Theorem 12.2.32.  Lemma 11.1.8. Let x be an involution in WI ∩ F f . Then one of the following holds: (1) Fx ≤ E(τ ). (2) x is of weight 1 or n − 1. (3) x = eI . (4) n is even, eI = 1 and x is of weight n/2. Proof.  Assume x = c = eI and x is not of weight 1 or n − 1. By 11.1.4, Ωx = Ω1 Ω2 and Fx◦ = F1 ∗ F2 with τi = (F1 , Ωi ) a quaternion fusion packet and ki = |Ωi | ≥ 2. Observe that from 11.1.4.5, either F1 is not isomorphic to F2 or n − k1 = k2 = k1 . In the former case Fi  Fx . In the latter case, n is even and k1 = k2 = n/2. Moreover if c = 1 then from 11.1.4.5, Z(Fi ) = xi where x1 = x and x2 = cx = x, so again Fi  Fx . Thus we may assume Fi  Fx for i = 1, 2. Now (1) holds by 3.4.13. 

11.2. Generation In this section we assume the following hypothesis: Hypothesis 11.2.1. τ = (F, Ω) is a quaternion fusion packet such that (1) F is transitive on Ω.  (2) ZS is of order nS and, setting WS = ZS and c = z∈ZS z, either m(WS ) = nS or m(WS ) = nS − 1 and c = 1. (3) For z ∈ ZS , |Ω(z)| ≤ 2 and in case of equality μ = μ(τ ) ∼ = Weyl(Dn ) for some n. ∼ Weyl(Dn ), so Notation 11.2.2. Suppose |Ω(z)| = 2. Then by 11.2.1.3, μ = by 3.3.14 there is a unique M ∈ M (τ ). Moreover n = 2nS or 2nS + 1 by 4.2.8.6, ¯ = ω and by 5.8.5, there is a surjective homomorphism π : G ¯ (Dn , m) → M with

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¯ As |Ω(z)| = 2, u ker(π) ≤ CW ¯ ∈ ker(π) by 5.8.5, so by 5.8.3.2, WΔ = ZΔ is ¯ (G). ˜ = M/CM (WS ) ∼ the core of the permutation module for M = μ/μ0 = Sym(I) ∼ = Sn on I = {1, . . . , n} (or that module modulo its center) with ZΔ a set of vectors of weight 2, where μ0 is the subgroup Q of section 5.8 and centralizes WΔ . Let WI ˜ , or that module modulo its center eI ; from be the permutation module for M 5.8.6 we may take WΔ ≤ WI  M , such that CM (WΔ ) centralizes WI and for each z ∈ ZS , z = e2i−1,2i for suitable 1 ≤ i ≤ nS , and Ω(z) = {Ki , Ki }. Let vi , vi be of order 4 in Ki ∩ W , Ki ∩ W , respectively, and v(z) = vi vi . From 5.8.6, v(z) is e2i−1 or e2i in WI . Notation 11.2.3. Let R ∈ F e , let G be a model for NF (R) and G∗ = G/R. Let Θ be the set of orbits of R on ZS . We say that R is exceptional if c = 1 and R is transitive on ZS , and we say that R is ordinary if R is not exceptional. Note that if R is exceptional then 1 < nS is a power of 2. For as R is transitive on ZS , nS is a power of 2. Further if nS = 1 then ZS = {c}, so c = 1. Also note that if R is exceptional and |Ω(z)| = 2 then n = 2nS as c = 1. If R is exceptional write ˆ for the set of R-invariant partitions of ZS of size 2, and when |Ω(z)| = 2 let Θ ˜ Θ  be the set of R-invariant partitions of I of size 2. For θ ⊆ ZS , set fθ = t∈θ t and  for α ⊆ I set eα = i∈α ei . If z ∈ ZS and |Ω(z)| = 1, write v(z) for an element of order 4 in O(z) with v(z)  CS (z). Lemma 11.2.4. Let θ ∈ Θ. Then (1) fθ ∈ Z(R), and (2) if R is ordinary then θ ∩ Z(R) = fθ . Proof. As R permutes θ, R centralizes fθ . Further as R ∈ F e , CS (R) = Z(R), so (1) holds. Suppose R is ordinary and let fσ ∈ Z(R) for some ∅ = σ ⊂ θ. As R is ordinary it follows from 11.2.1.2 that θ is the permutation module for R on θ. Therefore as R centralizes fσ , R acts on σ, contradicting R transitive on θ. Hence (2) holds.  Lemma 11.2.5. Assume R is exceptional. ˆ and if |Ω(z)| = 2 then CW (R)# = {eα : α ∈ (1) CWS (R)# = {fθ : θ ∈ π ∈ Θ} I ˜ π ∈ Θ}. ˆ such that {fi = fθ : 1 ≤ i ≤ β} is a basis for CW (R). For (2) Let θi ∈ πi ∈ Θ i S 1 ≤ γ ≤ β, let γ  θ(γ) = θi and ργ = θ(γ)R . i=1

Then ργ is an R-invariant partition of ZS into 2γ blocks of size nS /2γ , and Zγ = f1 , . . . , fγ consists of the fξ such that ξ is the union of blocks in ργ and {ξ, ZS −ξ} is R-invariant. (3) Choose β, γ, and fi , 1 ≤ i ≤ β as in (2) and set Eγ = fσ : σ ∈ ργ . For 1 ≤ i ≤ β set Ri = NR (θi ) and R(γ) = R1 ∩ · · · ∩ Rγ . Then R(γ) is the kernel of the action of R on ργ , and R/R(γ) ∼ = E2γ is regular on ργ . If γ > 1 then R(γ) = CR (Eγ ). ˆ such that fθ ∈ Zγ for θ ∈ π. Then π ∈ Θ ˆγ ˆ γ consist of those π ∈ Θ (4) Let Θ iff π is the set of orbits of the preimage in R of some hyperplane of R/R(γ) on ZS . ˆ γ and dim(Zγ ) = γ. Hence Zγ# = Θ

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Proof. Let θ ⊆ ZS such that 1 = fθ centralizes R. Then for r ∈ R, fθ = ¯ so π ∈ Θ. ˆ fθr = fθr , so θ r = θ or θ¯ = ZS − θ. It follows that R acts on π = {θ, θ}, ˜ ¯ } ∈ Θ. Thus (1) holds. Similarly 1 = eα centralizes R iff π = {α, α Choose θi , 1 ≤ i ≤ β as in (2); we prove (2) by induction on γ; when γ = 1, the result is trivial. Suppose the result holds at γ, so that ρ = θ R is a partition of ZS , where θ = θ1 ∩ · · · ∩ θγ . Suppose β = γ. If θγ+1 is the union of some set of blocks of ρ = ργ then fγ+1 ∈ Zγ as (2) holds at γ, contrary to the choice of the fi . Hence θγ+1 is not a union of blocks of ρ, so as λ = {θγ+1 , θ¯γ+1 } is the set of orbits of Rγ+1 on I, we have Rγ+1 transitive on ρ and |θγ+1 ∩ b| = k is independent of the choice of b ∈ ρ. Then as R is transitive on λ, also k = |θ¯γ+1 ∩ b|, so k = |θ|/2 and σ R is a partition of ZS , where σ = θγ+1 ∩ θ. This proves the first part of (2), and the second part follows from (4), which we prove in a moment. Assume the setup of (3). Then Ri is the stabilizer in R of θi and of index 2 in R, so the kernel X(γ) of the action of R on ρ = ργ is contained in R(γ) and R(γ)  R. Proceeding by induction on γ as in the previous paragraph, R(γ) fixes θ(γ), so as R(γ)  R, R(γ) = X(γ). Further for γ > 1, r ∈ R centralizes fθ(γ) iff r acts on θ(γ), so R(γ) = CR (Eγ ). As |R : Ri | = 2, R/R(γ) is elementary abelian. Then as R/R(γ) is transitive and faithful on ρ, R/R(γ) is regular on ρ. In particular |R/R(γ)| = |ρ| = 2γ , completing the proof of (3). If P is the preimage of a hyperplane of R/R(γ) then as R/R(γ) is regular on ρ, the set π of orbits of P on ZS is of order 2 and each orbit is the union of blocks of ρ, so fσ ∈ Zγ for σ ∈ π. Conversely if fξ ∈ Zγ then, changing the basis of Zγ , we may assume fξ = f1 , so ξ is an orbit of R1 , and R1 /R(γ) is a hyperplane of ˆ γ is the number 2γ − 1 of hyperplanes of R/R(γ) ∼ R/R(γ). Thus Zγ# = Θ = E2γ , so (4) holds.  Notation 11.2.6. We consider three cases: (I) For z ∈ ZS , |Ω(z)| = 2; in this case set W0 = WI . (II) For z ∈ ZS , |Ω(z)| = 1, nS = 4, R is exceptional, |WS : WS ∩ R| = 2, and WS ∩ R ≤ Z(R); in this case set W0 = Ω2 (K ∩ W ) : K ∈ Ω . (III) For z ∈ ZS , |Ω(z)| = 1 and case (II) does not hold; in this case set W0 = WS . In each case let W(R) consist of those involutions w ∈ CW0 (R) such that: (i) In case (I), w = eα for some α ⊆ I of order a positive power of 2. (ii) In case (II), w ∈ W0 . (iii) In case (III), w = fθ for some θ ⊆ ZS with |θ| a power of 2. Let U be the set of U ∗ ≤ G∗ such that S ∗ ∩ U ∗ is Sylow in U ∗ , U ∗ is not contained in a strongly embedded subgroup of G∗ over NS (R)∗ , U ∗ = u∗U for some u ∈ S∩G such that u∗ is an involution, and U ∗ ≤ CG∗ (w) for some w ∈ W(R). Until further notice we assume the following hypothesis: 

Hypothesis 11.2.7. (1) O 2 (CG (xG )) = R for each x ∈ W(R). (2) If U = ∅ then m2 (G∗ ) = 1 and U0 is not normal in G∗ for any nonempty U0 ⊆ U. Lemma 11.2.8. If R is exceptional then Eβ ≤ R.

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Proof. Assume R is exceptional but Eβ ≤ R and let γ be minimal subject to Eγ ≤ R. Observe that from 11.2.5.2, E1 = f1 ≤ CWS (R) ≤ R, so γ ≥ 2. By minimality of γ, Eγ−1 ≤ R. Next E2 is generated by the four conjugates of fθ(2) , which sum to 0, so |E2 | = 8. Thus |E2 : Z2 | = 2, so if γ = 2 then fθ(2) ∈ / R and [R, fθ(2) ] ≤ Z2 ≤ R, so fθ(2) ∈ G. In this case set y = fθ(2) . On the other hand suppose γ > 2. As Z(R) centralizes fξ for each ξ ∈ ργ−1 , Z(R) acts on ξ. Further ξ contains two blocks of ργ ; for each ξ ∈ ργ−1 , pick one of those blocks ζξ , and define  ζ= ζξ and y = fζ . ξ∈ργ−1

If y ∈ R then Z(R) centralizes y, so as Z(R) acts on ξ it also acts on ζξ , and then Eγ ≤ CS (Z(R)) = R by 11.2.7.1, contrary to the choice of γ. Therefore y ∈ / R, but as R acts on ργ−1 and ργ by 11.2.5.3, we have [R, y] ≤ Eγ−1 ≤ R, so y ∈ G. Thus in any event, y ∈ G − R. Set Z = f1G and H = y G . By 11.2.7.1,  O 2 (CG (Z)) = R, so [Z, y] = 1. Thus Z ≤ CR (Eγ ) = R(γ) by 11.2.5.2, so Z ≤ Ri for some 1 ≤ i ≤ γ, and then as R is transitive on ZS , Z ≤ Ri for any i. Hence Z is nontrivial on ρ2 , so E2 ≤ R and therefore γ = 2 by minimality of γ. Thus y = fθ(2) by construction. In paragraph two we saw that |E2 : Z2 | = 2, so E2 = Z2 y and hence E2∗ = y ∗ . Let h ∈ H with y ∗ , y ∗h not a 2-group, and set L = E2 , E2h and B = O2 (L). Then L ≤ H and [R, E2 ] = Z2 , so B = Z2 Z2h ≤ Z(R) as Z2 ≤ Z(R), and L  LR. If Z0 = Z2 ∩ Z2h = 1 then m(B) ≤ 3 and y induces a transvection on Z, so H ∗ ∼ = S3 by 2.4.1 in [Asc19]. But then Z0 ≤ CZ2 (H), contrary to 11.2.7.2. Therefore B = Z2 × Z2h and |y B | = |B : Z2 | = 4, so B is transitive on ρ2 . Observe that in case (I), we have nS > 2 (cf. 11.2.15), so n = 2nS ≥ 8 is a power of 2. Hence by 11.2.5.1, in case (I) each involution in CW0 (R) is of the form eα for some α ⊆ I of order a positive power of 2, and in case (III) each involution in CW0 (R) is of the form fθ for some θ ⊆ ZS of order a power of 2. Therefore in any case, each involution in CW0 (R) is in W(R). Next as [R, y] = Z2 , R = B × CR (L). Moreover if CR∩W0 (L) = 1 then 1 = CZ(R)∩W0 (L), so that from the previous paragraph, L∗ is in the set U of 11.2.6. Then by 11.2.7.2, m2 (H ∗ ) = 1 and H ∗ = U . Therefore: (a) We can choose h so that CW0 ∩R (L) = 1. Replace Z by Z2G . As Z2 = [R, y] and R = B × CR (L), we have |Z : CZ (y)| = 4. Set Z0 = CW0 ∩R (y). Suppose we can show that: (b) Z2 < Z0 . But for z0 ∈ Z0 , z0 = ab with a ∈ CZ (y) and b ∈ CR (H). For r ∈ R, z0r = abr , so either b = br or z0 z0r = bbr is a nontrivial element of CW0 ∩R (H). The latter contradicts (a), so br = b for each r ∈ R, and hence z0 ∈ CW0 ∩Z(R) (y) = Z1 . It follows that Z2 < Z1 . But R = B × CR (L), so CR (y) = Z2 × CR (L), and hence as Z2 < Z0 we have 1 = CZ1 (L) ≤ CW0 ∩R (L), contrary to (a). Therefore to complete the proof of the lemma, it remains to establish (b). Set Zy = CZ (y), let A be a complement to Z2 in Zy , and let D a complement to Zy in Z. Then D is regular on ρ2 with Zy the kernel of the action of Z on ρ2 . Suppose that Z is transitive on ZS and let s ∈ ZS , Zs a complement to CZ (s) on ZS , for each t ∈ ZS , [v(t), a] = 1 or t, in Z, and a ∈ CZ (s). As Z is transitive  independent of the choice of t. Thus as t∈ZS t = c = 1 and Zs is regular on ZS ,

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 in either case a centralizes w0 = u∈ZS v(s)u . Therefore Z = CZ (s)Zs centralizes w0 and w0 is of weight |ZS | ≥ |Z : CZ (y)| = 4, so if w0 ∈ W0 then by 11.2.7.1, w0 ∈ CW0 (Z) ≤ R and hence w0 ∈ Z0 − Z2 , establishing (b) in this case. Therefore / W0 , so case (II) or (III) holds. If nS = 4 then case (II) holds and w0 ∈ W0 , w0 ∈ so (b) holds here too. Thus nS > 4 and case (III) holds. So we may assume either Z is not transitive on ZS , or case (III) holds with transitive on ρ2 , we have nS > 4. nS > 4. In case (I), as D is  Let s ∈ θ(2) and w = x∈Z sx ; then w ∈ CW0 (Z) = W0 ∩ R by 11.2.7.1. If Z is not transitive on ZS then 1 = w and indeed w ∈ / Z2 , so (b) holds. Therefore we may assume that Z is transitive on ZS , so nS > 4 and case (III) holds. Then nS = |Z : CZ (s)| = 4|sA |.   Let CA (s) ≤ A1 of index 3 in A and set w1 = x∈A1 sx and w2 = d∈D (w1d . Then w2 centralizes DA1 and for a ∈ A − DA1 , w2a = w2 fZS = w2 as R is exceptional, so w ∈ Z0 − Z2 and (b) holds. This completes the proof of (b) and the lemma.  Lemma 11.2.9. If R exceptional and β > 1 then Z(R) acts on each member of ρβ . Proof. By 11.2.8, Eβ ≤ R, so Z(R) ≤ CR (Eβ ) = R(β) by 11.2.5.3. But R(β)  is the kernel of the action of R on ρβ by 11.2.5.3. Lemma 11.2.10. If R is ordinary then ZS ⊆ R. Proof. Assume otherwise; then there exists θ ∈ Θ such that θ ⊆ R, and we fix this θ and set f = fθ . By 11.1.4, f ∈ Z(R) and |θ| = 2a for some a > 0. Let Ra = R, θa = θ, and, proceeding recursively, let Ri be a subgroup of Ri+1 of index 2 with an orbit θi of length 2i on θi+1 . Pick b maximal subject to fθb ∈ / R and R set σ = θb . Then π = θb+1 is a partition of θ into 2a−b−1 blocks of size 2b+1 , and E = fp : p ∈ π ≤ R. Let Γ be a set of coset representatives for Rb+1 in R,  ζ= σ γ , and y = fζ . γ∈Γ 

If y ∈ R then as Z(R) acts on θb+1 , Z(R) acts on σ, so fσ ∈ O 2 (CG (Z(R)) = R by 11.2.7.1, contrary to the choice of σ. Therefore y ∈ / R. / R, Observe that [R, y] ≤ E ≤ R, so y ∈ G. Set Z = f G and H = y G ; as y ∈  H ∗ = 1. By 11.2.7.1, R = O 2 (CG (Z)), so y does not centralize Z. By 11.2.4.2, E ∩ Z ≤ E ∩ Z(R) = f . Therefore [Z, y] = f , so y induces a transvection on Z with center f . Thus by 2.4.1 in [Asc19], Z ∼ = E4 and H + = H/CH (Z) = GL(Z). Let h ∈ H − CH (f ), A = y E, and L = A, Ah . Then A is abelian with [R, y] ≤ E and H + = y + , y +h = L+ . It follows that L  LR with O2 (L) = EE h and E0 = E ∩ E h = Z(L). Hence if E0 = 1 then 1 = E0 ∩ Z(R) = E1 . Then as E ∩ Z(R) = f , we conclude that f ∈ Z(L), contradicting [f, H + ] = 1. Therefore E0 = 1, so B = O2 (L) = E × E h . We may take h ∈ L of order 3 inverted by y; then CB (h) = 1, so for each e ∈ E # , e, eh ∼ = E4 is h, y -invariant. Set x = f h ; from the previous paragraph, [x, y] = f . Also Π = {fp : p ∈ π} is a basis for E, and x acts on each p ∈ π, so p = σ γ ∪ σ γx , for γ ∈ Γ with σ γ ∈ p. Therefore CO(p) (x) = {llx : l ∈ O(σ γ )} ∼ = O(σ). Then P = CO(θ) (x) is the direct product of 2a−b−1 copies of O(σ) and Z(P ) = E. Therefore as E ∩ E h = 1, also P ∩ P h = 1. Further P ≤ CG (Z) = R, and as R acts on O(θ), P = CO(θ) (x)  R,

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so P, P h = P × P h . Thus [P, h] is nonabelian, whereas [R, L] = EE h is abelian, a contradiction.  Lemma 11.2.11. ZS ⊆ R. Proof. Assume otherwise; by 11.2.10, R is exceptional, so σ = θ(β) ⊆ R. The initial treatment of this case is much like that of the case R ordinary. By construction, |σ| = 2a for some a. Let Pa = NR (σ), σa = σ, and, proceeding recursively, let Pi be a subgroup of Pi+1 of index 2 with an orbit σi of length 2i R on σi+1 . Pick b maximal subject to fσb ∈ / R and set ξ = σb . Then π = σb+1 is a b+1 b+1 partition of ZS into nS /2 blocks of size 2 , and E = fp : p ∈ π ≤ R. Let Γ be a set of coset representatives for Pb+1 in R,  ξ γ , and y = fζ . ζ= γ∈Γ 

If y ∈ R then as Z(R) acts on σb+1 , Z(R) acts on ξ, so fξ ∈ O 2 (CG (Z(R)) = R by 11.2.7.1, contrary to the choice of ξ. Therefore y ∈ / R. Observe that [R, y] ≤ E ≤ R, so y ∈ G. Set Z = f1G . By 11.2.7.1, R =  O 2 (CG (Z)), so as y ∈ / R, y does not centralize Z. Let x ∈ Z − CZ (y). Then x acts on each p ∈ π, so for γ ∈ Γ with ξ γ ∈ p, ξ γx = ξ γ or p − ξ γ . As [x, y] = 1, x moves some ξ γ , and then as R is transitive on π and x ∈ Z(R), x moves every ξ γ . But  now yy x = c, so as c = 1, x centralizes y, a contradiction. Lemma 11.2.12. Z(R) acts on O(s) for each s ∈ ZS . Proof. By 11.2.11, Z(R) centralizes ZS , so the remark follows.



Lemma 11.2.13. Assume |Ω(z)| = 1 for z ∈ ZS . Then either |O(z) ∩ R| > 2 for some z ∈ ZS or the following hold: (1) R is ordinary. (2) R is transitive on ZS .  (3) Assume that τ satisfies the Inductive Hypothesis. Then τ ◦ = z∈ZS ρz , (1) where ρz = (Yz , Ω(z)) with Yz ∼ , G2 [m], or M12 . = L− 3 [m], L2 [2m] ◦  (4) If F is (P )Sp2nS [m], (P )SL2nS [m], or L2nS +1 [m] then nS = 1 and F ◦ is L− 3 [m]. (5) 11.2.6.II does not hold. Proof. Assume for each z ∈ ZS that O(z) ∩ R = z ; hence in particular v(z) ∈ / R. If R is ordinary then z ∈ θ ∈ Θ, and we set f = fθ and Z = f G . If R is exceptional let z  ∈ θ1 and set θ = ZS and Z = f1G . In any event set E = fj : j ∈ θ and y = j∈θ v(j). Assume y ∈ R. Then as R is transitive on θ, either Z centralizes the projection v(s) of y on O(s) for each s ∈ ZS , so v(s) ∈ R by 11.2.7.1, or R is exceptional and for some x ∈ Z, [v(s), x] = s for each s ∈ ZS . In the former case the lemma holds, so assume the latter.  Let σ = θ1 , u = z∈σ v(z), W1 = WS , y , and A = W1 , u . Then [R, u] ≤ / R and indeed [Z(R), u] ≤ f1 . Set W1 ≤ R, so A ≤ G. Also [x, u] = f1 , so u ∈ H = uG and G+ = G/CG (Z(R)). As G∗ has a strongly embedded subgroup we conclude from 2.4.1 in [Asc19] that Z ∼ = E4 and H + = GL(Z). Let h ∈ H of h order 3 and L = A, A . As A  S, [R, u] ≤ R ∩ A = W1 , so O2 (L) = W1 W1h

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and L  LR. Set W2 = W1 ∩ W1h ; as usual, W2 ≤ Z(L), so if WS ∩ W2 = 1 then 1 = WS ∩ Z(R), and then as L+ = H + we obtain a contradiction from 11.2.5.1 and 11.2.7.2. Therefore WS ∩ WSh = 1 and as WS  R, we have WS WSh = WS × WSh abelian and either W2 = 1 or we may take W2 = y and O2 (L) = y × WS × WSh is abelian. Hence O2 (L) centralizes WS and CWS WSh (h) = 1, so L acts on Wz = z, z h ∼ = E4 . Thus [z h , uy] = [z h , u] = z, a contradiction as z h acts on O(s) for each s ∈ ZS , so [z h , uy] ≤ fZS −σ .  Therefore y ∈ / R. By 11.2.7.1, R = O 2 (CG (Z)), so [y, Z] = 1. Let x ∈ Z − CZ (y) and j ∈ θ. By 11.2.12, [v(j), x] ∈ j , and as [y, x] = 1 there exists j ∈ θ with [v(j), x] = j. Then as R is transitive on θ and x ∈ Z(R), [v(k), x] = k for each k ∈ θ. But then if R is exceptional, [y, x] = c = 1, contrary to [y, x] = 1. Therefore R is ordinary, establishing (1). As R is ordinary, [Z(R), y] ≤ E ∩ Z = f by 11.2.4.2, so y induces a transvection on Z(R) with center f , and hence Z ∼ = E4 and H + = H/CH (Z(R)) = GL(Z)  G by 2.4.1 in [Asc19], where H = y . Moreover H += O 2 (G+ ). Suppose θ = ZS and let θ = β ∈ Θ and y0 = t∈β t. By symmetry between  θ and β, fβ = [Z(R), y0 ] and 1 = y0+ ∈ O 2 (G+ ) = H + , so y0+ = y+ . But then fβ = [Z(R), y0+ ] = [Z(R), y + ] = f + a contradiction. Therefore R is transitive on θ = ZS , proving (2). Set A = y, WS and L = A, Ah for h ∈ H of order 3 inverted by y. As usual, appealing to 11.2.4.2 and 11.2.7.2, Z(L) = 1, O2 (L) = WS × WSh , and L  LR. Then h acts on Wz = z, z h with [z h , y] = z. As O2 (L) is abelian, z h acts on O(s) for each s ∈ θ, and then also on v(s) . Then as [z h , y] = z, z h centralizes v(s) for s = z, while z h inverts v(z). Now h acts on each Wt and cv(z) centralizes  F = s =z Ws with Γ = cv(z) , ch inducing GL(Wz ) on Wz . Then Σz = cΓv(z) centralizes F and induces GL(Wz ) on Wz . As R is transitive on ZS , Σs centralizes Wz and acts as GL(Ws ) on Ws . Thus as z ∈ D(s), also z ∈ D(sh ), so sh centralizes O(z) by 3.1.14. Therefore CO(z) (z h ) ≤ CO(z) (f h ) ≤ CO(z) (Z) = O(z) ∩ R = z by 11.2.7.1. Hence z h ∈ D o (z) by 3.3.6. Let θz = ZS − {z} and Ez = CF (θz ). As z ∈ F f and R is transitive on ZS , for each t ∈ ZS we have t and θt ∈ Ff . Then O(θz )  Ez with O(z) ≤ CEz (O(θz )) = C, so Yz = [O(z)]Ez ≤ C. Then as Σz ≤ AutEz (Wz ), we have z h ∈ z C , so as z h ∈ D o (z), it follows from Theorem 5 that (1) , G2 [m], or M12 . (*) Yz ∼ = L− 3 [m], L2 [2m] As R is transitive on ZS and each s ∈ ZS is in F f , Ys = [O(s)]Es ∼ = Yz . Next Ws centralizes Wz O(z), so from the structure of Aut(K(Yz )) we have Ws inducing automorphisms on Yz in z . Then as Σz centralizes Ws , Ws centralizes (1) Yz . But if Yz is L− then Ws O(s) is Sylow 3 [m] or L2 [2m] in Ys , so it follows from 2.3 in [Asc11] that F contains the direct product Y0 = t∈ZS Yt of R-conjugates of Yz . Observe that (1) implies (5). If nS = 1 then (3) and (4), and hence also the lemma, holds by (*), so we may assume that nS > 1. Assume the hypothesis of (4). Then F ◦ = FS (L) for L = (P )Sp2nS (q), π ¯ be the covering group of L and U its dLet L (P )SLπ 2nS (q), or L2nS +1 (q). dimensional defining module, as in section 5.4. Let T = θz and observe that ¯ Now [U, T ] is of dimension 2(nS − 1) and CU (T ) Yz = FSY (Y ) for some Y ≤ L. is of dimension d − 2(nS − 1) = 2 or 3, so as Y is faithful on CU (T ) we conclude

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∼ L− [m]. Next Y0 = FS (Y0 ) with Y0 =  s ∈ ZS Ys and that d = 2NS + 1 and Yz = 0 3 Ys ∼ = L−π 3 (q), so 2NS + 1 = m(U ) ≥ 3nS , contradicting nS > 1. This proves (4). It remains to prove (3), so we may assume the hypotheses of (3). Then by 6.6.8, τz◦ = τ1 ∗ · · · ∗ τk is the central product of packets satisfying Theorem 1. Let s ∈ θz . Then z ∈ θs and ρs = (Ys , O(s)) is coconnected, so ρs ≤ τi = (Ci , Ωi ) for some i, and then Ys = CCi (O(θs )). As τi satisfies Theorem 1, we conclude that either ∼ − τi = ρs or Ys ∼ = L− 3 [m] and Ci = L2k+1 [m]. Let t ∈ θs − {z}; then as O(t) ≤ Ci , Yt , Ys ≤ CCi (O(ZS − {t, s})) ∼ is a contradiction as we’ve shown = L− 5 [m]. This  that Y0 ≤ F. Therefore the direct product ξz = s∈θz ρs is a subpacket of τz◦ , and τz◦ = O(z) × ξz . (1) If Yz is L− we’ve shown that Y0 ≤ F. We will show the same 3 [m] or L2 [2m] holds when Yz is G2 [m] or M12 . Then Hypothesis 8.1.8 is satisfied by (ρs : s ∈ ZS ), so (3) follows from 8.1.10. Hence it remains to assume Yz is G2 [m] or M12 and to produce Y0 . Let Tz be Sylow in Yz . We showed that Xs = Ws O(s) centralizes Yz , so Tz centralizes Xs . Then, arguing as above, Tz centralizes Ys , and then by 2.3 in [Asc11], Y0 exists, completing the proof.  Because of 11.2.13, during the remainder of the section we assume: Hypothesis 11.2.14. (1) |Ω(z)| = 2 for each z ∈ ZS . (2) For each K ∈ Ω, |K ∩ R| ≤ 2. , vs , vs of For s ∈ ZS , let Ω(s) = {Ks , Ks }, Vs = Ks ∩ W , Vs = Ks ∩ W    order 4 in Vs , Vs , respectively, and  v(s) = vs vs . For θ ⊆ ZS , set v(θ) = t∈θ v(t), Wθ = v(t), θ : t ∈ θ , and vθ = t∈θ vt . Lemma 11.2.15. If nS ≤ 2 then R is ordinary and c = 1. Proof. If ZS = {z} then c = z = 1, so R is ordinary. If ZS = {z1 , z2 } is of  z2 , so c = z1 z2 = 1, and again R is ordinary.  order 2 then z1 = Lemma 11.2.16. (1) For each s ∈ ZS , v(s) ∈ R. (2) WI ≤ R. ˜ = Sym(I), or that Proof. By 11.2.2, WI is the permutation module for μ module modulo eI , and the elements v(s), s ∈ ZS , are vectors of weight 1 in WI . Therefore WI = v(s), WS , en : s ∈ ZS . Observe that en ∈ Z(S) ≤ R if n is odd. Therefore (1) and 11.2.11 imply (2). / R. If R is ordinary then s ∈ θ ∈ Θ, and we set Assume s ∈ ZS with v(s) ∈ f = fθ and Z = f G . If R is exceptional set θ = ZS and Z = f1G . In any event  set E = θ and y = t∈θ v(t).  Assume y ∈ / R. By 11.2.7.1, R = O 2 (CG (Z)), so [y, Z] = 1. Let x ∈ Z − CZ (y) and t ∈ θ. By 11.2.12, [v(t), x] ∈ t , and as [y, x] = 1 there exists t ∈ θ with [v(t), x] = t. Then as R is transitive on θ and x ∈ Z(R), [v(t), x] = t for each t ∈ θ. But then if R is exceptional, [y, x] = c = 1, contrary to [y, x] = 1. Therefore R is ordinary. Observe r ∈ R permutes θ and hence also {v(t), v(t)t : t ∈ θ}, so [y, r] ∈ E ≤ R by 11.2.11. Hence y ∈ G. As R is ordinary, [Z(R), y] ≤ E ∩ Z(R) = f by 11.2.4.2,

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so y induces a transvection on Z with center f , and hence Z ∼ = E4 by 2.4.1 in [Asc19]. Next v(t) = vt vt with vt ∈ Vt , vt ∈ Vt of order 4. As [v(t), x] = t we may assume x centralizes vt and inverts vt . Thus vt centralizes Z = f, x , so vt ∈ R, contrary to 11.2.14.2. We’ve shown that y ∈ R. Then either Z(R) centralizes the projection v(s) of y on O(s) for s ∈ θ, so v(s) ∈ R by 11.2.7.1, or R is exceptional and for some x ∈ Z, [v(s), x] = s for all s ∈ ZS . In the former case the lemma holds, so assume the latter. Then v(s) induces a transvection on Z with center s, so by 2.4.1 in [Asc19], we have Z = f, x ∼ = E4 . As [v(s), x] = s we may assume x inverts vs and centralizes vs . But then vs ∈ CS (Z) = R by 11.2.7.1, contrary to 11.2.14.2.  Lemma 11.2.17. If R is ordinary let θ ∈ Θ and if R is exceptional let θ = ZS . / R. Then vθ ∈ G but vθ ∈ Proof. As R permutes θ, [vθ , R] ∈ WI , so vθ ∈ G by 11.2.16.2. Suppose vθ ∈ R and let Z = Z(R). Then Z centralizes vθ and acts on O(t) for each t ∈ θ by 11.2.12. Thus if R is ordinary, Z centralizes the projection vt of vθ on O(t), so vt ∈ R by 11.2.7.1, contrary to 11.2.14.2. Therefore R is exceptional. Let P = O(τ ) and P + = P/WS . Then Z centralizes vθ+ and acts on O(t), so Z centralizes the projection vt+ of vθ+ on O(t)+ . Thus for x ∈ Z, [vt , x] = t(x,t) for some (x, t) ∈ {0, 1}. Then [vθ , x] = fσ(x) , where σ(x) = {t ∈ θ : (x, t) = 1}, so as x centralizes vθ , σ(x) = ∅ or θ. By 11.2.14.2, vt ∈ / R, so there exists x ∈ Z with σ(x) = θ. Then Z0 = {x ∈ Z : σ(x) = ∅} is a hyperplane of Z centralizing vt for each t ∈ θ. Choose θ1 as in 11.2.5.2, let y = vθ1 , E = vθ , WS , and A = y, E . Then [R, y] ≤ E ≤ R, so y ∈ G. Suppose y ∈ R. Then, as in the previous paragraph, Z centralizes vt for t ∈ θ1 , so vt ∈ R by 11.2.7.1, contrary to 11.2.14.2. Therefore y∈ / R. / R and F ≤ Z(R) by 11.2.5.1, y is Set H = y G , f = f1 , and F = f G . As y ∈ nontrivial on F by 11.2.7.1, so there is x ∈ F − CF (y). By 11.2.12 either x acts on each Vs or on no Vs . In the first case [x, y] = f , while in the second [x, y] = v(θ1 ), contrary to 11.2.5.1. Thus y induces a transvection on F and Z(R) with center f , so by 2.4.1 in [Asc19], F ∼ = E4 and AutH (Z(R)) = GL(F ). As in the proof of 11.2.13, H = A, Ah for h ∈ H of order 3 inverted by y, and setting E0 = E ∩ E h , either O2 (H) = E × E h with CO2 (H) (h) = 1, or we may take E0 = vθ and O2 (H) = WS × WSh × E0 . Hence for each z ∈ ZS , Bz = z, z h ∼ = E4is an Hinvariant subspace of O2 (H) and [z h , y] = z. Therefore H acts on B = t∈ZS Bt . As [z h , y] = z and z h centralizes vθ ∈ O2 (H) since O2 (H) is abelian, it follows that z h inverts vz and centralizes vs for s ∈ ZS − {z}. By symmetry, vz centralizes sh , so vz induces a transvection on B with center z and axis z, Bs : s ∈ ZS − {z} . By symmetry, for each z = s ∈ ZS , vzh centralizes Bs , so Σ = vz , vzh centralizes Bs and induces GL(Bz ) on Bz . By symmetry, CF (Bz ) induces GL(Bs ) on Bs . Thus as z ∈ D(s), also z ∈ D(sh ), so sh centralizes O(z) by 3.1.14. Therefore CKz (z h ) ≤ CKz (f h ) ≤ CKz (F ) = Kz ∩ R = z by 11.2.7.1 and 11.2.14.2. Hence z h ∈ D o (z) by 3.3.6. Let θz = ZS − {z}, X = θz , and ρz = (CF (X), Ω(z)). By aa.2, ρz is a quaternion fusion packet, and as Σ centralizes X, z h ∈ Z(ρz ). Then as z h ∈ D o (z), it follows from 9.4.34, 9.4.35, and 9.5.33 that Yz = CF (O(θz ))◦ is Ω− 6 [m], Ω7 [m],

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or Ω− 8 [m]. This is a contradiction since as R is exceptional, c = 1, so we have  z = fθz ∈ X, and hence z is in the center of Yz . Lemma 11.2.18. R is ordinary. Proof. Assume R is exceptional and let θ = ZS and y = vθ . By 11.2.17, y ∈ G − R, so H = y G is not contained in R. Set E = Z(R) ∩ WI . By 11.2.4.1, ˜ ˜ and recall from iI.2 that CWI (R)# = {eσ : σ ∈ π ∈ Θ}. Let π = {σ, σ ¯ } ∈ Θ, ZS = {gi : 1 ≤ i ≤ nS }, where gi = e2i,2i−1 . If gi ∈ σ for some i, then as R ˆ and ¯ , so π corresponds to a member of Θ is transitive on ZS , each gj is in σ or σ eσ ∈ Zβ . Otherwise for each i, σ contains a unique member of {2i, 2i − 1}. We conclude that either E = Zβ or E = Zβ eξ for some ξ ⊆ I such that ξ contains exactly one member of {2i, 2i − 1} for each 1 ≤ i ≤ n/2. Note in the latter case, eξ = v(θ) for suitable choice of vt , vt , t ∈ θ. so we normalize and take v(θ) = eξ ∈ E. Set Z = E G . As [R, y] ≤ WI , [Z, y] ≤ Z ∩ WI ≤ E. By 11.2.7.1, y does not centralize Z, so there is x ∈ Z − CZ (y). As R is transitive on θ and x ∈ Z(R), Ktx = Kt for all t ∈ θ or Ktx = Kt for all t ∈ θ. Suppose the first case holds. As [y, x] = 1, [vt , x] = 1 for some t ∈ θ, and then as R is transitive on θ and x ∈ Z(R), x inverts vt for each t ∈ θ. But then [y, x] = t∈θ t = c = 1, a contradiction. Therefore Ktx = Kt for each t ∈ θ, so it follows that CZ (y) is a hyperplane of Z. Thus y induces a transvection on Z, so by 2.4.1 in [Asc19], Z = F × CZ (H) with F = [Z, H] ∼ = E4 and AutH (Z) = GL(F ). As R is exceptional, nS is a power of 2, and then by 11.2.15, nS ≥ 4. Then by 11.2.7.2, WI ∩ CZ (H) contains no vectors of weight nS . However as Ktx = Kt , [vt , x] = v(t)t(x,t) for some (x, t) ∈ {0, 1}, so [y, x] = v(θ)fσ(x) , where σ(x) = {t ∈ θ : (x, t) = 1}. Thus by our choice of notation in paragraph one, v(θ) ∈ E, so m(E) > 1. Thus CE (H) = 1, so as each  member of E # is of weight nS , we have a contradiction. Lemma 11.2.19. (1) For each θ ∈ Θ, Z(R) ∩ Wθ = fθ or fθ , v(θ) . (2) fθ is of weight 2|θ| in WI and v(θ) and fθ v(θ) are of weight |θ|. In particular if |θ| > 1 then each member of Z(R) ∩ Wθ is in W(R). Proof. Set Wθ+ = Wθ /θ . Then X = {v(t)+ : t ∈ θ} is a basis for Wθ+ . Further for r ∈ R and t ∈ θ, v(t)+r = v(tr )+ , so as X is a basis and R is transitive on θ, CW + (R) = v(θ)+ . Hence by 11.2.4.2, Y = Z(R) ∩ Wθ = fθ or fθ , y , θ where y = v(θ)fσ for some σ ⊆ θ. However, as in the proof of the previous lemma, in the latter case for suitable choice of vt , vt for t ∈ θ, we can choose σ = ∅, and (1) holds. Part (2) is a straightforward observation, given 11.2.2.  For θ ∈ Θ, set Hθ = vθG ≤ G, Aθ = Wθ vθ , and Lθ = Aθ , Ahθ for some h ∈ Hθ of odd order with vθ∗ , vθ∗h not a 2-group. Set Eθ = Z(R) ∩ Wθ and Fθ = EθG . Lemma 11.2.20. Let H = Hθ , F = Fθ , E = Eθ , and H + = H/CH (Z(R)). (1) R = CS (F ) and vθ is nontrivial on F . (2) For x ∈ Z(R) − CS (vθ ), either (a) for all t ∈ θ, [vt , x] = t and [vθ , x] = fθ , or (b) for all t ∈ θ, Ktx = Kt and [vθ , x] = v(θ) or v(θ)fθ . (3) CH (Z(R)) = CH (F ). (4) If E ∼ = E4 and H + = GL(F ). = Z2 then F ∼

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∼ E16 and H + ∼ (5) If |θ| > 1 and E ∼ = E4 then E = [F, vθ ] and either F = = S3 or D10 , or CF (H) = 1 and F/CF (H) is the natural module for H + ∼ = L2 (4). (6) If |θ| = 1 and E ∼ = S3 or F = E4 then either (5) holds or F ∼ = E8 and H + ∼ + ∼ is the natural module for H = A5 . (7) If |θ| > 1 then Lθ  RLθ , O2 (Lθ ) = Wθ × Wθh , |h| = p ∈ {3, 5}, and CO2 (Lθ ) (h) = 1. Proof. By 11.2.19, F contains fθ ∈ W(R), so part (1) follows from 11.2.7.1. Choose x as in (2) and let f = fθ , v = v(θ), and y = vθ . As x ∈ Z(R) and R is transitive on θ, either x acts on each Kt or x moves each Kt . In the first case as [y, x] = 1 and R ≤ CS (x) is transitive on θ, [vt , x] = t for all t ∈ θ, and hence (2a) holds. In the second case [v, x] = vfσ for some σ ⊆ θ, so (2b) holds by 11.2.19. Hence (2) is established. As [Z(R), y] ≤ F and H = y G , we have [Z(R), H] ≤ F , so O 2 (CH (F )) = 2 O (CH (Z(R)). Then (1) implies (3). Suppose E ∼ = Z2 . Then by 11.2.19, E = f and y induces a transvection on F with center E by (2), so (4) follows from 2.4.1 in [Asc19]. So assume E ∼ = E4 . Again by (2) either y induces a transvection on F or E = [F, y]. However in the first case F = [F, H] × CE (H) with CE (H) = 1, so 11.2.19 and 11.2.7.2 supply a contradiction when |θ| > 1. Further if |θ| = 1 then F ∼ = E8 and (6) holds in this case. Therefore we may assume E = [F, y] is of rank 2. Hence by 2.8.15, H + and its action on F are described in that lemma. If m(F ) = 6 or F is the L2 (4)-module for H + ∼ = L2 (4), then NH (E) is transitive on E # , a contradiction as NF (W ) controls fusion in WI while by 11.2.19, f is of weight 2|θ| while v is of weight |θ|. Hence if F/CF (H) is the natural module for H + ∼ = L2 (4) then CF (H) = 1 so (5) holds in this case. Thus we may assume m(F ) = 4 and H+ ∼ = Z2 /E9 , A5 , S3 , or D10 . In the first two cases H + = CH + (v), CH + (vf ) , contrary to 11.2.7.2 and 11.2.19 when |θ| > 1. Thus (5) and (6) hold. Next by (5) and (6), v + , v +h ∼ = D2p for |h+ | = p ∈ {3, 5}. The usual argument based on 11.2.19 and 11.2.7.2 show that L = Lθ = v, v h B  RL, where B = O2 (L) = Wθ × Wθh and CB (h) = 1. As |h+ | = p, k = hp centralizes F , and B = [B, k] × CB (k) with F ≤ CB (k). As L  RL, R = CR (k)[B, k], so if k = 1 then there is 1 = u ∈ C[B,k] (R), so u ∈ CB (R) = F , a contradiction. Therefore k = 1, so |h| = p, completing the proof of (7).  Lemma 11.2.21. R is transitive on ZS , so Θ = {ZS }. Proof. Suppose θi , i = 1, 2, are distinct members of Θ. Let vi = vθi , etc, and G+ = G/CG (Z(R)). As G+ has a strongly embedded subgroup and v1+ ∈ H1+  G+ with v1+ an involution, all involutions in G+ are in H1+ , so v2+ ∈ H1+ . Then H2+ = v2+G ≤ H1+ , so by symmetry, H1+ = H2+ = H + . Suppose first that v1+ = v2+ . Then [Z(R), v1 ] = [Z(R), v1+ ] = [Z(R), v2+ ] = [Z(R), v2 ]. But by 11.2.19, [Z(R), vi ] ≤ Ei , so E1 ∩ E2 = 1. By 11.2.19, Ei ≤ fi , v(i) , so as θ1 = θ2 we conclude that f1 = f2 = f generates E1 ∩ E2 and Θ = {θ1 , θ2 }. Then we conclude [Z(R), vi ] = f , so by 11.2.20, H + ∼ = S3 and F = Fi = f G ∼ = E4 for i = 1, 2. Suppose |θi | = 1 for i = 1, 2. Then as Θ = {θ1 , θ2 }, nS = 2, so c = 1 by 11.2.15, contradicting f1 = f2 . Thus we may assume |θ1 | > 1. Therefore by 11.2.20.7, L1 = B1 v1 , v1h  RL1 where h is of order 3, B1 = W1 × W1h , and CB1 (h) = 1. As v1+ = v2+ , u = v1 v2−1 ∈ R, so as L1  RL1 , [u, L1 ] ≤ B1 . Then

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∼ Z2 and u = u0 w1 for some w1 ∈ W1 . Y = L1 u = u0 × L1 where u0 = Z(Y ) = Now v2 = uv1 ∈ Y , so v1 , v2h contains a 3-element k such that L2 = A2 , Ak2 with + B1 L+ 2 = H . As usual L2  RL2 , so as k ∈ h , B1 = [B1 , k] ≤ L2 and then as v2 centralizes W1 , W1 ≤ CB2 (v2 ) = W2 . This is a contradiction as θ1 = θ2 . Therefore v1+ = v2+ . But if H + ∼ = D2p for p ∈ {3, 5} then v1+ and v2+ + + are Sylow groups of H + , so v1 = v2 , a contradiction. Therefore by 11.2.20, H+ ∼ = E4 is Sylow in H + . = A5 ∼ = L2 (4) and T + = v1+ , v2+ ∼ Suppose E1 ∩ E2 = 1. Then, as above Θ = {θ1 , θ2 } and f1 = f2 with c = 1. Then, again as above, nS = 2, so we may choose |θ1 | > 1, so by 11.2.20, F/CF (H) is the L2 (4)-module. Then as E1 ∩E2 = f1 , |CF (H)| = 2 and CF (v1 ) = CF (H)×F0 where F0 ∼ = E4 is NH + (T + )-invariant. But then as v(1) and v(1)f1 are of weight |θ1 | and f1 is of weight 2|θ1 |, we conclude f1 ∈ F0 , so E1 ∩ E2 = f1 . Therefore E1 ∩ E2 = 1, so it follows that CF (H) ∼ = E4 and F/CF (H) is the L2 (4)-module for H + with E = CF (T ) = E1 × E2 . Thus as Ei ≤ WI also E ≤ WI . N (E) As E2 ∈ E1 H , k = |θ1 | = |θ2 |. Now Ei has two members v(i) and v(i)fi of weight k and one member fi of weight 2k in WI , so the diagonal members of E consist of 4 members of weight 2k, four of weight 3k and one member e of weight / CF (H). However all 4k. As k = 2a , e is of weight 4k = 2a+2 , so by 11.2.7.2, e ∈ members of E − CF (H) are in orbits of length 3 under NL (E), so e is also of weight 2k or 3k, and hence n = 6k or 7k and eI = 1 in WI . Thus there are five diagonal members of weight 4k and four of weight 3k or 2k, so some member of weight 4k is  in CF (H), a contradiction. This completes the proof of the lemma.

During the remainder of the section we assume: Hypothesis 11.2.22. nS > 1. By 11.2.21, θ = ZS is the unique member of Θ. We write H, L, etc. for Hθ , Lθ , etc. and set f = fθ , v = v(θ), and y = vθ . Set B = WS WSh . Lemma 11.2.23. (1) B is the direct sum of the irreducible L-modules Bt = tL for t ∈ θ, of dimension 2, 4 for p = |h| = 3, 5, respectively. (2) If p = 3 then [f h , y] = f and [f h , vt ] = [th , y] = t. h (3) If p = 5 then [f h , y] = vf  and [f h , vt ] = [th , vt ] = v(t)t and Ktf = Kt = h Ktt . Proof. By 11.2.21, |θ| = nS , so by 11.2.22, |θ| > 1. Therefore by 11.2.20.7, B = WS ×WSh with WS = CB (y) and CB (h) = 1. It follows that each w ∈ WS# is in a unique irreducible L-submodule Bw of B, so Bw is of dimension 2, 4 for p = 3, 5, respectively. Thus (1) holds. In particular if p = 3 then Bf ∼ = E4 so [f h , y] = f , and hence [f h , vt ] = [th , y] = t, so (2) holds. So take p = 5. Then we may view Bf as the restriction of the + L orthogonal module for Ω− 4 (2) to L with f the set of singular points of Bf . Hence ⊥ the hyperplane f contains CBt (y) with f L − {f } ⊆ Bf − f ⊥ and [f ⊥ , y] = f . Therefore [f h , y] = f , so (3) follows from 11.2.20.2. 

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 Lemma 11.2.24. (1) Λ = AutF (B) contains a subgroup t∈θ Λt such that Λt centralizes Bs for s ∈ θ − {t} and Λt acts as L|Bt on Bt . (2) Bt centralizes O(s) for s ∈ θ − {t}. (3) If |h| = 3 then Ut = Bt , v(t), v(t)h centralizes O(s) for s ∈ θ − {t}, v(t)h

= Kt , and th ∈ D o (t). (4) If |h| = 5 then th ∈ D m (t). (5) If |h| = 5 then for some b(t) ∈ {v(t), v(t)t}, b(t)h acts on Kt and CKt (b(t)h ) = t . Kt

Proof. Set a = th and observe that (4) follows from 11.2.23.3. As B is abelian, a acts on O(s) for each s ∈ θ. By parts (2) and  (3) of 11.2.23, [a, y] = t, v(t)t for p = |h| = 3, 5, respectively. Therefore as y = s vs and a acts on O(s), we conclude that a centralizes vs for s ∈ θ − {t} and a inverts vt when p = 3. Hence vtL induces L|Bt on Bt and centralizes Bs for s ∈ θ − {t}, so Λt ≤ Λ and then (1) holds. Now Bt = tΛt , so as Λt centralizes Bs and t ∈ D(s), also a ∈ D(s), and hence (2) follows from 3.1.14. Suppose p = 3. Let α ∈ A(a); then z = aα ∈ ZS and as O(s) centralizes a we have O(s)α centralizing O(z). Then as v(t)h ∈ O(z) ≤ CS (O(s)α), we conclude that v(t)h centralizes O(s) = O(sα)α−1 , establishing the first statement in (3). By v(t)h

= Kt . 11.2.20.2, Kt Next if a ∈ / D o (t) then P = CKt (a)  is of order at least 4. But as O(t) centralizes h s for s ∈ θ − {t}, P centralizes r∈θ r h = f h , so P ≤ CS (Z0 ), where Z0 =  f, f h . Note that Z0 = f L while O 2 (CG (Z1 )) = R by 11.2.7.1, where Z1 =  f H . Thus O 2 (CG (Z0 )) = R if H = L. On the other hand if H = L then   AutH (Z(R)) ∼ = L2 (4) by 11.2.20 and O 2 (CG (Z0 )) = O 2 (CG (Z1 )) = R. Thus in 2 any event, O (CG (Z0 )) = R, so if S1 = CS (Z0 ) ≤ R then S2 = NS1 (R) ≤ R, a contradiction. Therefore as P ≤ R and |P | > 2, 11.2.14.2 supplies a contradiction. This completes the proof of (3). Assume p = 5. By (4), for some b ∈ {v, vf }, bh acts on Kt for each t ∈ θ. Then h [b , y] = f , so [b(t)h , y] = t = [b(t)h , vt ], where b(t) is the projection of b on Bt . Set P = CKt (b(t)h ). By (2), Bs centralizes O(t) for s ∈ θ − {t}, so P ≤ CS (bh ). But P  centralizes f and O 2 (CG (f, bh )) = R, so P ≤ R by 11.2.7.1. Therefore P = t by 11.2.14.2, establishing (5) and the lemma.  Lemma 11.2.25. Let t ∈ θ, Σ = θ − {t}, and Y = CF (O(Σ))◦ . (1) t and Σ are fully normalized and ξ = (Y, Ω(t)) is a quaternion fusion packet with μ(ξ) ∼ = E4 , S4 for n = 2nS , 2nS + 1, respectively. (2) v(t) ∈ / Z. (3) |h| = 3. − o ∼ (4) th ∈ DY (t), Y ∼ = L− 4 [m] if n is even, and Y = Ω7 [m] or Ω8 [m] if n is odd. (5) Let Xt = Λt , NO(t) (Ut )Ut Ut O(t) , u1 = v(t), u2 = tv(t), U (i) = tCXt (ui ) for i = 1, 2, and let Yt be the subsystem of Y on Ut O(t) generated by Ut Ot and NO(t) (U (i))NXt (U (i)) for i = 1, 2. Then Yt ∼ = L− 4 [m]. Proof. As R is transitive on ZS by 11.2.21, t and Σ are fully normalized. By construction, Ω(t) is the subset of Ω centralizing Q = O(Σ), so by 2.5.2, ξ ˜ = μ/μ0 ∼ is a quaternion fusion packet. From 11.2.2, M = Sym(I) with WI the permutation module or the permutation module modulo eI . Then μ(ξ)/μ(ξ)0 is

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the symmetric group on n − 2nS + 2 = 2, 3 letters, so μ(ξ) is E4 , S4 , respectively, establishing (1). As v(t) is of weight 1 in WI and t is of weight 2, v(t) ∈ / tNF (W ) and hence (2) follows as NF (W ) controls fusion in W . Set a = th and let Y be Sylow in Y. By parts (1) and (2) of 11.2.24, Bt ≤ CS (Q) and Λt acts as L|Bt on Bt and centralizes Q. Therefore a ∈ tY = Z(ξ). Assume |h| = 5. By 11.2.24.5 for some b ∈ {v(t), v(t)t}, bh acts on Kt and CKt (b) = t . Suppose μ(ξ) ∼ = E4 . Then by 9.4.34, Y is AE5 , P Sp4 [m], or L− 4 [m]. In the first two cases NY (Kt ) = O(t) so b ∈ O(t), contradicting CKt (b) = t . Similarly in the third case each involution in NY (Kt ) − O(t) is in Z(ξ), contrary to (2). ˇ 6 , AE7 , P Ω+ [m], Therefore μ(ξ) ∼ = S4 , so by 9.4.35 and 9.5.33, Y is AE6 , AE 6 + − Ω6 [m], Ω7 [m], Ω8 [m], Sp6 (2), or L3 (2)/E64 . In the last case, AutY (Bt ) is a 5 ˇ 6 , or AE7 then as D10 ∼ group, a contradiction. If Y is AE6 , AE = Λt ≤ AutY (Bt ), we conclude that Bt ≤ O2 (Y). But then CKt (bh ) = t , a contradiction. If Y is P Ω+ 6 [m] then Y has one class of involutions, contrary to (2). If Y is Sp6 (2) then b is of type b1 , so again CKt (bh ) = t . In the remaining cases, Y is Ωk [m] so Y is the 2-fusion system of L0 = Ω(U, q) as in Notation 5.3.2. Then from 5.3.4, b is of type i(π, 2) on U . As bh acts on Kt with CKt (bh ) = t , [U, bh ] ≤ [U, t] = U1 with [U, bh ] of dimension 2 and sign −π. This is a contradiction as [U, bh ] = [U, b]h , and hence is of sign π. This proves (3). o (t). Therefore if μ(ξ) ∼ By (3), |h| = 3, so by 11.2.24.3, a ∈ DY = E4 it follows − ∼ from 9.4.34 that Y = L4 [m]. Similarly if μ(ξ) ∼ = S4 then Y is Ω7 [m] or Ω− 8 [m] by 9.4.35 and 9.5.33, and hence (4) follows from (1). By 11.2.24.3, Ut = [Ut , Λt ] ≤ CS (Q), so Ut O(t) is contained in the Sylow group 2 Y of Y. If n is even then Y ∼ = L− 4 [m], so as |Ut O(t)| = 2m = |Y |, we have Ut O(t) = Y and then even Y = Yt by 9.4.36. Similarly if n is odd then Y is Ω7 [m] v(t)h o = Kt , 9.4.37 says that Yt ∼ or Ω+ = L− 8 [m], so as a ∈ D (t) and Kt 4 [m]. This proves (5).  By 11.2.25.3, |h| = 3. For t ∈ θ, define Ut = t, th , v(t), v(t)h and Yt ≤ Y as in 11.2.25.5. Thus Yt = O(t)Ut is Sylow in Yt and Yt ∼ = L− 4 [m]. Let Xt be a model for NYt (Ut ). In the remainder of the section we assume the following hypothesis: Hypothesis 11.2.26. Either − (1) F ◦ is (P )Ω+ 2n [m], Ω2n+1 [m], or Ω2n+2 [m], or (2) The Inductive Hypothesis is satisfied in τ Lemma 11.2.27. Let z ∈ Z ∩ F f . Then (1) τz◦ = τ1 ∗ τ2 , where τ1 = (O(z), O(z)) and τ2 = (C, Ω2 ) with Ω2 = Ω − Ω(z). (2) If n is even then C ∼ = Ω− 2n−2 [m]. (3) If n is odd then either Y ∼ = Ω7 [m] and C ∼ = Ω2n−3 [m] or Y ∼ = Ω− 8 [m] and − ∼ C = Ω2n−2 [m]. Proof. By 11.2.22, nS > 1, so n ≥ 4. If 11.2.26.2 holds then by 11.1.5.1, part (1) holds with C described in 11.1.5.3. If 11.2.26.1 holds this follows from the discussion in section 5.3.

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Choose t and Y as in 11.2.25, with t = z, and set Ξ = Σ − {z}. Then Y = CC (O(Ξ))◦ . If n is even then Y ∼ = L− 4 [m] by 11.2.25.4, so (2) follows from 11.1.5.3. [m] by 11.2.25.4, so (3) follows from 11.1.5.3.  If n is odd then Y is Ω7 [m] or Ω− 8 Lemma 11.2.28. (1) Ut = CYt (Bt ) and Xt /Ut is the subgroup of O(Ut , q) generated by transvections for some quadratic form q on Ut of sign +1, such that tXt , v(t)Xt ∪ (v(t)t)Xt is the set of singular,  nonsingular points of Ut , respectively. (2) There is a subsystem Y0 = t∈θ Yt of F. (3) n = 4 or 5. o (t) and by 11.2.25.5 Yt ∼ Proof. Let Ut = CYt (Bt ). By 11.2.25.4, th ∈ DY = t ∼ so by 9.4.16, Ut = E16 . But by construction, E16 ∼ = Ut ≤ Ut , so we conclude (1). that Ut = Ut . Then Xt is described in 9.4.16, which establishes  Set U = O2 (L); from 11.2.20.7, U = Wθ × Wθh = s∈θ Us . Let GU be a model for NF (CS (U )) ∩ NF (U ). From 9.4.16, Xt = Lt , NO(t) (Ut ) , where Ut = O2 (Lt ) and Λt = AutLt (Ut ), so that Lt /Ut ∼ = S3 . From 11.2.24, O(t) and Lt centralize Us for s ∈ θ − {t}, so Xt centralizesUs . Thus we may take Xt to be contained in GU , and we obtain a subgroup X = s∈θ Xs of GU . Let Tt = S ∩ Xt ≤ Yt . Then Yt = Ut O(t) = Tt O(t). As Xs centralizes Ut , Xs = FTs (Xs ) ≤ CFt (Ot ) and as Yt = Tt O(t), Xs centralizes Yt . Let u1 = v(t), u2 = tv(t), and U (i) = tCXt (ui ) . By 9.4.34, 9.4.22, 9.4.27, and 9.4.28, Yt = Yt O, NY (U (i)) : i = 1, 2 , where O ∼ = SL2 [m] ∗ SL2 [m] with O(t) Sylow in O. As Xs centralizes Yt it centralizes Yt O. From 9.4.21, NO(t) (U (i))NXt (U (i)) is a model for NYt (U (i)), so Xs centralizes NYt (U (i)) and hence also Yt . Let Σt = θ − {t}; by construction Yt ≤ CF (O(Σ  t )) and we just showed Yt centralizes Ts , so Yt centralizes St = O(Σt ) s =t Ts = s =t Ys . Therefore by 2.2 in [Asc11], F contains a direct  product s∈θ Y s , proving (2). Let X = s =z Ys ; then X = X ◦ is contained in the component C of Fz◦ appearing in 11.2.27. By 11.2.21, nS = 2a for some a, and a ≥ 1 by 11.2.22. Therefore n = 2nS = 2a+1 or 2nS + 1 = 2a+1 + 1. By 11.2.22, nS ≥ 2 so n ≥ 4. Now by 11.2.27, C is isomorphic to Ω− 2n−2 [m] or Ω2n−3 [m]. As C contains a copy of X , and X is the product of nS − 1 = 2a − 1 copies of L− 4 [m], 5.3.11.2 says that n = 4 or 5. completing the proof of (3) and the lemma. 

L− 4 [m],

By 11.2.28, n = 4 or 5, so nS = 2. Therefore ZS = {z, t} is of order 2 with zt = c. From Notation 11.2.2, we may take z = e1,2 , t = e3,4 , and v(z) = e1 . Let I1 be the elements of WI of weight 1; from 11.2.2, I1 = {ei : 1 ≤ i ≤ n}. Let e ∈ WI be of weight 1 with e ∈ F f . Let τe = (Fe , Ωe ) where Ωe = {K ∈ Ω : K ≤ CS (e)}. Lemma 11.2.29. Assume n = 4. Then (1) We may take e = e1 = v(z). (2) μ(τe ) ∼ = S4 . (3) ei : 2 ≤ i ≤ 4 are in Fe◦ . − ◦ ∼ ∼ (4) Ω− 6 [m] = C ≤ Fe = Ω7 [m] or Ω8 [m]. Proof. As n = 4, μ ∼ = Weyl(D4 ), and then from 11.2.2, WI is the permutation ˜ ∼ module for M = S4 . Now |S : CS (ei )| = 4 for each i ∈ I, so all members of I1 are fully centralized, so (1) holds. Further Ωe = {Kt , Kt } and CM˜ (e) ∼ = S3 , so (2) holds. Next e4 ∈ O(t) ≤ Fe◦ and CM (e) is transitive on I1 − {e}, so (3) holds.

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As v(z) ∈ O(z), it follows from 11.2.27 that e = v(z) centralizes C ∼ = Ω− 6 [m], so ◦ ◦ C ≤ Fe . Then as C ≤ Fe , (4) follows from 11.1.7.3 in case 11.2.26.2, and from the discussion in section 5.3 in case 11.2.26.1.  Lemma 11.2.30. Assume n = 5. Them (1) e = e5 . (2) μ(τe ) ∼ = Weyl(D4 ). (3) ei ∈ Fe◦ for 1 ≤ i ≤ 4. − (4) Fe◦ ∼ = Ω+ 8 [m], Ω9 [m], or Ω10 [m]. Proof. The proof is much like that of 11.2.29. As n = 5, μ ∼ = Weyl(D5 ), and ˜ ∼ then by 11.2.2, WI is the permutation module for M S or that module modulo = 5 eI . Let T be Sylow in M ; from the notational conventions of 11.2.2, T fixes e5 and is transitive on I1 − {e5 }, so (1) holds. Next Ωe = Ω and CM˜ (e) ∼ = S4 , so (2) holds. As ei ∈ O(z) or O(t) for i ≤ 4, and as Ω ⊆ Fe◦ , (3) holds. As in the proof of 11.2.29, e1 = v(z) centralizes C and F by 11.2.27.3, C is Ω7 [m] or Ω− and e ∈ F f , Fe◦ contains a copy 8 [m]. As e1 ∈ e of C. Then (4) follows from 11.1.7.3 in case 11.2.26.2, and from section 5.3 in case 11.2.26.1. For example as Fe◦ contains a copy of Ω7 [m] or Ω− 8 [m], only cases (iv) and (v) of 11.1.7.3 can occur.  − Lemma 11.2.31. Fe◦ contains a copy of Ω− 4 [m] × Ω6 [m].

Proof. By 11.2.28.2, F contains the direct product Y0 = Yz × Yt with Yz ∼ = Yt ∼ = Ω− 6 [m] by 11.2.25.5. Now e1 = v(z) is a non-2-central involution in Yz , so X = O 2 (CYz (e1 )) ∼ = Ω− 4 [m]. Then X × Yt ≤ Fe1 , so for α ∈ A(e1 ) with e1 α = e, Fe contains the copy X¯ × Y¯ of X × Yt , where X¯ = X α∗ and Y¯ = Yt α∗ . As Yt = Yt◦ , Y¯ ≤ Fe◦ . As e1 ∈ W , W α = W , so I1 = I1 α, and hence by 11.2.29.3 and 11.2.30.3, I1 − {e} ⊆ Fe◦ . In particular as e2 = e1 z ∈ X and e2 α = e, we conclude that X¯ = [e2 α]X ≤ Fe◦ , completing the proof.  Lemma 11.2.32. n = 4 or 5. Proof. By 11.2.29.4 or 11.2.30.4, Fe◦ ∼ = Ω7 [m], Ω8 [m], Ω9 [m], or Ω− 10 [m]. On − − ◦ the other hand by 11.2.31, Fe contains a copy of Ω4 [m] × Ω6 [m]. This contradicts 5.3.11.1.  We close this section with two theorems that summarize the various results from the section. Theorem 11.2.33. Assume Hypotheses 11.2.1 and R ∈ F e satisfies 11.2.7. In addition assume for each z ∈ ZS that (a) |Ω(z)| = 1. (b) O(z) ∩ R ≤ z . (c) One of the following holds: (1) F ◦ is (P )Sp2nS [m], (P )SL2nS [m], or L2nS +1 [m]. (2) τ satisfies the Inductive Hypothesis. (3) nS = 4 and m(WS ) = 3.  Then R is transitive on ZS and τ ◦ = z∈ZS ρz , where ρz = (Yz , Ω(z)) and Yz ∼ = (1) [m], L [2m] , G [m], or M . Further (c3) does not hold and if (c1) holds L− 2 2 12 3 then nS = 1 and F ◦ ∼ = L− 3 [m].

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Proof. This follows from 11.2.13.

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Theorem 11.2.34. Assume Hypothesis 11.2.1 and R ∈ F e satisfies 11.2.7. In addition assume for each z ∈ ZS that (a) |Ω(z)| = 2. (b) For each K ∈ Ω, |K ∩ R| ≤ 2. (c) Hypothesis 11.2.26 holds. Then |ZS | = 1. Proof. By (a) and (b), Hypothesis 11.2.14 holds. We may assume that |ZS | > 1, so Hypothesis 11.2.22 holds. By (c), Hypothesis 11.2.26 holds. Therefore by 11.2.28.3, n = 4 or 5. On the other hand n = 4 or 5 by 11.2.32, completing the proof. 

11.3. More generation In this section we assume: Hypothesis 11.3.1. (1) Hypothesis 11.2.1 holds; adopt Notation 11.2.2 and 11.2.3. (2) If |Ω(z)| = 1 for z ∈ ZS , nS = 4, and m(WS ) = 3 then μ is abelian or μ = Weyl(Φ) where Φ is of type An , Dn , or En for some n. Definition 11.3.2. Let W(τ ) consist of those involutions w such that one of the following holds: (1) |Ω(z)| = 2 for z ∈ ZS and w = eθ for some θ ⊆ I such that |θ| is a positive power of 2. (2) |Ω(z)| = 1 for z ∈ ZS and either (a) w = fθ for some θ ⊆ ZS such that |θ| is a power of 2, or (b) nS = 4, m(WS ) = 3, and w ∈ W∗ (τ ), where W1 (τ ) consists of the involutions w = v1 v2 v3 v4 such that vi generates Ω1 (Ki ∩ W ) and Ω = {K1 , . . . , K4 }, and W∗ (τ ) = W1 (τ ) unless μ is Weyl(A7 ), where W∗ (τ ) consists of the M -conjugates of members of W1 (τ ). Recall the definition of E(τ ) from 3.4.2 and define B = B(τ ) = E(τ ), Fx : x ∈ W(τ ) ∩ F f . Lemma 11.3.3. Assume the setup of 11.3.2.2.b and let w ∈ W∗ (τ ). Let W ∈ W (τ ) and M ∈ M (τ ). (1) If μ is abelian then there is α ∈ A(w) such that α is a NF (O(τ ))-map and wα ∈ W(τ ). (2) If μ is Weyl(Φ) with Φ = An for some n, then n = 7 and w ∈ xM , where ¯ of order 4, and x0 = fθ for x = c0 x0 ∈ Z(S) for some c0 ∈ Z(M ) with lift c¯0 in M some θ ⊆ ZS of order 2. (3) If μ is Weyl(Φ) with Φ = Dn for some n, then n = 4 or 5 and wM ∩ ZS ∩ f F = ∅. (4) If μ is Weyl(En ) for some n, then n = 6, w ∈ WΔ , and wM ∩ WS ∩ F f = ∅.

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Proof. As we are in the setup of 11.3.2.2.b, we have Ω(zi ) = {Ki } of order 1 and ZS = {z1 , . . . , z4 }. Assume first that μ is abelian. Then μ ∼ = E16 and η = {Vi : 1 ≤ i ≤ 4} where Vi = Ki ∩ W and vi is of order 4 in Vi .Set v = v1 · · · v4 and α ∈ A(v). If m > 8 then ηα = η, so ZS α = ZS and vα = i vi α ∈ v O(τ ) . Therefore v ∈ F f , so (1) holds in this case. So take m = 8. Still Vi α ≤ Kj(i) , so ηα ∈ η(τ ) and ZS α = ZS , so α is a NF (WS )-map and then α is induced in NF (O(τ )). This completes the proof of (1). Thus we may assume μ is Weyl(Φ) for Φ of type An , Dn , or En . Then by 4.3.8 ¯ → M where G ¯ is the universal group of we have a surjective homomorphism π : G ¯ 4.3.1 and ker(π) ≤ Z(G). Suppose Φ is Dn ; then as nS = 4 we have n = 4 or 5. As |Ω(z)| = 1, we have ker(π) = 1. Therefore π is an isomorphism. Now v = v1 · · · v4 ∈ z M by 5.8.7.6, so (3) holds. Suppose Φ is An . Again as nS = 4, n = 7 or 8. As m(WS ) = 3, we have z¯1 · · · z¯4 ∈ ker(π), so n = 7 and v is a projective involution. Working in SL8 (q), we find v¯ has eigenvalues λ and λ−1 of order 4, each of multiplicity 4, so for some ¯ of order 4, x c0 ∈ Z(M ) with lift c¯0 in M ¯ = v¯c¯0 has eigenvalues 1 and −1, each of multiplicity 4. Thus there is g ∈ M with xg = fθ with θ ⊆ ZS of order 2, and hence fθ ∈ Z(S), so (2) holds. Finally suppose Φ is En . Again as nS = 4 we have n = 6. Hence π is an isomorphism and v ∈ WΔ . Then as all involutions in WΔ are M -conjugate to some  member of WS in F f , (4) holds. Lemma 11.3.4. For each x ∈ W(τ ), Fx ≤ B. Proof. Suppose first that if 11.3.2.2.b holds with x ∈ W∗ (τ ), then μ is nonabelian. Let Q = W if |Ω(z)| = 2 or 11.3.1.2 holds with μ nonabelian, and let Q = O(τ ) otherwise. By construction of E(τ ) in 3.4.2, NF (Q) ≤ E(τ ) ≤ B. Observe also that Q is weakly closed in S with respect to S and x ∈ W(τ ) ⊆ Z(Q). Let α ∈ A(x). Then Qα = Q by the weak closure of Q, so xα ∈ Z(Q) and indeed xα ∈ W(x). Therefore Fxα ≤ B by 11.3.2. Hence for P ≤ CS (x), homFx (P, S) = =

(homFxα (P α, S))α−∗ ≤ homB (P α, S)α−∗ homBα−∗ (P, S) = homB (P, S),

so indeed Fx ≤ B. So assume instead that 11.3.2.2.b holds with x ∈ W∗ (τ ) and μ abelian. Then by 11.3.3.1 there is α ∈ A(x) with α a NF (O(τ ))-map and xα ∈ W(τ ). Now complete the argument as in the previous paragraph.  Notation 11.3.5. Let R consist of those R ∈ F e such that AutF (R) = AutB (R), and write R∗ for the members of R of maximal order. For R ∈ R, set W(R) = W(τ ) ∩ Z(R) and let G be a model for NF (R) and G∗ = G/R. Lemma 11.3.6. For each R ∈ R and K ∈ Ω, |K ∩ R| ≤ 2. Proof. As E(τ ) ≤ B the lemma follows from 3.4.5. ∗



Lemma 11.3.7. Assume R ∈ R .  (1) If x ∈ W(R) and Z = xG then O 2 (CG (Z)) = R. ∗ ∗ (2) Let U be the set of U ≤ G such that S ∗ ∩ U ∗ is Sylow in U ∗ , U ∗ is not contained in the strongly embedded subgroup of G∗ over NS (R)∗ , U ∗ = u∗U for

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some u ∈ S ∩ G such that u∗ is an involution, and U ≤ CG (w) for some w ∈ W(R). Then if U = ∅ then m2 (G∗ ) = 1 and for ∅ = U0 ⊆ U, U0 is not normal in G∗ . Proof. As x ∈ W(R), x ∈ Z(R), so Z = xG ≤ Z(R), and hence R ≤ O (CG (Z)).  Suppose H = O 2 (CG (Z)) = R and let T = S ∩ H. By 3.4.4.2, AutF (R) = AutH (R)AutNG (T ) (R). Further AutH (R) ≤ AutFx (R) ≤ AutB (R) by 11.3.4, so as AutF (R) = AutB (R), it follows that AutNG (T ) (R) ≤ AutB (R). But by the Alperin-Goldschmidt Fusion Theorem, there exists R ∈ F e containing a conjugate of T such that AutF (R ) = AutB (R ). As R < T , this contradicts the maximality of |R|, so (1) holds.  Let L = O 2 (G) and P ∈ Syl2 (L); then L∗ has a strongly embedded subgroup ∗ ∗ ∗ Y and U ≤ L for each U ∗ ∈ U. By 11.3.4, AutU (R) ≤ AutB (R) for each U ∗ ∈ U. By maximality of |R| and an argument in the proof of (1), AutNG (Q) (R) ≤ AutB (R) for each R < Q ≤ NS (R). Suppose that m2 (G∗ ) > 1 and U ∗ ∈ U. Then L∗ /O(L∗ ) is a Bender group and as U ∗ ≤ Y ∗ , we have L∗ = P ∗ , U ∗ , so AutL (R) = AutU (R), AutP (R) ≤ AutB (R) . Then as G = LNG (P ), we obtain a contradiction by arguing as in the proof of (1). Similarly for ∅ = U0 ⊆ U, U0 is not normal in G∗ , so (2) holds.  2

Hypothesis 11.3.8. Hypothesis 11.3.1 holds and for each z ∈ ZS , |Ω(z)| = 1. Moreover one of the following holds: (1) τ satisfies the Inductive Hypothesis. (2) F ◦ is a central image of Sp2nS [m], SL2nS [m], or L2nS +1 [m]. (3) (nS , m(WS ) = (4, 3). Hypothesis 11.3.9. Hypothesis 11.2.1 holds and for each z ∈ ZS , |Ω(z)| = 2. Moreover either (1) τ satisfies the Inductive Hypothesis, or − (2) F ◦ is (P )Ω+ 2n [m], Ω2n+1 [m], or Ω2n+2 [m]. We prove the next two theorems together. Theorem 11.3.10. Assume Hypothesis 11.3.8. Then either (1) F = B(τ ), or  (2) S is transitive on ZS and τ ◦ = z∈ZS ρz , where ρz = (Yz , Ω(z)) and (1) , G2 [m], or M12 . Moreover (nS , m(W )) = (4, 3) and if Yz ∼ = L− 3 [m], L2 [2m] 11.3.8.2 holds then F ◦ ∼ = L− 3 [m]. Theorem 11.3.11. Assume Hypothesis 11.3.9. Then either (1) F = B(τ ), or (2) |ZS | = 1. Assume F is a counter example to Theorem 11.3.10 or Theorem 11.3.11. Then F = B, so R = ∅ by the Alperin-Goldschmidt Fusion Theorem. Pick R ∈ R∗ . By 11.3.7, the pair F, R satisfies Hypothesis 11.2.7. We appeal to Theorems 11.2.33 and 11.2.34, so to complete the proof it remains to verify the hypotheses of those theorems. By 11.3.6, condition (b) of each of the theorems is satisfied. Hypothesis 11.3.8 says conditions (a) and (c) of Theorem 11.2.33 are satisfied, while Hypothesis 11.3.9

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says conditions (a) and (c) of Theorem 11.2.34 are satisfied. This completes the proof of Theorems 11.3.10 and 11.3.11.

11.4. Essentials and normal subsystems In this section we assume the following hypothesis: Hypothesis 11.4.1. F is a saturated fusion system on a finite 2-group S and F0  F with nontrivial Sylow group S0 . Notation 11.4.2. For X ≤ S, set X0 = X ∩ S0 . Let R ∈ F f rc and G a model for X = NF (R). Set X0 = NF0 (R0 ) and G∗ = G/R. Conjugating in F, we may assume R0 ∈ F f . Lemma 11.4.3. R0 = 1. Proof. As S0 = 1, also 1 = Z(S) ∩ S0 . Moreover as R ∈ F c , CS (R) ≤ R.



Lemma 11.4.4. (1) R0 ∈ F0f rc . (2) Set V = R0 CS (R0 ). Then V0 = R0 and V ∈ F f c , so NF (V ) has a model GV .

(3) There is a model G0 for X0 normal in GV . (4) [G0 , CS (R0 )] ≤ Z(R0 ). (5) Set Rc = R0 CR (R0 ). Then, conjugating in F, we may take Rc ∈ F f , so Rc ∈ F f c and hence NF (Rc ) has a model Gc . (6) We may take G0  Gc and G = NGc (R) ≤ Gc . (7) O2 (Gc ) ≤ R. (8) NG0 (R)  G. (9) R0 = O2 (NG0 (R)), so NG0 (R)∗ ∼ = NG0 (R)/R0 and O2 (NG0 (R)∗ ) = 1. Proof. As R ∈ F f rc we have O2 (G) = O2 (X ) = R, and hence R0  X , so X acts on O2 (NF (R0 )) = Q. Then NQ (R) ≤ O2 (G) = R, and hence Q ≤ R. Therefore O2 (NF0 (R0 )) ≤ Q0 ≤ R0 , so R0 ∈ F0r . Set X0 = CS0 (R0 ) and Y0 = NX0 (R). Then [Y0 , R] ≤ R0  G, so Y0 ≤ O2 (G) = R and hence Y0 = Z(R0 ). Then X0 = Z(R0 ), so as R0 ∈ F f , also R0 ∈ F0c , completing the proof of (1). Next V0 = S0 ∩ R0 CS (R0 ) = R0 (S0 ∩ CS (R0 )) = R0 CS0 (R0 ) = R0 by (1). As R0 ∈ F f and R0 = V0 , also V ∈ F f , while by construction CS (V ) ≤ CS (R0 ) ≤ V , so V ∈ F f c . Thus (2) holds. As V0 = R0 , from Notation 4.1 in [Asc08a], NF (V ) is the system D(R0 ) defined there, and E(R0 ) = NF0 (R0 ). Then by 6.10 in [Asc08a], NF0 (R0 )  NF (V ), and hence by Theorem 2 in [Asc08a], there is a model G0 for NF0 (R0 ) normal in GV , establishing (3). By (3), G0  GV , so [G0 , CS (R0 )] ≤ CG0 (R0 ) = Z(R0 ) by (1), proving (4). As R0 = V0 also Rc ∩ S0 = R0 . Then as R0 and R are in F f , we may take Rc ∈ F f . Set S1 = CS (Rc ) and S2 = NS1 (R). Observe that S1 ≤ CS (R0 ), so [S2 , R] ≤ CR (R0 ) ≤ Rc . Then as R ∈ F f rc , we have S2 ≤ R, so S2 ≤ CR (Rc ) ≤ Rc ≤ R, and hence S2 = S1 . Therefore S1 = S2 ≤ Rc , so Rc ∈ F f c , proving (5).

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By (4), G0 acts on Rc , so we may take G0 ≤ Gc and then G0  Gc by Theorem 1 in [Asc08a]. As R0  G, also Rc = R0 CR (R0 )  G, so we may take G = NGc (R), completing the proof of (6). As R ∈ F f rc , R = O2 (G), so as G = NGc (R) by (6), we conclude that (7) holds. Similarly (8) follows from (6). As G0 is a model for NF0 (R0 ), we have NS0 (R0 ) Sylow in G0 . Therefore R ∩ NG0 (R0 ) ≤ R ∩ S0 = R0 . Hence R0 = O2 (NG0 (R)) by (8), so (9) holds.  Lemma 11.4.5. Assume R ∈ F e and R0 = S0 . Let H = NG0 (R), K ∗ the subgroup of G∗ generated by the involutions in G∗ , and M ∗ a strongly embedded subgroup of G∗ containing (S ∩ G)∗ . (1) K ∗ is the subgroup of G∗ generated by the involutions in H ∗ and K ∗ is transitive on such involutions. (2) M ∗ ∩ H ∗ and M ∗ ∩ K ∗ are strongly embedded in H ∗ and K ∗ , respectively.   (3) Either O 2 (G∗ ) = O 2 (H ∗ ) or m2 (G∗ ) = 1. Proof. By 11.4.4.8, H ∗  G∗ . Set S1 = NS0 (R0 ) and S2 = NS1 (R). As R0 = S0 , also R0 = S1 , so R0 = S2 . Hence as S2 is Sylow in H, H ∗ contains an involution. But as R ∈ F e , G∗ has a strongly embedded subgroup M ∗ . Therefore G∗ is transitive on its involutions, so as H ∗  G∗ , we have K ∗  H ∗ . As K ∗  G∗ is of even order, M ∗ ∩ K ∗ is strongly embedded in K ∗ , so K ∗ is transitive on its involutions, and hence (1) and (2) hold.  As M ∗ is strongly embedded in G∗ either O 2 (G∗ ) = K ∗ or m2 (G∗ ) = 1, so (3) follows from (1).  Hypothesis 11.4.6. F0 = F1 ∗ · · · ∗ Fk is a central product of subsystems permuted by AutF (S0 ). Let Si be Sylow in Fi and for X ≤ S, set Xi = X ∩ Si . Set I = {1, . . . , k}. Lemma 11.4.7. Assume Hypothesis 11.4.6 and define G0 as in 11.4.4.3. Then (1) R0 = R1 · · · Rk is a central product. (2) X0 = NF1 (R1 ) ∗ · · · ∗ NFk (Rk ). (3) For i ∈ I, Ri ∈ Fif rc . (4) For i ∈ I there is a model Gi for NFi (Ri ) such that G0 = G1 ∗ · · · ∗ Gk is a central product of the Gi for i ∈ I and Gc permutes the Gi via conjugation. + Proof. For i ∈ I, set Zi = Z(Fi ) and set Z = Z 1 · · · Zk and F0 = F0 /Z. As F0 is a central product, Fi ∩ Fi ≤ Zi , where Fi = j =i Fj . Therefore F0+ = F1+ × · · · × Fk+ is a direct product. Further Z ≤ O2 (F0 ) ≤ R0 by 11.4.4.1. + + + + Let  Pi be the projection of R0 on Si and Pi the preimage of Pi in Si . Set P = i Pi . From the definition of the direct product in Chapter 2 of [Asc11], we have NF + (R0+ ) ≤ NF + (P + ), so by 11.4.4.1, P + ≤ R0+ and hence P ≤ R0 . Hence 0 0 Pi = Ri and then (1) holds. Next [NFi (Ri ), R0 ] = [NFi (Ri ), Ri Ri ] = [NFi (Ri ), Ri ] ≤ Ri , so NFi (Ri ) ≤ X0 . Hence Y0 = i NFi (Ri ) ≤ X0 . As G0 acts on Ri , also X0 ≤ Y0 , so (2) holds. Now NSi (R0 ) = NSi (Ri ), so as R0 ∈ F0f , also Ri ∈ Fif . Further CSi (Ri ) = CSi (R0 ), so as R0 ∈ F0c by 11.4.4.1, we have Ri ∈ Fic . Finally O2 (NFi (Ri )) ≤ O2 (NFi (R0 )) ≤ O2 (NF0 (R0 )) = R0 by 11.4.4.1, so Ri ∈ Fif rc , completing the proof of (3).

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By (1), R0 = Ri CR0 (Ri ). Applying 11.4.4.3 to the pair R0 , Ri in the role of Gi for NFi (Ri ) with Gi  G0 . As Ri cenR, R0 , we conclude there is a model  tralizes Fi , Ri centralizes Gi = j =i Gj . Now [Gi , Gi ] ≤ Gi ∩ Gi ≤ CG0 (Ri Ri ) = CG0 (R0 ) = Z(R0 ), so [Gi , Gi ] = 1. Therefore the central product X of the subgroups Gi is a normal subgroup of G0 . Set T0 = NS0 (R0 ) and let Ti be Sylow in Gi . Thus T0 = T1 ∗ · · · ∗ Tk is Sylow in G0 . As T0 ≤ X, it follows from a Frattini argument that G0 = XY where Y is a Hall 2 -subgroup of NG0 (T0 ). For y ∈ Y , cy ∈ AutX0 (R0 ), so by (2), cy = α1 · · · αk where αi = βi × 1 centralizes Ri and βi ∈ AutFi (R0 ). Now αi = cyi for some yi ∈ CG0 (Ri ) of odd order. Then yi ∈ O 2 (CG0 (Ri )) ≤ Gi as O 2 (CX0 (Ri )) ≤ NFi (Ri ) by (2). Therefore y ∈ X, so G0 = X. This also shows that O 2 (CG0 (Ri )) = O 2 (Gi ). By 11.4.4.6, G0  Gc , so by a Frattini argument, Gc = G0 NGc (T0 ). Thus to show that Gc permutes the Gi and complete the proof of (4), it suffices to show that each g ∈ NGc (T0 ) permutes the Gi . As F0  F, α = cg = ϕφ with ϕ ∈ AutF (S0 ) and φ ∈ homF0 (T0 ϕ, S0 ). By 11.4.6, ϕ permutes the Fi , so Si ϕ = Siσ for some σ ∈ Sym(I). Thus Ti α = Ti ϕφ ≤ (Siσ ∩ T0 ϕ)φ ≤ Siσ , so Ti α ≤ Siσ ∩ T0 = Tiσ and hence Ti α = Tiσ . Then also Rig = Ri α = Riσ . Hence as O 2 (CG0 (Ri )) = O 2 (Gi ), we conclude that    O 2 (Gi )g = O 2 (Giσ ). Also O 2 (Gi ) = RiG0 , so O 2 (Gi )g = O 2 (Giσ ), and hence g  permutes the Gi , completing the proof of (4) and the lemma. Lemma 11.4.8. Assume Hypothesis 11.4.6 with R ∈ F e and R0 = S0 . Let H = NG0 (R), I+ = {i ∈ I : Ri = Si }, and I− = I − I+ . Then (1) R is represented on I via Rir = Rir for i ∈ I and r ∈ R, and R is transitive on I− . (2) Let 1 ∈ I− and Q = NR (1). Set G!c = Gc /R0 and E = QF1 . Then CG!1 (Q! ) = NG1 (Q)! . (3) Let Σ be a  set of coset representatives for Q in R and define α : NG1 (Q)! → ! ! CG!0 (R ) by g α = σ∈Σ g !σ . Then α is an isomorphism and H ∗ = CG!0 (R! ). (4) Q ∈ E e . Let GQ be a model for NE (Q). (5) GQ /Q ∼ = H ∗. Proof. The first statement in (1) is trivial. By construction,  R acts on I− . Suppose I− = I1 ∪ I2 is a nontrivial R-partition and let Xj = i∈Ij Gi and Hj = NXj (R). Then Hj  H and, as in the proof of 11.4.5, Hj∗ contains an involution a∗j . As Hj∗ , j = 1, 2, are distinct normal subgroups of H ∗ , a∗1 and a∗2 are not conjugate in H ∗ , contrary to 11.4.5.1. This completes the proof of (1). As T1 = NS1 (R0 ) is Sylow in G1 and R0 = R ∩ S0 , we have Q ∩ T1 R0 = R0 , so Q! ∩ T1! = 1. Therefore CG!1 (Q! ) = NG!1 (Q! ) = NG1 R0 (Q)! = NG1 (Q)! , since R0 ≤ Q. This proves (2).  By (1) and construction, the map β : g ! → ( σ∈Σ g !σ is an isomorphism of CG!1 (Q! ) with CG!0 (R! ). Then by an argument in the proof of (2), CG!0 (R! ) = NG0 (R)! = H ∗ . Thus (3) holds. Observe that by 11.4.5.2, H ∗ has a strongly embedded subgroup, so by (3), NG1 (Q)! has a strongly embedded subgroup. In particular NS1 (Q) is Sylow in NG1 (Q), so Q is fully normalized in QNF1 (R0 ). Next R0 = Q0 , so NS0 (Q) ≤ NS0 (R0 ). Then as R0 ∈ F f and Q is fully normalized in QNF1 (R0 ), it follows that Q ∈ E f . Next CQS1 (Q) ≤ CQS1 (R1 ) ≤

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295

QCS1 (R1 ) ≤ QR1 ≤ Q, so Q ∈ E f c . Thus NE (Q) has a model GQ , and GQ = NQG1 (Q) = QNG1 (Q). Then GQ /Q ∼ = NG1 (Q)! ∼ = H ∗ has a strongly embedded e subgroup, so Q ∈ E and (4) holds. Moreover we have also proved (5). 

11.5. Generating Ωd [m] In this section we assume the following hypothesis: Hypothesis 11.5.1. τ = (F, Ω) is a quaternion fusion packet such that F ◦ = − ◦ O (F) is isomorphic to (P )Ω+ 2n [m], Ω2n+1 [m], or Ω2n+2 [m] with n ≥ 2 and F is + not (P )Ω4 [m]. Let S be Sylow in F. 2

Notation 11.5.2. As in section 5.3, we write F ◦ = FSL (L) where L = Ω(U, q) is a group of isometries of an orthogonal space (U, q) over a finite field F of odd order. (We are abusing notation in the case F ◦ = P Ω+ 2n [m], where L should be the ◦  projective group P Ω+ (F ).) We write F = (P )Ω 2n+a [m], where 0 ≤ a ≤ 2 and 2n = sgn(U, q) if a is even. Given a nondegenerate subspace A of U , set LA = CL (A⊥ ). Thus LA acts faithfully as Ω(A, q) on A. From 5.3.4, τ ◦ = (FSL (L), Ω) is a Lie packet of L. From that lemma, for K ∈ Ω, [U, z(K)] is nondegenerate of dimension 4 and sign +1, and there is a to SL2 (F ); fundamental subgroup L(K) of Lsubnormal in CL (z(K)) isomorphic  set O(K) = FK (L(K)), Oz(K) = K∈Ω(z(K)) O(K), and Oτ = K∈Ω O(K). If a = 0 set Λ = ∅. So assume for the moment that a = 1 or 2. Suppose first that n is even, and let Λ = {1} or {1, 2} for a = 1 or 2, respectively. Now U = U1 ⊥ · · · ⊥Un/2 ⊥U0 is an orthogonal direct sum decomposition, where Ui = [U, si ] is 4-dimensional of sign +1, ZS = {s1 , . . . , sn/2 }, and dim(U0 ) = a with sgn(U0 ) = −1 if a = 2. In this case set A = Un/2 + U0 , so A is of dimension 4 + a and sign −1 if a = 2, so LA ∼ = Ω5 (F ) or Ω− 6 (F ). Set z = sn/2 and let SA = S ∩ LA m be Sylow in LA . For λ ∈ Λ, let tλ ∈ DL (z), Bλ = z, tλ , v , and Rλ = CS (Bλ ). A Thus tλ ∈ z LA interchanges the two members K1 and K2 of Ω(z), and t1 and t2 are not conjugate in CLA (z) if a = 2. As in 9.1.6, v = v1 v2 where vi is of order 4 in Ki . Choose notation so that z ∈ F f ; this insures that Bλ and Rλ are in F f . If o (z) and assume S induces automorphisms in O(U, q) on L. This a = 2 let s ∈ DL A insures that, setting Us = CSA (z, s ), we have NS (Us ) = NSA (Us )CS (Us ). Then by 9.4.22, 9.4.27, and 9.4.28, we may choose the tλ so that FNS (LA ) (NS (LA )LA ) = NS (LA )Oz , AutLA (Rλ ) : λ ∈ Λ . Note that SA plays the role of T in these lemmas and |SA : O(z)| = 2a , with SA Oz the 2-fusion system of CLA (z). Suppose next that n is odd. Here U = U1 ⊥ · · · ⊥U(n−1)/2 ⊥U0 with dim(U0 ) = a + 2 and sgn(U0 ) = −1 if a = 2. This time pick A = U0 . Then LA = Ω3 (F ) or Ω− 4 (F ). Take Λ = {1} and Bλ a 4-subgroup of LA . Set Rλ = CS (Bλ ) and observe Bλ and Rλ are in F f . Again if a = 2 assume that S induces automorphisms in O(U, q) on L, which insures that if we pick A0 of dimension 5 or 6, s, and Us as earlier, then NS (Us ) = NSA0 (Us )CS (Us ). Then, setting B = U(n−1)/2 + U0 , it follows from 9.5.9 that FNS (LB ) (NS (LB )LB ) = NS (LB )O, FNS (LB ) (NS (LB )M (LB )), AutLA (Rλ ) : λ ∈ Λ ,

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where we apply 9.5.9 to τB = (FB , Ω(z)), where SB = NS (LB ) and FB = FSB (SB LB ), observing that NFB (WB ) = FSB (SB M (LB )), and when a = 2 that the term Fb is 9.5.9 is contained in E(τB ), AutLA (Rλ ) . In this section we prove: Theorem 11.5.3. Assume Hypothesis 11.5.1 and adopt Notation 11.5.2. If F◦ ∼ = Ω− 2n+2 [m] assume S induces automorphisms in O(U, q) on L. Then F = SOs , FS (SM ), AutLA (Rλ ) : λ ∈ Λ S . Assume the theorem is false and pick a counter example with d = dim(U ) minimal. As in Notation 11.5.2, F ◦ ∼ = (P )Ω2n+a [m]. Set Y = SOτ , FS (SM ), AutF (Rλ ) : λ ∈ Λ S . Lemma 11.5.4. (1) n > 3. (2) F ◦ is not (P )Ω+ 8 [m]. Proof. By Hypothesis 11.5.1, n ≥ 2 and n > 2 if a = 0. Assume first that n = 2. Then F ◦ = Ω5 [m] or Ω− 6 [m] for a = 1, 2, respectively. Therefore the lemma holds from the discussion in 11.5.2, contrary to the choice of τ as a counter example. − Assume next that n = 3. Then F ◦ is P Ω+ 6 [m], Ω7 [m], or Ω8 [m]. In the latter two cases the lemma holds from the discussion in 11.5.2; in the first case it holds by 9.5.9. Finally F ◦ is not (P )Ω+ 8 [m] by 10.2.14.3 and 2.1.15; the first reference handles [m] case, and then the second lifts the result to Ω+ the P Ω+ 8 8 [m]. This completes the proof of the lemma.   Lemma 11.5.5. Let θ ⊆ ZS and f = fθ = t∈θ t with 1 = f ∈ F f . Then either (1) Ff ≤ Y, or (2) |θ| = n/4 and L = P Ω+ 2n (F ). Proof. Assume (2) does not hold and let |θ| = k and U1 = [U, f ] and U2 = CU (f ). Thus d1 = dim(U1 ) = 4k and U2 = U1⊥ is of dimension d2 = 2n + a − 4k = 2(n − 2k) + a. Let Li = LUi have Sylow group Ti and Fi = FTi (Li ). Let T be Sylow in Ff . Claim CL (f ) acts on Ui for i = 1, 2. If not d1 = d2 , so 2n + a = 8k ≡ 0 mod 8. Thus if n is even then a = 0 and k = n/4. Further f = (−I) · f , so L = P Ω+ 2n (F ), contrary to the assumption that (2) does not hold. Similarly if n is odd then a = 2. But now U1 is of sign +1 while I2 is of sign −1, contradicting U1 conjugate to U2 . This completes the proof of the claim. Now Ff = T F1 F2 , and by the claim, Fi  Ff for i = 1, 2. By minimality of d, T F1 = T Oτ1 , FT (T M1 ) T , and if d2 > 4 then ¯ λ ) : λ ∈ Λ , T F2 = T Oτ , FT (T M2 ), AutL ¯ (R 2

A

Bλg

−1

¯ ¯λ , R ¯ λ = CT (B ¯λ ), and R ¯ g ≤ Rλ ; where A¯ ≤ U2 , g ∈ M such that Ag = A, =B λ this is possible as Bλ ∈ F f . But T Oτi ≤ SOτi ≤ Y and FT (T M2 ) ≤ FS (SM ) ≤ Y, −1 ¯ λ ) ≤ AutY (Rλ ), as g ∈ M , FS (SM ) ≤ Y, and R ¯ g ≤ Rλ . Therefore and AutLA¯ (R λ when d2 > 4 we have Ff ≤ Y by 1.3.2 in [Asc19].

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So assume d2 ≤ 4. If d2 = 4 and sgn(U2 ) = 1 then T F2 ≤ SOτ . If d2 ≤ 2 then T F2 = T ≤ Y. Finally if d2 = 3 or d2 = 4 and sgn(U2 ) = −1, then n is odd and a = 1 or 2. Then T F2 = AutLA (Rλ ) T ≤ Y, completing the proof.  Lemma 11.5.6. (1) |ZS | = nS is a power of 2. (2) L = P Ω+ 2n (F ). (3) Let θ ⊆ ZS be of order nS /2 and f = fθ ∈ F f . Then Ff ≤ Y. Proof. From 3.4.11.2, E = E(τ ) = NF (O(ξ)), SOτ , NF (W ) : ξ ∈ Ξ , where W ∈ W (τ ) and Ξ consists of those ξ ⊆ ZS with O(ξ) ∈ F f and NS (O(ξ)) transitive on ξ. Recall that B = B(τ ) = E, Fx : x ∈ W(τ ) ∩ F f , where W(τ ) consists of those fζ with ζ ⊆ ZS of order a power of 2. By 11.5.4 and Theorem 11.3.11, we have B = F. Thus by the choice of τ as a counter example, B is not contained in Y, so NF (O(ξ)) or Fx is not contained in Y for some ξ ∈ Ξ or some x = fζ ∈ F f with |ζ| a power of 2. In either case Fy ≤ Y for some y = fζ ∈ F f with |ζ| a power of 2. Now the lemma follows from 11.5.5.  Notation 11.5.7. By 11.5.6, L = P Ω+ 2n (F ) and n is a power of 2. Hence a = 0 so from 11.5.2, Λ = ∅ and hence Y = SOτ , FS (SM ) . Choose θ = θ1 as in 11.5.6.3; thus |θ| = nS /2 with n = 2nS a power of 2. It is convenient to work in L = Ω(U, q) rather than in the projective group P L. Thus we regard fθ as a member of L. Now Π1 = {θ1 , ZS − θ1 } is an S-invariant partition of ZS , and U = U1 ⊥U1⊥ , where U1 = [U, fθ ] and U1⊥ = CU (fθ ) are of dimension n = d/2. Proceeding recursively, assume Πk is an S-invariant partition of ZS into 2k blocks of size nS /2k . Pick θk ∈ Πk and let θk+1 ⊆ θk be of order nS /2k+1 such S that |NS (θk ) : NS (θk+1 )| = 2, and set Πk+1 = θk+1 , so that Πk+1 is an S-invariant k+1 k+1 partition of ZS into 2 blocks of size nS /2 . Continuing this process, we arrive at θr = {z} for some z ∈ ZS ; that is nS = 2r . Let Πk = {θk,1 , . . . , θk,2k } and Uk,i = [U, fθk,i ]. Thus  U = Uk,1 ⊥ · · · ⊥Uk,2k and we set Uk = {Uk,1 , . . . , Uk,2k }, Lk,i = LUk,i and Lk = i Lk,i the direct product of 2k copies of Ω+ (F ). Let U k = {U1 , . . . , Uk } and Gk the stabilizer in SL of U k . 4·2r−k Set Dk = FS (Gk ). Lemma 11.5.8. Gk = SLk . Proof. By construction, SLk ≤ Gk and Lk is contained in the kernel X of the k action of Gk on Uk . By 2.1.16, GU k is contained in a Sylow 2-group of Sym(Uk ). For x ∈ X of odd order, x|Uk,i ∈ Lk,i as Lk,i acts faithfully as Ω(Uk,i , q) on Uk,i . Therefore X/Lk is a 2-group, completing the proof.  ∼ Ω+ (F ) has Sylow 2-group O(si ) for that si ∈ ZS Lemma 11.5.9. (1) Lr,i = 4 such that si ∈ Z(Lr,i ) and Ut,i = [U, si ]. (2) Oτ = FO(τ ) (Lr ). (3) Dr = FS (Gr ) = SOτ ≤ Y. Proof. By construction, Ur,i = [U, si ] for some si ∈ ZS , and then (1) follows. In particular by (1), O(τ ) is Sylow in Lr and (2) holds. By 11.5.8, Gr = SLr , so  FS (Gr ) = FS (SLr ) = SOτ , establishing (3). Lemma 11.5.10. E(τ ) = Y.

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Proof. From 3.4.11.2, E = E(τ ) = NF (O(ξ)), SOτ , NF (W ) : ξ ∈ Ξ , where W ∈ W (τ ) and Ξ consists of those ξ ⊆ ZS with O(ξ) ∈ F f and NS (O(ξ)) transitive on ξ. Now NF (W ) = FS (SM ), so Y = SOτ , FS (SM ) ≤ E with E = Y, NF (O(ξ)) : ξ ∈ Ξ . Thus it remains to show that N = NF (O(ξ)) ≤ Y for ξ ∈ Ξ. But N ≤ Fx , where x = fξ , and by 11.5.5 either Fx ≤ Y or (x, ξ) = (1, ZS ) or (f, θ), and we may assume one of the latter two cases holds. Suppose (x, ξ) = (1, ZS ). Then CS (O(τ )) ≤ O(τ ) so N has a model G. Let H be the kernel of the action of G on Ω. Then W ≤ T ∈ Syl2 (H) and O 2 (H) = 1 if m > 8, while if m = 8 then FO(ξ) (O 2 (H)) = Oτ . Thus if m > 8 then W  G, so N ≤ FS (SM ) ≤ Y. So take m = 8. Then N = FS (SH), FS (NG (T )) . But FS (SH) = SOτ ≤ Y and W  NG (T ), so FS (NG (T )) ≤ FS (SM ) ≤ Y. Thus the lemma holds in this case, so we may assume instead that (x, ξ) = (f, θ). Now N = FS (X), where X = NL1,1 (O(ξ))L1,2 S. Therefore N = FS (SL1,2 ), NN (S2 ) , where S2 is Sylow in L1,2 . By minimality of d, FS (SL1,2 ) ≤ Y. Further NN (S2 ) ≤  NN (O(τ )), which is contained in Y by the previous case. Notation 11.5.11. Observe that D1 = Ff . Therefore by 11.5.6.3, D1 ≤ Y. On the other hand by 11.5.9.3, Dr ≤ Y. Thus there exists a least k such that Dk ≤ Y. Set D = Dk , ρ = (D, Ω), let S0 = S ∩ Lk , and write D0 for FS0 (Lk ). I set Si = S ∩ Lk,i and Li = FSi (Lk,i ). From Let I = {1, . . . , 2k } and for i ∈  11.5.7, Lk = i∈I Lk,i , so D0 = i∈I Li is a direct product. Set Ωi = Ω ∩ Si and ρi = (Li , Ωi ). Observe: Lemma 11.5.12. The pair D, D0 satisfies Hypotheses 11.4.1 and 11.4.6. Notation 11.5.13. As in 11.4.2 and 11.4.6, for X ≤ S and 0 ≤ i ≤ 2k , write Xi for X ∩ Si . As D ≤ Y, it follows from the Alperin-Goldschmidt Fusion Theorem that the set R of subgroups R ∈ D e such that AutD (R) ≤ AutY (R) is nonempty. Pick R ∈ R and let G be a model for ND (R) and set G∗ = G/R. By 11.5.12 and 11.4.4.1, R0 ∈ F f rc , so there is a model G0 for ND0 (R0 ). By 11.4.4, we can embed G and G0 in a certain group Gc with G0  Gc and G = NGc (R). Thus we can form H = NG0 (R) and by 11.4.4 H  G with R0 = O2 (H) and H ∗ ∼ = H/R0 . Set Q = NR (Lk,1 ) and X = FQS1 (QLk,1 ). Let Σ be a set of coset representatives for Q in R with 1 ∈ Σ. Write R∗ for the set of those R ∈ R of maximal order. Lemma 11.5.14. For each K ∈ Ω, |K ∩ R| ≤ 2. Proof. Assume otherwise; then from 3.4.5, AutD (R) = AutE(ρ) (R). But E(ρ) ≤ E(τ ) and by 11.5.10, E(τ ) ≤ Y, contrary to the choice of k.  Lemma 11.5.15. (1) For each i ∈ I, Ri = Si . (2) R is transitive on {Lk,i : i ∈ I}. (3) Q ∈ X e . Let GQ be a model for NX (Q). (4) GQ /Q ∼ = H ∗. (5) Let W1 ∈ W (ρ1 ) and W1,I the subgroups of W1 defined in 11.1.2 that is the permutation module for Sym(I) ∼ = μ(ρ1 )/μ0 (ρ1 ). Define α : W1,I ∩ Z(Q) → W ∩ R

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 by α : x → σ∈Σ xσ . Then xα ∈ Z(R) and if x is of weight p in W1,I then xα is of weight 2k p in W . (6) If p < 2|ZS (ρ1 )| then CD (xα) ≤ Y. Proof. If Ri = Si then Ωi ⊆ R, contrary to 11.5.14. Thus (1) holds. By 11.5.12, we can appeal to results in section 11.4. In particular (2) follows from (1) and 11.4.8.1, (3) follows from 11.4.8.4, and (4) follows from 11.4.8.5. From 11.5.13, 1 ∈  Σ is a set of coset representatives for Q in R. If σi ∈ Σ with R1 σi = Ri then R0 = i∈I Ri = i∈I R1σi , so for x ∈ W1,I ∩ Z(Q) of weight p, xα centralizes Q, Σ = R and xα is of weight |Σ|p = |I|p = 2k p in W . Thus (5) holds. Assume the hypothesis of (6) and set y = xα. Observe that |ZS (ρ1 ) = nS /2k = 2r−k . As p < 2|ZS (ρ1 )|, y = 1 so y is an involution. As 2k p is even, y is fused in M to fξ for some ξ ⊆ ZS with |ξ| = 2k−1 p. Then by 11.5.5, either Fy ≤ Y or 2k−1 p = nS /2 = 2r−1 , and we may assume the latter, so that p = 2r−k . Then R ∪ (θk − θk+1 )R . But conjugating in M , we may take x = fθk+1 , so Πk+1 = θk+1 then CSLk (y) = CS (y)Lk+1 , so as Dk+1 ≤ Y, again CD (y) ≤ Y.  Lemma 11.5.16. ζ = (X , Ω1 ) satisfies Hypothesis 11.2.1 and if R ∈ R∗ then ζ, Q satisfies Hypothesis 11.2.7. Proof. Visibly ζ satisfies Hypothesis 11.2.1 with |Ω(z)| = 2 and μ(ζ) ∼ = W eyl(Dn/2k ). So assume R ∈ R∗ . By 11.5.15.4, Q ∈ X e . From 11.2.6, W(Q) consists of those involutions x ∈ Z(Q) ∩ W1,I such that x = eα for some α ⊆ I(ζ) of weight a positive power of 2. By 11.5.15.6, for each x ∈ W(Q), CD (xα) ≤ Y. Let x ∈ W(Q), y = xα and Z = y G . As R ∈ R∗ and CD (y) ≤ Y, the proof of  11.3.5.1 shows that O 2 (CG (Z)) = R. Set Zx = xGQ . Then Z is a full diagonal   subgroup of σ∈Σ Zxσ and the isomorphism of 11.5.15.4 maps O 2 (CGQ (Zx ))/Q to   O 2 (CG (Z))/R = 1, so O 2 (CGQ (Zx )) = Q. Hence ζ, Q satisfies 11.2.7.1. Similarly if ρ, R satisfies the setup of 11.2.7.2 and U = ∅ then the proof of 11.3.5.2 shows that m2 (G∗ ) = 1 and U is not normal in G∗ . Let B ∗ be the subgroup of G∗ generated by its involutions. By 11.4.5.2, B ∗ is the subgroup of H ∗ generated by its involutions. Thus the isomorphism of 11.5.15.4 maps the subgroup ∗ ∗ of G∗Q = GQ /Q generated by involutions to B ∗ , so m2 (G∗Q ) = 1 and UQ = BQ . BQ Therefore ζ, Q satisfies 11.2.7.2. This completes the proof of the lemma.  We are now in a position to obtain a contradiction, completing the proof of Theorem 11.5.3. Namely choose R ∈ R∗ . By 11.5.16, the pair ζ, Q satisfies Hypotheses 11.2.1 and 11.2.7. Further condition (a) of Theorem 11.2.34 is satisfied, as is condition (b) by 11.5.14. Condition (c) of 11.2.34 holds as X = FQS1 (SLk,1 ). But then by Theorem 11.2.34, |ZS (ζ)| = 1, a contradiction. This completes the proof of Theorem 11.5.3. In this last lemma of the section we do not assume Hypothesis 11.5.1. Lemma 11.5.17. Assume τ = (F, Ω) is a quaternion fusion packet with F ˆ 9 , Spin9 [m], or Spin− [m]. Let S be Sylow in F, M = M (τ ), isomorphic to AE 10 and U = Z(M ).

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ˆ 9 then Y = Aut(F) is the split extension of E = O2 (Y ) by (1) If F is AE Sym(I) ∼ = S9 with E the core of the 9-dimensional permutation module for Sym(I). (2) U ∼ = E4 and Z(S) = Z(F) ∼ = Z2 , so S is transitive on U − Z(F). (3) There exists no automorphism α of F of order 4 centralizing M . Proof. Observe that μ(τ ) ∼ = Weyl(D4 ) (cf. 11.6.2 and 5.3.4) and M ∼ = ω ¯ (D4 , m) (cf. 5.1.16, 5.3.9, and 5.3.10), so U ∼ = E4 by 5.8.4. Then as Z = Z(F) is of order 2, if Z(S) = Z then S is transitive on U − Z(S), so that (2) holds. ˆ 9 and let G ˆ be a model for F. By 5.1.16, Q = O2 (F) ∼ Suppose F is AE = 3 ∗ ˆ Q8 , so as F (G) = Q we have Z(S) = Z(Q) = Z, and hence (2) holds in this case. As Q ∼ = Q38 , Out(Q) ∼ = O6− (2). Now Y ≤ Aut(Q) and indeed Y /Inn(Q) = NOut(Q) (OutGˆ (Q)) with OutGˆ (Q) ∼ = A9 absolutely irreducible on Q/Z, so we have Y /Inn(Q) ∼ = S9 , establishing (1). ∼ ˆ Next α induces an automorphism α of G = G/Z = AE9 of order 4 centralizing M/Z, and by (1), α ∈ Y = ESym(I). Then M E/E is the stabilizer of the partition Λ = {{1, 2}, {3, 4}, {5, 6}, {7, 8}, {9}} of I, so CSym(I) (M E/E) = t where t = (1, 2)(3, 4)(5, 6)(7, 8). Hence α ∈ tE, a contradiction as t does not centralize K ∩ E for K ∈ Ω. So we may take F to be Spin9 [m] or Spin− 10 [m]. Then F is tamely realized by −π ∼ ˆ L = Spin9 (q) or Spin10 (q) for suitable q by 3.5 in [AO16], and then by 2.22.3 in [AO16], α is induced by an automorphism α of L = Ω9 (q) or Ω−π 10 (q). Let V be the defining unitary space for L; then Aut(L) = Γ = ΓO(V ). Now M = M (τ ) ≤ ˆ with X ∼ X ≤L = Spin+ 8 (q). Let T ∈ Syl2 (X) and X = FT (X). As α centralizes O(τ ) ≤ M , α centralizes CX (z) for z ∈ ZS by 2.6.12, so by 11.5.3, α centralizes X = CX (z), M . Thus α centralizes X, a contradiction as a Sylow 2-subgroup of CΓ (X) is elementary abelian. It remains to show that Z = Z(S); as M ∼ ¯ (D4 , m) we have U = Z(T ). = ω If F ∼ = Spin9 [m] then S = T x where x induces a reflection on X, and hence ˆ ˆ induces GL(U ) on U and xL ˆ ∈ Thus indeed [U, x] = Z as Out(L) / O2 (Out(L)). Z = Z(S) in this case. When F is Spin− [m] we have Z = Z(S) by 5.9.4 and its 10 proof. 

11.6. Generating AEk In this section we assume the hypothesis and notation of section 5.1, except we write k for the integer “n” in that section, and instead write k = 2n + a for ˇ k and a ∈ {0, 1}. Assume k ≥ 7; thus n ≥ 3. In particular G = AE ∼ = AE + ∼ G = G/E = Alt(I) = Ak . For X ⊆ G, write F ix(X) for the set of fixed points of X ∗ on I and write M ov(X) for the set of points moved by X ∗ . ˜ and S ∈ Syl2 (G) as in As in Notation 5.1.5, let z = e1,2,3,4 . Choose K, K, G 5.1.6, set Ω = K ∩ S, and F = FS (G). From 6.3.6.3, τ = (F, Ω) is a quaternion ˜ of order 2. Choose notation so that z ∈ F f , and fusion packet with Ω(z) = {K, K} if k is odd, so that F ix(S) = {k}. ˜ i : 1 ≤ i ≤ s}, where From 5.1.9 we may choose notation so that Ω = {Ki , K ˜ i ) = eλ n = 2s, n = 2s + 1 for n even, odd, respectively, and z(Ki ) = zi = z(K i with λi ∈ P¯2 , where P¯2 is a partition of the set Δ2 of the first 4s points of I into s ˜ i ) = λi . Choose notation so that z = z1 . blocks of size 4, such that M ov(Ki K

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Lemma 11.6.1. (1) For 1 ≤ i ≤ s, there is Lzi  CG (zi ) with O(zi ) = O2 (Lzi ), Ozi = FO(zi ) (Lzi ). Lzi = O 2 (Lzi ), Lzi /O(zi ) ∼ = Z3 , and M ov(Lzi ) = λi . Set   (2) Set Lτ = Lzi : i and Oτ = Ozi : i . Then Lτ = i Lzi and Oτ = i Ozi . Proof. Part (1) is 5.1.6.2 and (1) implies (2).



Again by 5.1.9, there is a partition P¯1 of the set Δ1 of the first 2n points of I into n blocks of size 2, such that P1 = P¯1 ∪ (I − Δ1 ) is a partition of I that is a refinement of P¯2 ∪ (I − Δ2 ), and ZΔ = {eα∪β : α, β ∈ P¯2 and α = β}. If k and n are even let P2 = P¯2 . If k is odd and n is even, let P2 = P¯2 ∪ {k}. If k is even and n is odd, let P2 = P¯2 ∪ {k1 , k2 }. If k and n are odd let P2 = P¯2 ∪ {k − 1, k − 2} ∪ {k}. Thus in each case, P2 is a partition of I and P1 is a refinement of P2 . Moreover P1 and P2 are S-invariant as Ω and ZΔ are S-invariant. ˜ + = (1, 2)(3, 4), (1, 3)(2, 4) is the root 4From the proof of 5.1.6, K + = K + group of G moving λ1 . Similarly Ki+ is the root 4-group with M ov(Ki+ ) = λi .

Lemma 11.6.2. Let W = W (τ ), and M1 = NG (P1 ). (1) M1 = NG (W ). (2) M1+ = M + . (3) μ(τ ) ∼ = Weyl(Dn ). Proof. Observe M1+ permutes the set T of transpositions t in Sym(I) such that M ov(t) ∈ P¯1 , and W + is the subgroup of index 2 in T that is the core of the permutation module for M + /W + = Sym(T ) = Sym(P¯1 ) ∼ = Sn . As W is weakly closed in S, it follows that (1) holds. Next M = K M1  M1 and + M1+ = K +M1 = M + , establishing (2). Next define WI as in Notation 11.1.2, and observe that W ∩E = WI = eα : α ∈ P¯1 is the permutation module for M1 /W E ∼ = Sn on {eα : α ∈ P¯1 }, so |W | = 22n−1 . If k = 2n is even then E/WI is the core of the permutation module for M1 /W E, so E = [E, M ] ≤ M as M  M1 . Then μ = μ(τ ) = M/W is an extension of E/WI by Sn , so (3) holds in this case by 4.3.8.1. On the other hand if k = 2n + 1 is odd then E/WI is the permutation module so M ∩ E = [E, M ] is of index 2 in E and again (3) holds. Moreover we’ve shown that if k is even then M = M1 , which we record as:  Lemma 11.6.3. If k is even then M = NG (P1 ) contains S. Lemma 11.6.4. Assume k is not congruent to 3 modulo 4. Then (1) NG (O(τ )) = Lτ NG (O(τ )W ) = NG (P2 ). (2) MF (O(τ )) = SOτ , NF (O(τ )W ) . ¯ 2 = NG (P¯2 ). By construction, Proof. Let O = O(τ ), M2 = NG (P2 ), and M ¯ ¯ 2 , so M2 = NG (O). M2 = NG (O). But as k is not congruent to 3 modulo 4, M2 = M Similarly by construction, Lτ and Lτ W are normal in M2 , with OW Sylow in Lτ W , so by a Frattini argument, M2 = Lτ NG (OW ), completing the proof of (1). Then (1) implies (2) as G is a model for F.  Notation 11.6.5. Set SM = S ∩ M , so that SM is Sylow in M . Set B = SM Oτ , FSM (M ) .

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Lemma 11.6.6. If k is even then S is Sylow in M and F = B. Proof. Assume k is even. By 11.6.3, S is Sylow in M and M = NG (P1 ). By the Alperin-Goldschmidt Fusion Theorem, F = NF (S), AutF (B) : B ∈ F e . By 5.2.5 in [Asc10], NG (S) = S, so B = AutF (B) : B ∈ F e . Let B ∈ F e , and define B∗ (G+ , S + ) as in section 5 of [Asc10]. Then B + ∈ B∗ (G+ , S + ), so by 5.11 in [Asc10], NG (B) ≤ NG (Pi ) = Mi for i = 1, 2. Therefore NF (B) ≤ FS (Mi ). If i = 1 then NF (B) ≤ FS (M ) ≤ B by 11.6.3, so it remains to show FS (M2 ) ≤ B. But by 11.6.4, FS (M2 ) = SOτ , NF (O(τ )W ) ≤ B as  NF (O(τ )W ) ≤ NF (W ) = FS (M ). ˜ CG (z) and assume k ≥ 9. Then D ∼ Lemma 11.6.7. Let D = (Ω − {K, K}) = ˇ AEk−4 . ˜ Proof. Observe that CG (z) is the stabilizer in G of λ1 with Γ = Ω − {K, K} + trivial on λ1 . Therefore D ≤ Gλ1 and Gλ1 = Alt(I − λ1 ). Further for J ∈ Γ, J ≤ O 2 (D), so D = O 2 (D), and hence D ∩ E ≤ ej : j ∈ I − λ1 . The result follows.  Lemma 11.6.8. Assume k is odd and set L = K Gk . Then (1) SM ∈ Syl2 (L) and S = SM E. Indeed if k ≥ 9 then S = SM (E ∩ D), where D is as in 11.6.7. ˇ 2n . (2) L ∼ = AE (3) B = FSM (L). +

+Gk , we have Gk = LE. Then as E is the 2n-dimensional Proof. As G+ k = K permutation module for L+ ∼ = A2n , it follows that L ∩ E = [E, L] is of index 2 in E, and then that (2) holds. Then as M ≤ L, we conclude from 11.6.6 that SM is Sylow in L and (3) holds. Also |S : SM | = 2 with S = SM E. Finally if k ≥ 9 then D ∩ E ≤ [L, E], so S = SM (D ∩ E), completing the proof of (1) and the lemma. 

Lemma 11.6.9. Assume k is odd and let λ ∈ P¯1 such that R = Sλ is of maximal order. Then (1) F = SB, O 2 (AutF (R)) . (2) R ∈ F e . (3) λ is in the shortest orbit of S on P¯1 . (4) If n is odd then λ is the unique S-invariant member of P¯1 . Proof. As in the proof of 11.6.6, F = AutF (B) : B ∈ F e , and for each B ∈ F e , B + ∈ B∗ (G+ , S + ). By 11.6.8.3, we may choose B with NG (B) ≤ Gk . < Partially order such B by the transitive extension ∼ of the relation B1 ≺ B2 if B1 ≤ B2 and Σ1 = AutF (B1 ) = AutB (B1 ), NΣ1 (AutB2 (B1 ) . Pick B maximal < with respect to ∼. As k is odd, F ix(B) = ∅, so we may appeal to 5.10 in [Asc10]. As NG (B) ≤ Gk , F ix(B) = {k}, so B + appears in one of the three cases in 5.10.3; in particular J = F ix(B) is of order 3. If k = 7 we check directly that |F e | = 2 and the lemma holds. Thus we may assume k > 7, so that case (ii) or (iii) of 5.10.3 holds. As |J| = 3 and G is 3-transitive on I, we may assume J = λ ∪ {k} for some λ ∈ P¯1 . Hence B ≤ Sλ . Suppose case (ii) holds. Then n is odd and λ satisfies (3) and (4). Let X be of order 3 in G with M ov(X) = J. Then [Sλ , X] = Q is a 4-subgroup of

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E, so Q ≤ E ≤ B and NG (B) = NSL (B)B2 X, where B2 = NS (B). Therefore AutF (B) = AutB (B)AutB2 X (B), so by maximality of B, B = B2 , so that B = Sλ . By (4), |S : B| = 2 and as k > 7, NG (B) = SX, so (2) holds. Suppose on the other hand that (iii) holds. Then B ∈ Syl2 (GJ ), so B = Sλ . As B ∈ F f and NG (P¯1 ) is transitive on P¯1 , it follows that (3) holds, so in particular if n is odd then we are back in case (ii), so that B is unique and (1), and hence also the lemma holds. Hence we may assume n is even, so all choice for B are in case (iii). Then as NG (P¯1 ) ≤ LE, B is unique under conjugacy in SB, so (1) holds, and B = Sλ with |Sλ | maximal. Further (3) implies (2) and (4) holds vacuously, completing the proof.  Lemma 11.6.10. Assume k is odd and G = AEk . Let S ∈ Syl2 (G). Then O2 (G) = E = J(S). Proof. This follows from B.2.7 and B.3.2.4 in [AS04].



CHAPTER 12

|Ω(z)| = 2 and the proof of Theorem 6 In Chapter 12 we prove Theorem 6 under the additional hypothesis that the Extended Inductive Hypothesis is satisfied. Then, at the end of the paper when Theorem 1 is established, the extra assumption becomes unnecessary. In any event throughout Chapter 12 we assume τ = (F, Ω) is a quaternion fusion packet such that F ◦ is transitive on Ω, |Ω(z)| ≥ 2 for z ∈ ZS (τ ), and τ satisfies the Inductive Hypothesis. The case where |Ω(z)| > 2 has already been treated in Theorem 10.2.13, so we may assume |Ω(z)| = 2. We begin in section 12.1 with the case μ = μ(τ ) ∼ = Weyl(D4 ). Then in section 12.2 we determine those packets with μ ∼ = Weyl(Dn ) for some n ≥ 5. By Theorems 9.3.24 and 9.4.33, we may assume D∗ (z) = D(z) and Z(τ ) ∩ O(z) = {z}. Therefore, strengthening the Inductive Hypothesis to the Extended Inductive Hypothesis, Hypothesis 12.3.1 holds. Hence (cf. 12.3.4) μ is the direct product of r copies of Weyl(Dn ) for some n ≥ 3. Finally we complete the proof of Theorem 6 (under the additional assumption that the Extended Inductive Hypothesis holds) in Theorem 12.3.8 by proving r = 1. Note that the case μ ∼ = S4 was = Weyl(D3 ) ∼ treated in Theorem 9.5.33. These threads are drawn together in Theorem 12.3.50 and the proof of the (weak version) of Theorem 6 formally appears in section 12.4. 12.1. |Ω(z)| = 2, μ isomorphic to Weyl(D4 ) In this section we assume the following hypothesis: Hypothesis 12.1.1. τ = (F, Ω) is a quaternion fusion packet such that (1) For z ∈ ZS , |Ω(z)| = 2. (2) μ ∼ = Weyl(D4 ). (3) τ satisfies the Inductive Hypothesis Notation 12.1.2. By 12.1.1.2 and 4.2.8.7, F ◦ is transitive on Ω. By 4.2.8.6, |Ω| = 4, so |ZS | = 2 by 12.1.1.1. Observe that Hypotheses 9.1.1 and 9.1.8 are satisfied, and adopt Notation 9.1.2 and 9.1.6. Let t ∈ ZS − {z}, Ω(t) = {K3 , K4 }, and set c = zt and WI = v M . Let μ0 = (dK∩W dK2 ∩W )μ , O = O(z), and b = v3 v4 . For a ∈ {c, b, bc}, let Ta be the Sylow group of Fa◦ . Observe also that Hypothesis 11.1.1 is satisfied. Lemma 12.1.3. (1) Z(M ) = c . (2) μ0 ∼ = E8 is the core of the 4-dimensional permutation module for μ/μ0 ∼ = S4 , μ0 centralizes WI , and WI is the 4-dimensional permutation module for μ/μ0 . (3) c ∈ Z(S). (4) CS (z) = CS (t) is of index 2 in S and z ∈ F f . 305

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(5) For each a ∈ bM ∪ (bc)M , |S : CS (a)| = 4 and a ∈ F f . (6) z M ∼ = E8 with ZΔ = z M − c of order 6. M (7) z ∩ D∗ (z) − ZS = {v11 · · · v44 : i ∈ {1, −1}} is of order 4 and ZΔ = ZS ∪ (z M ∩ D ∗ (z)). Proof. Part (1) follows from 5.8.7.2. Next μ0 ∼ = E8 is the core of the permutation module for μ/μ0 ∼ = S4 from the structure of the reflection group M ∗ = M/W = μ ∼ = Weyl(D4 ) (cf. Remark 4.3.4). The remaining statements in (2) follow from 11.1.2. As S acts on M , (3) follows from (1). Part (4) holds as |ZS | = 2 and F is transitive on Ω. Similarly (5) and (6) follow from (2). By (2), for some choice of i , x = v11 · · · v44 ∈ Σ = D∗ (z) ∩ z M − ZS , and then as O(τ ) is transitive on such elements, all are in Σ, so (7) holds.  Lemma 12.1.4. (1) τc = (Fc , Ω) is a quaternion fusion packet with μ(τc ) ∼ = Weyl(D4 ). ˇ 8, Ω ˇ + (2), or Ω+ [m]. ˇ 3 (2)/23+6 , AE (2) Either Fc◦ = FTc (M ) or Fc◦ ∼ =L 8 8 Proof. This is a special case of 11.1.6.



Lemma 12.1.5. For a ∈ {b, bc} let ηa = (K ∩ W )CM (a) , Wa = ηa , and Ma = K CM (a) . Then (1) τa = (Fa , Ω(z)) is a quaternion fusion packet with ηa ∈ η(τa ), Ma is a / Mb , and Z(Mb ) = bc . model for [K]NFa (Wa ) , μ(τa ) ∼ = S4 , Z(Mbc ) = bc , b ∈ + ◦ ◦ ∼ ˇ = FTbc (Mbc ) or Fbc or Ω AE (2) Either Fbc = 6 6 [m]. (3) One of the following holds: (i) Fb◦ = FTb (Mb ). ˇ 6 or Ω+ [m]. (ii) Fb◦ ≤ Fc◦ and Fb◦ ∼ = AE 6 (iii) Fb◦ ∼ = L3 (2)/E64 , Sp6 (2), AE7 , Ω7 [m], or Ω− 8 [m]. ∼ Weyl(D3 ) Proof. From 11.1.7.1, τa is a quaternion fusion packet with μ(τa ) = ∼ = S4 . As a centralizes O by construction, Ω(z) = CΩ (a). Therefore ηa ∈ η(τa ) and Ma is a model for [K]NFa (Wa ) . By 5.8.7.4, b ∈ c Mb , so either b ∈ Mb or b ∈ cMb . But as c ∈ Z(M ), Mb = Mbc , so b ∈ cMb iff bc ∈ Mb = Mbc . By 5.8.7.3, M is determined up to isomorphism by Hypothesis 12.1.1, so to prove bc ∈ Mb and complete the proof of (1), we may work in the 2-fusion system F of L = Ω(U, q), where (U, q) is an 8-dimensional orthogonal space of sign +1 over F = Fq , with q ≡ π mod 4 and π = ±1. Then B = [U, b] and A = B ⊥ are of sign π and dimension 2 and 6, so LA ∼ = Ωπ6 (F ) has center generated by an involution i of type i(6, π). In particular i = b, so i = bc and hence indeed bc ∈ Mb . Parts (2) and (3) follow from 11.1.7.3 together with the fact that bc ∈ Z(Mb ) =  Z(Mbc ) from (1). Lemma 12.1.6. (1) Fz◦ = O(z) ∗ C where ρ = (C, Ω(t)) is a quaternion fusion packet with C = C ◦ and μ(ρ) ∼ = E4 . (2) Either C = O(t) or C ∼ = AE5 , P Sp4 [m], or L− 4 [m]. Proof. This is a special case of 11.1.5. Lemma 12.1.7. (1) Fc ≤ E(τ ). (2) Fb , Fbc ≤ E(τ ).



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(3) E(τ ) = SO(τ ), NF (O), NF (O(τ )), NF (W ) = Fz , Fc . (4) F = E(τ ) = Fz , Fc . ˜ ˜ ˜ (5) Let F˜ = Ω9 [m] or Ω− 10 [m]. Then F = Fz , Fc . Proof. The first equality in (3) follows from the definition of E = E(τ ) in 3.4.2, while E ≤ Fz , Fc as SO(τ ), NF (O(τ )), NF (W ) ≤ Fc while NF (O) ≤ Fz . Thus to complete the proof of (3), it suffices to prove (1) and Fz ≤ E. But O(z)  Fz , so by 1.3.2 in [Asc19], Fz = SO(z), NF (O) . Then as SO(z) ≤ SO(τ ), we have Fz ≤ E, so (1) implies (3). Let a ∈ {b, c, bc}. By 1.3.2 in [Asc19], Fa = CS (a)Fa◦ , NFa (Ta ) . Further NFc (Tc ) ≤ NF (O(τ )) and NFa (Ta ) ≤ NFa (O) for a ∈ {b, bc}, so to prove (1) and (2) it remains to show that X = CS (a)Fa◦ ≤ E. But if a = c then by 12.1.4 and 10.1.15.2, X = NX (z, c ), FS (SM ) and O(τ )  NX (z, c ), so by 1.3.2 in [Asc19], NX (z, c ) = SO(τ ), NX (O(τ )) ≤ E, proving (1). So take a = b or bc. Here by 12.1.5, 9.3.20, and 9.5.9, X = NX (z, a ), FCS (a) (CS (a)Ma ) , and O(z)  NX (z, a ), so by 1.3.2 in [Asc19], NX (z, a ) = CS (a)O(z), NX (O) ≤ E, proving (2). By Theorem 11.3.11, F = B, where B = E, Fz , Fc , so (4) follows from (3). A subset of these arguments also proves (5).  Lemma 12.1.8. Assume m = 8 and set S0 = CS (O), R0 = OS0 , let Gz be a model for NFz (R0 ), Hz = W Gz , and Lz = O 2 (Hz ). Then either Lz = 1 or O(τ ) = O2 (Lz ), Lz is of index 3e in (SL2 (3) ∗ SL2 (3))2 for some 0 ≤ e ≤ 3, and Lz centralizes CS (O(τ )). Proof. By 9.1.11.2, either the lemma holds or O = O2 (Lz ) and Lz is of index 1 or 3 in SL2 (3) ∗ SL2 (3). As W is nontrivial on J/z(J) for J ∈ Ω and S is transitive on Ω, the latter case does not hold.  Lemma 12.1.9. (1) If C = O(t) then c ∈ Z(F). (2) If Fc◦ = FTc (M ) or Fa◦ = FTa (Ma ) for some a ∈ {b, bc} then c ∈ Z(F). ˇ + (2) then c ∈ Z(F). ˆ 3 (2)/23+6 or Ω (3) If Fc◦ ∼ =L 8 (4) If Fb◦ ∼ = L3 (2)/E64 or Sp6 (2) then c ∈ Z(F). (5) If m = 8 and c ∈ / Z(F) then Lz = L1 × L2 where L1 ∼ = L2 , O = O2 (L1 ), O(t) = O2 (L2 ), and L1 is of index k = 1 or 3 in SL2 (3) ∗ SL2 (3), with k = 3 iff C∼ = AE5 . Proof. If C = O(t) then c = zt ∈ Z(Fz ), so by 12.1.7.4, F = Fz , Fc = Fc , proving (1). Assume c ∈ / Z(F). Then by (1) and 12.1.6, C ∼ = AE5 , P Sp4 [m] or L− 4 [m]. In particular if m > 8 then O(z) ∼ = O(t) ∼ = SL2 [m] ∗ SL2 [m], while if m = 8 then a model L2 for O 2 (CC (t)) is of index k ∈ {1, 3} in SL2 (8) ∗ SL2 (8), with k = 3 iff C ∼ = AE5 , and L2 centralizes CS (O(t)). Then L2  Lz and if k = 1 then O 2 (Aut(O(t)) = O 2 (AutF (O(t)). As S is transitive on Ω we conclude, using 12.1.8, that Lz = L1 × L2 , where L1 = Ls2 for s ∈ S − CS (z). Therefore (5) holds. Notice L1 centralizes b and bc, and Lz centralizes c. Suppose Fc◦ = FTc (M ) or Fa◦ = FTa (Ma ) for a ∈ {b, bc}. Then as O(z) centralizes b, c, z , it follows that O(z) = O, so as O(z) ∼ = SL2 [m] ∗ SL2 [m] when

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m > 8, we conclude that m = 8. But then as L1 centralizes b, c, z , L1 acts on W ∩ O and hence centralizes O, again a contradiction. This proves (2). ˇ + (2). Then m = 8 and in 12.1.8, e = 3 or 1 in the ˆ 3 (2)/23+6 or Ω Assume Fc◦ ∼ =L 8 respective case. This is a contradiction as Lz is of index 1 or 32 in (SL2 (3)∗SL2 (3))2 by (5). Thus (3) holds. Finally assume Fb◦ ∼ = L3 (2)/E64 or Sp6 (2) and let Lb be a model for O 2 (CFb◦ (z)). Then m = 8 and by 9.5.14.3, 9.5.15.10, and 9.5.16, case (iii) of 9.5.11.3 holds, so that O2 (Lb ) = O × B with B ∼ = E4 . This is a contradiction as Lb ≤ Lz and Lz = L1 × L2 with Li of index 1 or 3 in SL2 (3) ∗ SL2 (3) by (5). So (4) holds.  In the remainder of the section we assume: Hypothesis 12.1.10. Hypothesis 12.1.1 holds and c ∈ / Z(F). Lemma 12.1.11. (1) C ∼ = AE5 , P Sp4 [m], or L− 4 [m]. + ◦ ∼ ˇ (2) Fc = AE8 or Ω8 [m]. ◦ ∼ ˇ (3) Fbc = AE6 or Ω+ 6 [m]. ◦ ∼ (4) Fb = AE7 , Ω7 [m], or Ω− 8 [m]. Proof. This follows from the descriptions of the various possible systems in 12.1.4, 12.1.5, and 12.1.6, together with the exclusions in 12.1.9, unless 12.1.5.3.ii holds, and we may assume the latter. Let s ∈ S interchange z and t, and set d = bs , so that d ∈ O and (Fb◦ )c∗s = Fd◦ . Therefore as c ∈ Z(S) and Fb◦ ≤ Fc◦ in 12.1.5.3.ii, it follows that Fd◦ ≤ Fc◦ . Then as d ∈ O, we have C ≤ Fd◦ ≤ Fc◦ , a contradiction.  Notation 12.1.12. Set S0 = CS (O), R0 = OS0 , and if m = 8 define Gz , Hz , and Lz as in 12.1.8 and L1 as in 12.1.9.5. If m > 8 set O = O(z) and if m = 8 set O = FO (L1 ). Set Oc = O 2 (CFc◦ (O(t)z )), and for a ∈ {b, bc} set Oa = O 2 (CFa◦ (z)). Lemma 12.1.13. (1) O = Oa for a ∈ {b, bc, c}. (2) O centralizes S0 . (3) Either O ∼ = SL2 [m] ∗ SL2 [m] or m = 8, C ∼ = AE5 , and L1 is of index 3 in SL2 (3) ∗ SL2 (3). ◦ ∼ ˇ 8 iff F ◦ ∼ ˇ (4) If m = 8 then C ∼ = AE5 iff Fc◦ ∼ = AE bc = AE6 iff Fb = AE7 . (5) O  Fz . Proof. Suppose m > 8. Then by 12.1.11.1, C ∼ = P Sp4 [m] or L− 4 [m], so ∼ ∼ O(t) = SL2 [m] ∗ SL2 [m], and hence as O(z) = O(t), (3) holds. By definition in 12.1.12, O = Oz . Therefore (5) holds by 2.6.10 and (2) holds by 2.6.11. Part (1) follows from (3) and inspection of the local structure of the systems in 12.1.11 when m > 8. Finally (4) is vacuously true. So we may assume m = 8. Then from 12.1.12, O = FO (L1 ). But from 12.1.8 and 12.1.9.5, L1 centralizes S0 , so (2) holds. Let a ∈ {b, bc, c} and set Wc = W . By definition in 12.1.12, Oa = O 2 (CFa◦ (z)) ˇ 6 or AE7 and if a ∈ {b, bc}, so from 12.1.11, either Oa ∼ = SL2 [8]∗SL2 [8] or Fa◦ ∼ = AE 2 Oa is of index 3 in SL2 [8] ∗ SL2 [8]. Similarly Oc = O (CFc◦ (z O(t)), so by 12.1.11 ˇ 8 and Oc is of index 3 in SL2 [8] ∗ SL2 [8]. either Oc ∼ = SL2 [8] ∗ SL2 [8] or Fc◦ ∼ = AE ◦ Also Fa  Fa , so [Wa , CS0 (a)] ≤ CWa (O) ≤ CFa◦ (Oa ) from 12.1.11, and hence Oa

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centralizes CS0 (a). Then as |S0 : CS0 (a)| ≤ 2, it follows that Oa centralizes S0 , so a model La for Oa is contained in CLz (S0 ) = L1 . Conversely L1 centralizes a, so L1 = [L1 , Wa ] ≤ La . Therefore L1 = La , so (1) holds, and (3) and (4) follow from (1) and the structure of Oa just described. It remains to prove (5). By (2), O centralizes S0 . Further as W is nontrivial on J/z(J) for J ∈ Ω, O 2 (AutF (O(τ ))) = AutLz (O(τ )), so O 2 (AutF (O)) = AutL1 (O) = AutO (O). Then (5) follows from 2.6.14.  Lemma 12.1.14. If C ∼ = AE9 . = AE5 then F ◦ ∼ ˇ 8 . Set Uz = Proof. Assume C ∼ = AE = AE5 . Then m = 8 and by 12.1.13.4, Fc◦ ∼ ◦ O2 (C), Uc = O2 (Fc ), and U = Uz Uc . Now B = O(t) ∩ Uz is a hyperplane of Uz and the unique CS (z)L2 -invariant E8 -subgroup of O(t). Therefore B = O(t) ∩ Uz ≤ Uc , so Uc ≤ CS (B). However as C ∼ = AE5 , there is no transvection in AutFz (Uz ) centralizing B, so Uc centralizes Uz and hence U is abelian. Let Y = Fc◦ . Now U = CUC (Uz )  U C, and similarly U  U Y. By 2.5.2, τU = (NF (U ), Ω) is a quaternion fusion packet. As Y ≤ NF (U ), μ(τU ) ∼ = W eyl(D4 ). As B ≤ Uz ∩ Uc is a hyperplane of Uz , either Uz ≤ Uc and U = Uc is of rank 7, or Uz ≤ U and m(U ) = m(Uc ) + 1 = 8. In either case as a model for Y is irreducible on Uc /c and indecomposable on Uc , and as C does not centralize c, a model for X = NF (U )◦ is irreducible on U and U = z X . Also z ∈ U ≤ O2 (X ) and K ∩ O2 (X ) ≤ K ∩ O2 (Y) = z , so we conclude from Theorem 2 that X ∼ = AE9 . Thus to complete the proof of the lemma, it suffices to show that X  F, since then F ◦ = X . First L = OC  Fz using 12.1.13.5, so by 1.3.2 in [Asc19], Fz = CS (z)L, NFz (L) for L Sylow in L. As O, C ≤ X , U  CS (z)L. As NFz (L) ≤ NFz (O(τ )) ≤ CFz (c) and Uc  Fc , and as Uz  Fz , it follows that U = Uz Uc  NFz (L). Thus U  Fc . Next by 1.3.2 in [Asc19], Fc = SY, NF (Tc ) . Also Y ≤ X and U = Uc Uz Uzs for s ∈ S − CS (z), so U  SY. Similarly NF (Tc ) ≤ NF (O(τ )) ≤ NF (z, c ) and NF (z, c ) acts on U = Uc Uz Uzs . Therefore U  Fc . Hence by 12.1.7, U  Fz , Fc = F, completing the proof.  Because of 12.1.14, during the remainder of the proof we assume: Hypothesis 12.1.15. Hypothesis 12.1.1 holds, c ∈ / Z(F), and O ∼ = SL2 [m] ∗ SL2 [m]. Notation 12.1.16. Let T0 be Sylow in C and set T = T0 Tc . Set E1 = CT (z)OC, F1 = Fz , E2 = T Fc◦ , F2 = Fc , and Y = E1 , E2 .

F.

Lemma 12.1.17. (1) Ei  Fi for i = 1, 2. (2) Y is F-invariant, so in particular T is strongly closed in S with respect to

Proof. We first prove E1  F1 . Set T1 = CT (z), let L = OC, and let L be Sylow in L. Thus L = OT0 and, using 12.1.13.5, L  F1 . Next NF1 (L) ≤ NF1 (O(τ )) ≤ CF1 (c), and if m > 8 then NFc◦ (O(τ )) = Tc , so CTc (z)  NF1 (L). Therefore E1  F1 by Theorem 1.5.2 in [Asc19] when m > 8.

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So take m = 8 and let X = FO(τ ) (Lz ). Then NFc◦ (O(τ )) = Tc X , so CTc (z)X  NF1 (L). Then as NL (L) = T0 X , again E1  F1 by Theorem 1.5.2 in [Asc19]. So in any event, E1  F1 . We next show that E2  F2 . First NFc (Tc ) ≤ NFc (O(τ )) ≤ NF (z, c ), and from our proof that E1  F1 , either CT (z)  NFz (O(τ )) or m = 8 and CT (z)X  NFz (O(τ )). Then T  NFc (Tc ), so E2  F2 by Theorem 1.5.2 in [Asc19]. This completes the proof of (1). By 12.1.7, F = F1 , F2 . Hence the tuple Ei , Fi , i = 1, 2, satisfies Hypothesis 1.4.1 in [Asc19]. Therefore to establish (2), it suffices by 1.4.4 in [Asc19] to verify Hypothesis 1.4.3 in [Asc19]. Condition (1) of 1.4.3 is satisfied by (1). As NF (T ) ≤ NF (O(τ )) ≤ Fc and T is Sylow in E2  F2 , Σ = AutF (T ) ≤ Aut(E2 ). Similarly Σ permutes {z, t} and hence as E1  F1 , Σ permutes E1 , E1 , where E1 = E1 c∗x for x ∈ S − CS (z). Thus condition (2) of 1.4.3 of [Asc19] is satisfied by 1.4.6 in [Asc19]. Therefore it remains to verify the extension condition 1.4.3.4 of [Asc19]. As T is Sylow in E2 , the extension condition is trivially satisfied by E2 , so it remains to choose α ∈ AutF1 (T1 ) and show that α extends to T . But α acts on O(τ ) and hence centralizes c, so as E2  Fc and T is Sylow in E2 , α = ϕφ for some ϕ ∈ AutF (T ) and φ ∈ homE2 (T1 ϕ, T ). But T1 = CT (z, c ) is AutF (T )invariant, so T1 ϕ = T1 . Then as O(τ ) is weakly closed in S, z, c φ = z, c , so T1 φ = CT (z, c )φ = T1 . Finally T = NE2 (T1 ), so φ extends to ψ ∈ AutE2 (T ), and then ϕψ extends α to T , completing the proof.  Lemma 12.1.18. Let X = O 2 (CC (b)) and X Sylow in X . (1) X ∼ = L2 [m], L2 [2m] for C ∼ = P Sp4 [m], L− 4 [m], respectively. Further bt ∈ Z(X). ◦ ∼ + ◦ (2) Set ¯b = bt. Then bz = ¯bc ∈ bcO(t) , Fbc ). = Ω6 [m], and X ≤ CFbz (Fbz − − ◦ ◦ ∼ ∼ (3) C = CFv (z) and Fv = Ω7 [m], Ω8 [m] for C = P Sp4 [m], L4 [m], respectively. (4) T M ∼ = P Sp4 [m], L− = ω(C4 , m), 2ω(C4 , m) for C ∼ 4 [m], respectively. In particular M0 = T M and T are determined up to isomorphism. Proof. By 12.1.11 and 12.1.15, C is P Sp4 [m] or L− 4 [m], so (1) follows from the structure of C. We next prove (2). Visibly bz = ¯bc ∈ bcO(t) . By (1), X is a (solvable) component of CFz (b O), while CFz (b O) = CFbz (O) by 2.2 in [Asc10]. Now as ◦ ∼ + bz ∈ (bz)F , we have Fbz = Ω6 [m] by 12.1.11.3, 12.1.13.4, and 12.1.15. Thus from + ◦ ◦ (O)) = 1, so X ≤ F the structure of Ω6 [m], O 2 (CFbz bz . Then by 2.2.6.3, X cen◦ ∼ tralizes Fbz . Indeed unless X = L2 [8], X is a component of CFbz (z), so (2) follows from 10.11.3 in [Asc11]. So assume C ∼ = P Sp4 [8]. By 2.2.1.4, X  NCFbz (z) (X), so as X centralizes Tbz ◦ (O)  NF and Tbz O = NFbz (O), X centralizes Tbz . Therefore as X centralizes bz ◦ Fbz , (2) follows from 2.2 in [Asc11]. Similarly C = CFz (v)◦ and CFz (v) = CFv (z) by 2.2 in [Asc10], so C = CFv (z)◦ . As v ∈ bM , it follows from 12.1.11.4, 12.1.13.4, and 12.1.15 that Fv◦ ∼ = Ω7 [m] or Ω− [m]. By 12.1.5.1, vc ∈ M so as O(t) ≤ M and c = zt, also vz ∈ Tv . Then v v 8 [m]. We conclude that (3) holds. C = CFv (z)◦ ∼ = P Sp4 [m] or L− 4 Let r = bt be an involution in X, J1 = K, J2 ∈ K M with z(J2 ) ∈ / {z, t}, and J3 = K3 . From the structure of C, K3r = K4 . Therefore from 5.9.2.5, the subgroup

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M0 = J1 , J2 , J3 , r of G is M r and isomorphic to ω(C4 , m). If C ∼ = P Sp4 [m] then T = Tc r , so M0 = T M and (4) holds in this case. So assume C ∼ = L− 4 [m], and choose y ∈ X with y the cyclic subgroup of X of index 2; thus X = y, r and T = Tc T0 = Tc y, r . Working in Fv◦ ∼ = Ω− 8 [m] y 2 and using 9.5.24.7 and 9.5.29, we may choose y so that y = [r, v¯3 ], k3 = k3 v¯3 for v3 . Now by 5.9.3, M0 = J1 , J2 , J3 , y, r = M y, r ∼ k3 ∈ K3 − W and K3 ∩ W = ¯ = 2ω(C4 , m) with T Sylow in M0 , so that T M = M0 . This completes the proof of (4) and the lemma.  Notation 12.1.19. If C ∼ = P Sp4 [m] set F˜ = Ω9 [m], while if C ∼ = L− 4 [m] set − ˜ ˜ ˜ ˜ ˜ F = Ω10 [m]. Let S be Sylow in F and let τ˜ = (F , Ω) be the Lie packet of F˜ . Let ˜ = ˜ ˜ a model of N ˜ (W ˜ ). η˜ ∈ η(˜ τ ), W η , and G F ˜ ∼ ˜ = T M , S˜ = T , Ω ˜ = Ω, and Lemma 12.1.20. G = T M , so we may take G ˜ NF˜ (W ) = FT (T M ). Proof. By 12.1.18.4, there is an isomorphism α : T M → ω, where ω is ω(C4 , m) or 2ω(C4 , m), such that Ωα = Γ, where ρ = (W, Γ) is the quaternion fusion packet described in 4.3.7.5 and section 5.9 with ω the model for W. Simi˜ → T M with Ω ˜ = Γ. larly by 5.3.9.2, and 5.3.10.2, there is an isomorphism β : G ˜ → T M is an isomorphism with Ωγ ˜ = Ω, so identifying G ˜ with Thus γ = βα−1 : G T M via γ, the lemma holds.   Remark 12.1.21. We will show that F˜ = Y. Hence Y = O 2 (Y) is saturated, so as the proof of 12.1.17 showed that the tuple Fi , Ei , i = 1, 2 satisfies Hypothesis 1.4.3 in [Asc19], the tuple also satisfies the hypothesis of 1.4.5 in [Asc19]. Then by that lemma, Y  F, so F ◦ = Y = F˜ .

Lemma 12.1.22. Let P = O · Oc∗s for s ∈ S − CS (z). (1) T P˜ = T P. ˜ (2) Fc◦ ∼ = Ω+ 8 [m] and Fc = E2 . Proof. By 12.1.11.2, 12.1.13.4, and 12.1.15, Fc◦ ∼ = Ω+ 8 [m]. Then NE1 (z, c ) = CT (z)P as Oc∗g = O 2 (CC (t)), while by 10.1.15.2, E2 = NE1 (z, c ), FT (T M ) and ˜ ) . Therefore given 12.1.20, (1) implies (2). Thus it F˜c = NF˜c (z, c ), NF˜ (W remains to prove (1), which follows from Theorem 2.7.3.  ◦ ◦ ◦ ≤ Fc◦ , so F˜bz = Fbz . Lemma 12.1.23. (1) Fbz ◦ ˜ (2) Fbz = Fbz X . (3) F˜z = E1 . ◦ ∼ + Proof. By 12.1.18.2, B = Fbz = Ω6 [m] and X centralizes B. Then by 9.3.20, = Tb O, FTbz (Mbz ) , so as Tb O and FTbz (Mb ) are contained in Fc◦ , (1) follows using 12.1.22.2. Next T0 = (Tc ∩ T0 )X, so CT (bz) = CTc (bz)X = Tbz X. Then CT (bz) is Sylow in BX . As X is the fusion system on X generated by O 2 (Aut(U )) as U varies over the 4-subgroups of X, X˜ = X . Then by (1), ◦ F˜bz = CT (bz)F˜bz , CT (bz)X˜ = CT (bz)B, CT (bz)X = BX ,

◦ Fbz

proving (2). It remains to prove (3). First E1 = CT (z)OC = CT (z)O, CT (z)C and CT (z)O ≤ ˜ = CT (z)O. Therefore it suffices to show that W = T P, so from 12.1.22.1, CT (z)O

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∼ P Sp4 [m] or L− [m]. Therefore by 9.4.22, 9.4.27, ˜ Now C = CT (z)C = CT (z)C˜ = W. 4  and 9.4.28, W = CT (z)O , AutW (Ri ) : i ∈ I where O = Oc∗x for x ∈ S − CS (z) ˜  = CT (z)O . If and the Ri are suitable members of W e . By 12.1.24.1, CT (z)O C ∼ P Sp [m] then we are in case (3) or (4a) of 9.4.22, so the Ri are in R1 . If = 4 − C ∼ L [m] we are in case (1) of 9.4.22, so again the R are in R1 . Therefore = 4 i Ri = CT (z, t, b, s ) for suitable s ∈ Dm (t) ∩ tC . But then by 9.4.21, AutW (Ri ) centralizes b or bt, and hence also bz or bzt. Therefore by (2), AutW (Ri ) = AutW ˜ (Ri ), so (3) holds.  ∼ P Sp4 [m] then F ◦ ∼ Lemma 12.1.24. (1) If C = = Ω9 [m]. − ◦ (2) If C ∼ [m]. = L4 [m] then F ∼ = Ω− 10 Proof. By Remark 12.1.21 it suffices to show that Y = F˜ . But this is a consequence of 12.1.7.5, 12.1.22.2, and 12.1.23.3.  Theorem 12.1.25. Assume Hypothesis 12.1.1. Then one of the following holds: ˇ 3 (2)/23+6 , AE ˇ 8, Ω ˇ + (2), or (1) c ∈ Z(F) and either F ◦ = FT (M ) or F ◦ ∼ =L 8 + Ω8 [m]. (2) F ◦ ∼ = AE9 . (3) F ◦ ∼ = Ω9 [m]. (4) F ◦ ∼ = Ω− 10 [m]. Proof. If c ∈ Z(F) then (1) holds by 12.1.4.2. Thus we may assume Hypothesis 12.1.10 holds. Then by 12.1.13.3, either O ∼ = AE5 , = SL2 [m] ∗ SL2 [m] or C ∼ and in the latter case (2) holds by 12.1.14, so we may assume the former. Therefore Hypothesis 12.1.15 is satisfied. Then C is isomorphic to P Sp4 [m] or L− 4 [m] by 12.1.11.1, so (3) or (4) holds by 12.1.24. 

12.2. More |Ω(z)| = 2 In this section we assume Hypothesis 11.1.1 with n ≥ 5. Adopt Notation 11.1.2. In addition define C as in 11.1.5.1. Notation 12.2.1. Set O = O(z), S0 = CS (O), and if m = 8 let Gz be a model for NFz (OS0 ) and Lz = O 2 (W Gz ). If m > 8 set O = O(z) and if m = 8 set O = FO (OLz ). Similarly if m = 8 let Gτ be a model for NF (O(τ )CS (O(τ )), identify  NGz (O(τ )) with a subgroup of NGτ (OS0 ) using 2.2 in [Asc10], and set Lτ = t∈ZS (Lt ), where Lt = Lgz for g ∈ Gτ with z g = t. If m > 8 set Oτ = O(τ ), while if m = 8 set Oτ = FO(τ ) (O(τ )Lτ ). Set c = eI . Lemma 12.2.2. (1) If m > 8 then either O(z) = O or O(z) ∼ = SL2 [m]∗SL2 [m]. (2) If m = 8 then Lz centralizes S0 and either Lz = 1 or O is Sylow in Lz and Lz is of index 1 or 3 in SL2 (3) ∗ SL2 (3). (3) If m = 8 then O 2 (AutF (O)) = AutLz (O) and O  Fz . (4) If m = 8 then for each t ∈ ZS , Lt ∼ = Lz , Lτ centralizes CS (O(τ )), and AutLτ (O(τ )) = O 2 (Γ), where  Γ= NAutF (O(τ )) (O(t)). t∈ZS

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Proof. Parts (1) and (2) follow from 9.1.11. Suppose m = 8. For Vj ∈ Δ with zj ∈ D∗ (z) − D(z) and w a generator of Vj , w induces an outer automorphism on K and K  , so the automorphism cw induced on O by w inverts a Sylow 3-subgroup of AutF (O). Therefore the first statement in (3) holds, and the second follows from the first and 2.6.14.2. Then as O  Fz , the groups Lt , t ∈ ZS are well defined and isomorphic to Lz , and the remaining statements in (4) follow from (2) and (3).  Lemma 12.2.3. One of the following holds: (1) F = E(τ ). (2) n = 2k + 1 and F = E(τ ), Fx , where x is the element eI−{n} of WI ∩ F f of weight 2k . (3) n = 2k , c = 1, and F = E(τ ), Fc . (4) n = 2k , c = 1, and F = E(τ ), Fx , where x ∈ WI ∩ F f is of weight 2k−1 . Proof. Assume otherwise. By Theorem 11.3.11, F = B(τ ). Therefore by 11.3.2, F is generated by E = E(τ ) and the set of subsystems Fx , as x varies over the set Γ of involutions x ∈ WI ∩ F f of weight a positive power of 2. Thus there is x ∈ Γ with Fx ≤ E. In particular wt(x) = 2k for some k ≥ 1, while by 11.1.8, x satisfies one of conditions (2), (3), or (4) of that lemma. But now we conclude that the corresponding conclusion of our lemma holds.   Lemma 12.2.4. Let θ ⊆ ZS and set fθ = t∈θ t. (1) NF (O(θ)) ≤ CF (fθ ). (2) O(θ) ∈ F f iff fθ ∈ F f . (3) If O(θ) ∈ F f then NF (O(θ)) ≤ NF (O(τ )), NS (O(θ))CF (O(θ))◦ . (4) If fθ ∈ F f then CF (fθ ) ≤ CS (fθ )CF (fθ )◦ , NF (O(τ )) . (5) If m > 8 then NF (O(τ )) = SX , NF (XW ) , where X = CF (O(τ )) and X = O(τ )CS (O(τ )). (6) Suppose m = 8. Then NF (O(τ )) = SX , NF (XW ), SOτ . Proof. Set P = O(θ) and f = fθ . As AutF (P ) permutes θ, (1) holds. Further ˜ = Sym(I), or the image by 11.1.2, WI is the core of the permutation module for μ of that module modulo eI , so either CS (f ) = NS (P ), or n ≡ 0 mod 4, c = 1 and |θ| = n/4 with |S : NS (P )| = 2. In either case (2) holds by 11.1.3.7. Assume P ∈ F f ; then f ∈ F f by (2). Adopt the notation of 11.1.4 with f in the role of x. Then by 11.1.4.3, Ff◦ = F1 ∗ F2 with F2 = CF (P )◦ . By 1.3.2 in [Asc19], NF (P ) = NS (P )F2 , NF (T ) where T is Sylow in P F2 . Then NF (T ) ≤ NF (O(τ )) as O(τ ) ≤ T . This proves (3). Similarly by 1.3.2 in [Asc19], Ff = CS (f )Ff◦ , NFf (R) where R is Sylow in ◦ Ff . Then as O(τ ) ≤ R, NFf (R) ≤ NF (O(τ )), so (4) holds. Finally let Q = O(τ ), X = CF (Q), and X = QCS (Q). Recall from 12.2.1 that G = Gτ is a model for NF (X). Observe that FS (SCG (Q)) ≤ SX . By 1.3.2 in [Asc19], NF (Q) = SX , NF (X) , so to prove (5) and (6) it suffice to show W  G in (5) and NF (X) = SOτ , NF (XW ) in (6). Let Y be the kernel of the action of G on Σ = {O(t) : t ∈ ZS }. Then W ≤ Y , so G = Y NG (W ) by a Frattini argument. Suppose m > 8. Then Aut(O) is a 2-group, so O 2 (Y ) ≤ H = CG (Q). Set SQ = CS (Q). Then [W, SQ ] ≤ W ∩ SQ ≤ CSQ (W ), so W is quadratic on SQ . Therefore Φ(W ) centralizes SQ /Φ(SQ ), so [Φ(W ), H] ≤ SQ and hence Φ(W )SQ  W H. Let η0 = {Φ(V ) : V ∈ Δ−Ω}. Then as m > 8, H permutes η0 , so H acts on W1 = η0 .

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Then as W centralizes W1 , we have [H, W ] ≤ CH (W1 ). But CH (W1 ) acts on each member of Δ, so O 2 (CH (W1 )) ≤ CH (W ). Therefore O 2 (Y ) ≤ H ≤ NG (W ), so G = Y NG (W ) = NG (W ), completing the proof of (5). So assume m = 8. Then by 9.1.11.2, CG (Q) ≤ NG (W ). Therefore G =  Lτ NG (W ), so NF (X) = SOτ , NF (XW ) , establishing (6). Lemma 12.2.5. The following are equivalent: (1) c ∈ Z(Fz ). (2) cz ∈ Z(C). ˇ 2n−4 , or Ω+ [m], or n = 6 and C ∼ (3) C is isomorphic to ω(Dn−2 , m), AE = 2n−4 + 3+6 ˆ ˆ L3 (2)/2 or Ω8 (2). Proof. First c ∈ Z(Fz ) iff cz ∈ Z(Fz ). Further from 11.1.5.2, cz ∈ C, so if c ∈ Z(Fz ) then cz ∈ Z(C). But from 11.1.5.3, |Z(C)| ≤ 2, so cz ∈ Z(C) iff Z(C) = cz iff cz ∈ Z(Fz ). Thus (1) and (2) are equivalent. Also from 11.1.5.3, Z(C) = 1 iff C is listed in (3), in which case Z(C) = Z(M2 ) = cz by 11.1.5.2 and the structure of the systems in (3). Therefore (2) and (3) are equivalent.  Lemma 12.2.6. If c ∈ Z(Fz ) then c ∈ Z(F). Proof. We may assume 1 = c ∈ Z(Fz ). By definition of c, c = Z(M ), so as FS∩M (M )  NF (W ), c ∈ Z(NF (W )). As c ∈ Z(S), c centralizes SO(τ ) by 2.6.12. Let x = fξ ∈ F f for some ∅ = ξ ⊆ ZS . Then x = eθ for some θ ⊆ I with k = |θ| = 2|ξ|. Claim that Fx ≤ Fc . If k = n then x = c and the claim is trivial. Assume k = n − 1. Then by 11.1.4.3, Fx◦ = F1 ∗ F2 with x = Z(F1 ). Observe that as c = 1, cx = x, so even if k = n/2, we have Fi  Fx for i = 1, 2. Now |θ2 | = n − k > 1, so M2 = 1 and cx = Z(M2 ) by 11.1.4.5. Let t ∈ ξ ∩ F1f . Then CF1 (t)◦ = O(t) ∗ C1 . From 11.1.3.7, there is g ∈ M with α = c∗g ∈ A(t) and tα = z. Then (C1 F2 )α∗ ≤ C is centralized by tα = z, and hence also by c as c ∈ Z(Fz ). But O(τ )α = O(τ ), so α centralizes c. Hence ct centralizes C1 F2 , so c centralizes F2 and then as cx ∈ M2 we have cx = Z(F2 ). Thus cx ∈ Z(Fx ), so c ∈ Z(Fx ), establishing the claim in this case. We interject a short digression before completing the proof of the claim. As c ∈ Z(Fx ), also c ∈ Z(NF (O(ξ)) by 12.2.4.1; that is c ∈ Z(NF (O(ξ))) for each ξ ⊆ ZS with |ξ| = (n − 1)/2. Therefore by 3.4.11.2, c ∈ Z(E(τ )) unless possibly / Z(Fx ) for x ∈ WS of weight n − 1 in WI . Hence if n = 2a + 1 n = 2a + 1 and c ∈ then c ∈ Z(F) by 12.2.3, so we may assume n = 2a + 1. Finally assume n = k + 1, and set Y = Fx◦ . By 12.2.4.4, we have Fx ≤ CS (x)Y, NF (O(τ )) . By parts (5) and (6) of 12.2.4, NF (O(τ )) = SCF (O(τ )), N , where N = NF (XW ) and X = O(τ )CS (O(τ ), or m = 8 and N = NF (XW ), SOτ . Now CF (O(τ )) ≤ Fz ≤ Fc , c ∈ Z(NF (W )), and O(τ ) centralizes CS (O(τ )) by 12.2.2.4, so NF (O(τ )) centralizes c. Therefore E(τ ) ≤ Fc by 3.4.11.2, and it remains to show that Y centralizes c. Recall that n − 1 = 2a Set Y + = Y/x and τx+ = (Y + , Ω+ ). By 12.2.3, Y + = E(τx+ ), NY + (y + ) where y ∈ WS is of weight (n − 1)/2 in WI . Therefore Y = E(τx ), NY (Y ) , where Y = x, y . As we saw earlier, E(τ ) centralizes c, so E(τx ) ≤ E(τ ) ≤ Fc . Also NY (Y ) = CY (Y )◦ Tx for Tx Sylow in Y, and CY (Y )◦ ≤ Fy◦ ≤ Fc , so indeed Y, and hence also Fx , centralizes c.

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We’ve shown that E(τ ) centralizes c. Hence c ∈ Z(F) by 12.2.3 unless n = 2a + 1, where F = E(τ ), Fx for x of weight n − 1 in WI . But we also saw that c ∈ Z(Fx ), completing the proof.  Because of 12.2.6 and 11.1.6, during the remainder of the section we assume: Hypothesis 12.2.7. If c ∈ Z(F) then c = 1. Lemma 12.2.8. Let m = 8, x = eθ ∈ WI ∩ F f , and define the factorization = F1 ∗ F2 as in 11.1.4.3. Take z = zi and assume {i, i + 1} ⊆ θj for some j, so that O ≤ Fj . Assume |θj | ≥ 3. (1) O 2 (AutF (O)) = O 2 (AutFj (O)). (2) One of the following holds: (i) Fj ∼ = ω(Dk , m), AutF (O) is a 2-group, and O = O. ˇ k or AEk and O is of index 3 in SL2 [8] ∗ SL2 [8]. (ii) Fj ∼ = AE ∼ (iii) Fj = Ωk [8] and O ∼ = SL2 [8] ∗ SL2 [8].

Fx◦

Proof. By hypothesis, x ∈ S0 , so Lz centralizes x by 12.2.2.2. Hence Σ = O 2 (AutF (O)) = AutLz (O) = O 2 (AutFx (O)) = Σx by 12.2.2.3. Suppose Fj ∼ = ω(Dk , m). Then W ∩ O  CFx (z), so Σx is a 2-group, and hence O = O, so that (1) holds in this case, as does (2i). Suppose next that Fj ∼ = Ωk [8]. Then Σx ∼ = SL2 (3)∗SL2 (3) = O 2 (Aut(O)) = Γ, so (1) holds and O ∼ = SL2 [8] ∗ SL2 [8], so (2iii) holds. ˇ k or AEk . Then Σj = O 2 (AutF (O)) is of index 3 in SL2 (3) ∗ Suppose Fj is AE j SL2 (3). Further E = O ∩ O2 (Fj ) ∼ = E8 is Σx -invariant, so Σ = Σx = Γ, so Σ = Σj and O is of index 3 in SL2 [8] ∗ SL2 [8]; in particular (1) holds, as does (2ii). If a = |θj | ≥ 5 then by induction on n (cf. Theorem 12.2.32), Fj is in one of the three generic classes of examples considered already, so (1) holds in this case. Therefore we may assume a = 3 or 4 and Fj is not generic. Suppose a = 4. Then Fj is described in Theorem 12.1.25, and hence as Fj is ˇ 3 (2)/23+6 or O ˇ + (2). Then (cf. 12.1.12 and the proof of 12.1.13) not generic, Fj ∼ =L 8 3 Lz is of index 3 or 3 in (SL2 (3) ∗ SL2 (3))2 , contrary to 12.2.2.2. Finally suppose a = 3. Then Fj is described in Theorem 9.5.33, so Fj is L3 (2)/E64 or Sp6 (2). But then (cf. 9.5.11.3, 9.5.15, and 9.5.16) |Σ|3 = 3 or 33 and O is not Sylow in Lz , contrary to 12.2.2.2. This completes the proof of (1), and in the process we have also proved (2).  Lemma 12.2.9. Let vz , vz be elements of K ∩W, K  ∩W of order 4, respectively, and set v = vz vz . (1) If n is odd then v(n) ∈ F f is of weight 1. (2) If n is even then v ∈ F f is of weight 1. (3) Let x ∈ WI ∩ F f be of weight 1 and g ∈ M such that α = c∗g ∈ A(v) with vα = x. Then Cα∗ = CF (x, zα )◦ = CFx◦ (zα)◦ and one of the following holds: (i) Fx◦ ∼ = ω(Dn−2 , m), c = 1, and z ∈ C. = ω(Dn−1 , m), C ∼ ˇ 2n−4 , c = 1, and z ∈ C. ˇ 2n−2 , C ∼ (ii) Fx◦ ∼ = AE = AE (iii) Fx◦ ∼ / C. = AE2n−1 , C ∼ = AE2n−3 , c = 1, and z ∈ + ∼ (iv) Fx◦ ∼ [m], C Ω [m], c = 1, and z ∈ C. = Ω+ = 2n−2 2n−4 / C. (v) Fx◦ ∼ = Ω2n−3 [m], c = 1, and z ∈ = Ω2n−1 [m], C ∼ − ∼ (vi) Fx◦ ∼ [m], C Ω [m], c =  1, and z ∈ / C. = = Ω− 2n 2n−2

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Proof. Part (1) follows from 11.1.7.2. Suppose n is even. In Notation 11.1.2 ˜ = Sym(I). Then ζ(dj ) = ej,j+1 with z = ei,i+1 for we chose d˜j = (j, j + 1) ∈ μ some i by 11.1.3.1, and v = ei or ei+1 by 11.1.3.2. Thus, using 11.1.3.5, CS (v) is of index 2 in CS (z), and as z ∈ F f , also v ∈ F f , proving (2). Pick x as in (3); by 11.1.3.7 it is possible to pick g and α as in (3). By 2.2 in [Asc10], CFz (v)α∗ = CF (z, v )α∗ = CF (zα, x ) = CFx (zα), so as C = CF (z, v )◦ , Cα∗ = CF (x, zα )◦ = CFx◦ (zα)◦ . Define ρ = (C, Ω − Ω(z)) as in 11.1.5; by 11.1.5.1, μ(ρ) ∼ = Weyl(Dn−2 ). Now Fx◦ is described in 11.1.7.3, and from the structure of the systems listed there, either CFx◦ (zα)◦ is as described in (3), or one of the following holds: ˇ 3 (2)/23+6 , and C ∼ (vii) n = 5, Fx◦ ∼ = ω(D3 , m). =L + ∼ ˇ ˇ (viii) n = 5, Fx◦ ∼ Ω (2), and C AE = 8 = 6. By Hypothesis 12.2.7, c ∈ Z(F) iff c = 1, while by 12.2.6, c ∈ Z(F) iff c ∈ Z(Fz ). Then by 12.2.5, c = 1 iff cz ∈ Z(C) iff z ∈ C. Moreover, unless one of the exceptional cases (vii) or (viii) holds, the cases where cz ∈ Z(C) are listed in 12.2.5.3. Thus to complete the proof of (3) it remains to show that neither (vii) nor (viii) holds. But this follows from 12.2.8.2.  Lemma 12.2.10. If O = O then F ◦ = FS (M ). Proof. Assume O = O and set Q = O(τ ). By 12.2.8.2 and 12.2.9.3, case (i) of 12.2.9.3 holds. Choose x as in 12.2.9.3; then Fx◦ ∼ = ω(Dn−1 , m), C ∼ = ω(Dn−2 , m), c = 1, and z ∈ C. As O = O, O(τ ) = Q, so SO(τ ) = S ≤ NF (W ). Also by 12.2.2.3, AutF (O) is a 2-group, so by parts (5) and (6) of 12.2.4, NF (Q) = SCF (Q), NF (QW ) . Now CF (Q) ≤ Fz1 ∩Fzn−1 . We may choose notation so that z = z1 . As C ∼ = ω(Dn−2 , m), Fz◦ = O ∗ C = Mz with Wz = V1 V1 · V3 V4 · · · Vn−1 Vn and CF (Q) ≤ NF (Wz ). Similarly Wzn−1 = V1 V2 · · · Vn−3 Vn−1 Vn , and CF (Q) ≤ NF (Wzn−1 ). Then as W = Wz Wzn−1 , CF (Q) ≤ NF (W ). Of course NF (W Q) ≤ NF (W ), so NF (Q) ≤ NF (W ). Let θ ⊆ ZS with O(θ) ∈ F f and define f = fθ as in 12.2.4. By 12.2.4.2, f ∈ F f , so by 12.2.4.4, Ff ≤ CS (f )Ff◦ , NF (Q) . By 12.2.8, CS (f )Ff◦ = FCS (f ) (CS (f )Mf ) ≤ NF (W ), so Ff ≤ NF (W ). Further by 12.2.4.1, NF (O(θ)) ≤ Ff ≤ NF (W ). Therefore E(τ ) ≤ NF (W ). If n = 2k and x ∈ WS ∩F f is of weight 2k−1 in WI then x = fθ where |θ| = n/4, so Fx ≤ NF (W ). Further c = 1, so by 12.2.3, F = E(τ ), Fx ≤ NF (W ). Similarly if n = 2k +1 and x = eI−{n} then x = fθ where |θ| = (n−1)/2, so Fx ≤ NF (W ) and then F ≤ NF (W ) by 12.2.3. Finally in the remaining cases, F = E(τ ) ≤ NF (W ) by 12.2.3. Hence in any case W  F, so the lemma follows from 3.3.13.3.  ˇ 2n−4 then F ◦ ∼ Lemma 12.2.11. (1) If C ∼ = AE2n . = AE ◦ (2) If C ∼ = AE2n+1 . = AE2n−3 then F ∼ ˇ 2n−4 or AE2n−3 . Then m = 8 and applying 12.2.8 Proof. Assume C is AE to z = x ∈ ZS , O is of index 3 in SL2 [8] ∗ SL2 [8]. Hence there is a unique W Oinvariant E8 -subgroup E(z) of O. As AutW O (O)  AutF (O), E(z)  Fz . Thus for t ∈ ZΔ , we can define E(t) = E(z)g for g ∈ M with c∗g−1 ∈ A(t) and z g = t. Set U = E(t) : t ∈ ZΔ . Thus U  M , and using 11.1.3.5, also U  NF (W ).

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Next for x ∈ WI ∩ F f we obtain the factorization Fx◦ = F1 ∗ F2 of 11.1.4.3, ˇ k or AEk . Hence from and if j ∈ {1, 2} with |θj | ≥ 3, then by 12.2.8, Fj ∼ = AE the structure of such systems, O2 (Fj ) = E(t) : t ∈ Z(τj ) . Therefore O2 (Fj ) ≤ U and indeed U = CUFj (O2 (Fj ))  U Fj . Thus for each θ ⊆ ZS and fθ ∈ F f , CF (fθ )◦ ≤ NF (U ). Similarly U = CULz (E(z))  U Lz , so also U  U Lτ and hence U  U Oτ . We may choose notation so that z = z1 . Observe ZΔ = Z(τz ) ∪ Z(τz )g for g ∈ M with z g = zn−1 . Thus U = O2 (C)O2 (C)g , so in particular CF (O(τ )) ≤ NF (U ). Then by 12.2.4.6, NF (O(τ )) ≤ NF (U ). Choose f = fθ as above. We’ve seen that CS (f )Ff◦ and NF (O(τ )) act on U , so Ff ≤ NF (U ) by 12.2.4.4. Then NF (O(θ)) ≤ NF (U ) by 12.2.4.1. Hence E(τ ) ≤ NF (W ), and then U  F by 12.2.3. Now the lemma follows from Theorem 2.  In light of 12.2.8.2, 12.2.10, and 12.2.11, it remains only to consider cases (iv), (v), and (vi) of 12.2.9.3, where from 12.2.8 we have O ∼ = SL2 [m] ∗ SL2 [m]. Thus for the remainder of the section we assume: Hypothesis 12.2.12. Hypothesis 12.2.7 holds and O ∼ = SL2 [m] ∗ SL2 [m]. Equivalently, as we have already observed, by 12.2.9.3: Lemma 12.2.13. Let x ∈ WI ∩ F f be of weight 1. Then one of the following holds: ∼ + (1) Fx◦ ∼ = Ω+ 2n−2 [m], C = Ω2n−4 [m], c = 1, and z ∈ C. ◦ ∼ / C. (2) Fx = Ω2n−1 [m], C ∼ = Ω2n−3 [m], c = 1, and z ∈ − ∼ (3) Fx◦ ∼ [m], C Ω [m], c =  1, and z ∈ / C. = = Ω− 2n 2n−2 Notation 12.2.14. Let Ξ be the set of nonempty subsets ξ of ZS such that either  (a) ξ is a union of orbits of S on ZS , fξ = t∈ξ t = 1, and if ξ = ZS then S is transitive on ZS , or (b) n is power of 2, c = 1, and |ξ| = n/4 with O(ξ) ∈ F f . For ξ ∈ Ξ, define Bξ (τ ) = CF (fξ ), NF (W ) . Lemma 12.2.15. Let ξ ∈ Ξ, f = fξ , and B = Bξ (τ ). (1) f ∈ Z(S). (2) NF (O(τ )) ≤ B. (3) SO(τ ) ≤ B. (4) Let θ ⊆ ZS with Q = O(θ) ∈ F f , and let T be Sylow in NF (Q). Let Σ be a nontrivial orbit of T on θ and g ∈ M such that α = c∗g ∈ A(O(Σ)). Then if T αCF (O(Σ)α)◦ ≤ B, we have NF (Q) ≤ B. Proof. In case (a) of 12.2.14, ξ is S-invariant, so f ∈ Z(S). In case (b), O(ξ) ∈ F f , so S acts on {O(ξ), O(ZS − ξ), and as c = 1, we have f = fZS −ξ . Hence f ∈ Z(S), so that (1) holds. Set P = O(τ ). Then SCF (P ) ≤ Ff ≤ B, and SO(τ ) ≤ Ff ≤ B, so (2) follows from parts (5) and (6) of 12.2.4. Similarly SO(τ ) ≤ Ff , so (3) holds.

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Assume the hypothesis of (4). By 12.2.4.3, NF (Q) ≤ NF (P ), T CF (Q)◦ and NF (P ) ≤ B by (2), so to prove (4) it remains to show that T CF (Q)◦ ≤ B. But as α is a B-map and (T CF (Q)◦ )α∗ ≤ T αCF (O(Σ)α)◦ , T CF (Q)◦ ≤ B if  T αCF (O(Σ)α)◦ ≤ B. Lemma 12.2.16. Let ξ ∈ Ξ, f = fξ , and B = Bξ (τ ). Then either (1) F = B, or (2) S is transitive on ZS , ξ = ZS , and F = B, NF (O(θ))◦ , where θ ⊆ ZS is of order |ZS |/2 and O(θ) ∈ F f . Proof. Assume F = B and set Γ = ZS . Claim NF (O(θ)) ≤ B for some θ ⊆ Γ with O(θ) ∈ F f . For if not, then E(τ ) ≤ B by 12.2.15.3. Then by 12.2.3, one of the exceptional cases (2)-(4) of that lemma holds. However in each case S is transitive on Γ, so either ξ = Γ or case (4) holds and ξ is in 12.2.14.b. In each case f is the element appearing in 12.2.3 such that F = E(τ ), Ff , so that F = B, contrary to assumption. This establishes the claim. By the claim and 12.2.15.4, there is θ ⊆ Γ such that Q = O(θ) ∈ F f , θ is an orbit of a Sylow group T of NF (Q), and Y = CF (Q)◦ ≤ B. Let ξ  = Γ − ξ. In case (a) of 12.2.14, θ ⊆ ξ or ξ  , say θ ⊆ ξ. Further if θ = ξ then Y ≤ Ff ≤ B, a contradiction. So θ ⊂ ξ in case (a). Similarly in case (b), conjugating in M we may take θ ⊂ ξ. ¯ and τ¯ = (T Y, Ω). ¯ = {J ∈ Ω : z(J) ∈ θ}, ¯ From 11.1.4, τ¯ is a Set θ¯ = Γ − θ, Ω ∼ quaternion fusion packet with μ(¯ τ ) = W eyl(Da ), where a = n − 2|θ|. Observe that ¯ = a/2 or (a − 1)/2 for n even, odd, respectively. |θ| Set M Γ = NM (Γ)/CM (Γ); as WI is the core of the permutation module for μ ˜ = Sym(I), or the image of that module modulo its center, we conclude that M Γ = Sym(Γ). Therefore Γ = Γ1 ∪· · ·∪Γk , where |Γi | = 2ai with 0 ≤ a1 < · · · < ak and S Γ = S1Γ × · · · × SkΓ , where SiΓ is transitive on Γi and fixes the remaining orbits pointwise. Next θ ⊆ Σ an orbit of S on Γ; thus Σ = Γi for some i. As Q ∈ F f , i is minimal subject to |θ| ≤ 2ai , and θ S is a partition of Σ. Thus in case (a), Sj ≤ T for j = i and |Σ| = |S : T ||θ|. In case (b), |ξ| = |NS (ξ) : T ||θ|. ¯ Then as T acts on ξ  and ξ − θ with θ = ξ, Suppose T is transitive on θ.  we conclude ξ = ∅, so ξ = Γ. Thus by 12.2.14, case (a) of 12.2.14 holds with S transitive on Γ; in particular Σ = Γ. Further T is transitive on ξ − θ, so as Σ = Γ and |Σ| = |S : T ||θ|, it follows that |S : T | = 2 = |Γ|/|θ|, and hence 12.2.16.2 holds. ¯ Therefore we may assume T is not transitive on θ. ¯ ¯ = (4, 2) Suppose a < 5. Then as |θ| = a/2 or (a−1)/2, we conclude that (a, |θ|) ¯ (3, 1), or (2, 1), and in each case the stabilizer T of θ in S is transitive on θ, contrary to the previous paragraph. Therefore a ≥ 5. ¯ by induction on ¯ As a ≥ 5 and as T is not transitive on θ, Let ξ¯ = ξ ∩ θ. ◦ τ ). However Bξ¯(¯ τ ) ≤ Ff , NF (W ) ≤ B, contradicting Y ≤ B. This n, Y = Bξ¯(¯ completes the proof.  Notation 12.2.17. Let ξ ∈ Ξ and assume either (I) F = Bξ (τ ) and set f = fξ , or (II) F = Bξ (τ ), so that (by 12.2.16) S is transitive on ξ = ZS . Here set f = fθ , where θ ⊆ ZS is of order |ZS |/2 and O(θ) ∈ F f .

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In each case f = eθ1 for some θ1 ⊆ I of even order k; indeed k = 2|ξ| in (I) while k = 2|θ| in (II). Set θ2 = I − θ1 and define the factorization Ff◦ = F1 ∗ F2 as in 11.1.4. Let Ti be Sylow in Fi for i = 1, 2. Lemma 12.2.18. (1) μ1 ∼ = Weyl(Dk ) and F1 ∼ = Ω+ 2k [m] with f = Z(F1 ) and T1 ≤ M1 . (2) μ2 ∼ = Weyl(Dn−k ) and one of the following holds: (i) 12.2.13.1 holds, F2 ∼ = Ω+ 2(n−k) [m], and T2 ≤ M2 . (ii) 12.2.13.2 holds, F2 ∼ = Ω2(n−k)+1 [m], and |T2 : T2 ∩ M2 | = 2. (iii) 12.2.13.3 holds, F2 ∼ = Ω− 2(n−k)+2 [m], and |T2 : T2 ∩ M2 | = 4. (3) If k = 2|ZS | then k is a power of 2 and n = k + 1. (4) If k = 2|ZS | then either n − k ≥ 3 or n = k + 2 is even and k ≥ 4. (5) Either f ∈ Z(S) or 12.2.17.II holds and |S : CS (f )| = 2. Proof. By 11.1.4.5, μ1 ∼ = Weyl(Dk ), μ2 ∼ = Weyl(Dn−k ), f = Z(M1 ), f c = Z(M2 ). In particular f ∈ Z(F1 ), so as O ∼ = SL2 [m] ∗ SL2 [m], F1 ∼ = Ω+ 2k [m] by induction on n. By 5.3.8, T1 ≤ M1 . Thus (1) holds. Similarly as μ2 ∼ = Weyl(Dn−k ), we conclude by induction on n that F2 is  Ω2(n−k)+a [m] for some 0 ≤ a ≤ 2 and suitable . Let y ∈ WI ∩ F1f be of ◦ weight 1; then C1 = CF1 (y)◦ ∼ = Ω+ 2k−2 [m] and CF (y, f ) = C1 ∗ F2 . By 11.1.3.7 ∗ there is g ∈ M such that α = cg ∈ A(y). Set x = yα. By 2.2 in [Asc10], CF (y, f )α∗ = CFx (f α), so C1 α∗ ∗ F2 α∗ = CFx (f α∗ )◦ = CFx (w)◦ , where w = f αx ∈ Fx◦ is of weight k − 1. Therefore as μ(τx ) ∼ = Weyl(Dn−1 ), CFx (w)◦ = G1 ∗ G2 with ρi = (Gi , Γi ) a quaternion fusion packet, μ(ρ1 ) ∼ = Weyl(Dk−1 ), and μ(ρ2 ) ∼ = ∗ ∗ Weyl(Dn−k ). Therefore G1 = C1 α and G2 = F2 α . Then G1 ∼ = Ω+ 2k−2 [m], so as Fx◦ ∼ = Ω2(n−2)+a [m] by 12.2.13, it follows that G2 ∼ = Ω2(n−k)+a [m]. By 5.3.8, 5.3.9, a and 5.3.10, |T2 : T2 ∩ M2 | = 2 , establishing (2). Suppose k = 2|ZS |. Then 12.2.17.I holds with ξ = ZS , so by 12.2.14, S is transitive on ZS . Thus k is a power of 2 and n = k or k + 1. In the latter case (3) holds. In the former, c = f = 1, so c ∈ / Z(F) by 12.2.7. But as 12.2.17.I holds, F = Bξ (τ ) = Ff , NF (W ) , whereas f = c generates Z(M )  NF (M ), so F = Ff , contradicting c ∈ / Z(F). Hence (3) holds. Suppose k = 2|ZS |. In 12.2.19.II, k = |ZS | and n = 2k or 2k + 1, so n − k = k or k + 1. If n = 2k then n is a power of 2 so n ≥ 8 and n − k = k ≥ 4. If n = 2k + 1 then as n ≥ 5, n − k = k + 1 ≥ 3. Thus (4) holds in 12.2.17.II, so assume 12.2.19.I holds. Then f = fξ and by assumption, ξ = ZS , so |θ2 | = 2|ZS − ξ| ≥ 2. Finally n − k = |θ2 | or |θ2 | + 1 for n even or odd, respectively, so (4) holds. In 12.2.17.I, f = fξ and ξ is S-invariant, so f ∈ Z(S). In 12.2.17.II, f = fθ with |S : NS (θ)| = 2. Thus (5) holds.  Notation 12.2.19. Let TM = S ∩ M be Sylow in M , and let Tf be Sylow in Ff◦ . Set T = TM Tf . Recall from Notation 11.1.2 that {Ki : i ∈ I} ⊆ K M and Vi = Ki ∩W ∈ η with W = V1 · · · Vn . Recall from 11.1.4 that for i = 1, 2, Wi = ηi . Lemma 12.2.20. (1) |W : W1 W2 | = 2. (2) CT (f )Ff◦  Ff . (3) Ff = CS (f )Ff◦ , NFf (W ) . Proof. Let b = 2, 1 for c = 1, c = 1, respectively and set U = W1 W2 . From 4.3.7.2, |W1 | = (m/2)k /2, |W2 | = (m/2)n−k /2, and |W | = (m/2)n /2b. Further

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W1 ∩ W2 ≤ Z(F1 ) = f , so W1 ∩ W2 = f , 1, for b = 2, 1, respectively. Therefore |U | = (m/2)n /4b = |W |/2, proving (1). We next prove (2). By 1.5.2 in [Asc19], it suffices to show that CT (f )  N = NFf (Tf ). As U is weakly closed in Tf with respect to Ff , N acts on U . Observe that there is a unique i ∈ I with Vi ≤ U , and by (1) |Vi : U ∩ Vi | = 2, so si = z(Ki ) ∈ U . Let X = CFf (Ff◦ ), X Sylow in X , and α ∈ A(si ) with si α = z. Then Y = Vi Vi acts on X with |Y : Y ∩ U | = 2 and Y ∩ U centralizes X , while Oz  Fz , so [X, Y ] ≤ Y ∩ U ∩ X = 1 and then Y centralizes X by 9.5.2 in [Asc11]. Therefore W = Y U centralizes X . Now F ∗ (N ) ≤ O2 (N )X so as W centralizes X and [W, O2 (N )] ≤ U  N , it follows that W ≤ O2 (N ). Then as W is weakly closed in S with respect to F, W  N . Then as FTM (M )  NF (W ), O2 (M )  N . Finally μ ˜ = M/O2 (M ) ∼ = Sn and Nμ˜ (T˜f ) ≤ Nμ˜ (T˜M ), so CT (f )  N , completing the proof of (2). Further by 1.3.2 in [Asc19], Ff = CS (f )Ff◦ , N , and we showed W  N , so (3) holds too.  Lemma 12.2.21. Assume 12.2.13.1 holds and let G1 = Ff , G2 = NF (W ), E1 = CT (f )Ff◦ , E2 = FT (M ), and Y = T Oτ , FT (M ) . If case II of 12.2.17 holds set x = fξ , G3 = Fx , and E3 = T Fx◦ . Set Σ = {1, 2} in case I and Σ = {1, 2, 3} in case II of 12.2.17. (1) F = Gσ : σ ∈ Σ . (2) Eσ  Gσ for each σ ∈ Σ. (3) Y ∼ = P Ω+ 2n [m]. (4) Y = Eσ : σ ∈ Σ . (5) T = TM , and T is Sylow in Ei for i = 1, 2, unless case II of 12.2.17 holds, where CT (f ) is of index 2 in T . Proof. As 12.2.13.1 holds, c = 1. By 3.3.13, E2  G2 , and by 12.2.20.2, E1  G1 . Thus (2) holds in case I. Further in case I, f = fξ and F = Bξ (τ ) = G1 , G2 by 12.2.16, so (1) also holds in this case. Suppose case II holds. Then S is transitive on ξ = ZS and by 12.2.16, F = Fx , Ff , NF (W ) , so (1) holds. To complete the proof of (2), we must show that E3  G3 . From 12.2.18.3, n = 2|ZS | + 1 is odd. As c = 1, x is of weight 1, so from 12.2.9.1 we may take x = v(1) = e1 . Now Wx = V2 · · · Vn and W = Wx V1 . Set W4 = V4 · · · Vn , Γ = {K2 , K2 }, and ρ = (CF (W4 ), Γ). Then Γ = CΩ (W4 ), so ρ is a quaternion fusion packet by 2.5.2. Also μ(ρ) ∼ = = Cμ (W4 ) = d1 , d2 , d2 ∼ Weyl(D3 ), and we may take K = Kn , so ρ is a subpacket of C. Therefore as   12.2.13.1 holds, X = CF (W4 )◦ ∼ = Ω+ 6 [m]. Then CX (Wx ) = CX (V2 V2 ) = V1 V2 V2 , so W = V1 Wx  CFx (Wx ). By 1.3.2 in [Asc19], Fx = SFx◦ , NFx (Tx ) . Also NFx (Tx ) ≤ NFx (Wx ) ≤ NFx (W ). Then as T = Tx O2 (M ), T  NFx (Tx ), so E3 = T Fx◦  Fx = G3 by 1.5.2 in [Asc19]. This completes the proof of (2). Next Ff◦ = F1 F2 . Then Tf = T1 T2 , so by parts (1) and (2) of 12.2.18, Tf ≤ M1 M2 ≤ M . Hence T = Tf TM = TM . ˜ Ω) ˜ the Lie packet for F˜ , η˜ ∈ η(˜ ˜ = (F, τ ), etc. Let Let F˜ = P Ω+ 2n [m], τ ∼ ∼ ˜ ω = ω(Dn , m). By 5.3.5.4, NF˜ (W ) = ω/Z(ω), and as c = 1, M = ω/Z(ω) from ˜ ), and as T = TM is Sylow in M , T = S˜ and 11.1.2. Thus we may take M = NF˜ (W ˜ ˜ ˜ Ω = Ω, etc. By 11.5.3, F = T Oτ˜ , FT (M ) , and by Theorem 2.7.3, T Oτ˜ = T Oτ . Therefore Y = F˜ ∼ = P Ω+ 2n [m], proving (3).

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By definition T = TM is Sylow in M , so T is Sylow in E2 . If case I holds then f ∈ Z(S) by 12.2.18.5, and as Tf ≤ T , T is Sylow in T Ff◦ = E1 . In case II, |TM : CTM (f )| = 2, so |T : CT (f )| = 2, completing the proof of (5). ∼ + From 12.2.18, F1 ∼ = Ω+ 2k [m] and F2 = Ω2(n−k) [m]. Therefore by 11.5.3, E1 = CT (f )Ff◦ = CT (f )Oτ , FCT (f ) (CT (f )Mf ) ≤ Y. Hence in case I where T = CT (f ), we have Y = E1 , E2 . Therefore (4) holds in case I, so assume case II holds. There E3 = T Fx◦ with Fx◦ ∼ = Ω+ 2n−2 [m], so again by 11.5.3, E3 = T Oτ , FT (T Mx ) ≤ Y,  and hence Y = E1 , E2 , E3 , completing the proof. Lemma 12.2.22. If 12.2.13.1 holds then F ◦ ∼ = P Ω+ 2n [m]. Proof. We show that Y  F. Then F ◦ = Y, and 12.2.21.3 completes the proof. To prove Y  F, we verify that the tuple Eσ , Gσ , σ ∈ Σ, satisfies Hypothesis 1.4.3 in [Asc19], and then appeal to 1.4.5 in [Asc19]. By 12.2.21.1, F = Gσ : σ ∈ Σ , so Hypothesis 1.4.1 in [Asc19] is satisfied. By 12.2.21.2, Eσ  Gσ for each σ ∈ Σ, so 1.4.3.1 in [Asc19] is satisfied. As T = TM and W and O(τ ) are normal in NF (TM ), AutF (T ) is contained in Aut(T Oτ ) and Aut(FT (M )), so 1.4.3.2 in [Asc19] follows from 1.4.6 in [Asc19]. Except in case II with σ = 1, T is Sylow in Eσ , so the extension condition 1.4.3.3 in [Asc19] is satisfied. Thus Hypothesis 1.4.3 in [Asc19] is satisfied except possibly in case II.  Finally Y is saturated with Y = O 2 (Y) by 12.2.21.3. Therefore the lemma follows from 1.4.5 in [Asc19], except in case II where we must verify the extension condition when σ = 1. But if α ∈ AutFf (CT (f )) then as W ≤ CT (f ), α ∈ AutF2 (CT (f )). Further AutF2 (CT (f )) ≤ NF2 (T ), so the proof is complete.  In light of 12.2.22, during the remainder of the proof we assume: Hypothesis 12.2.23. Hypothesis 12.2.12 holds, as does case (2) or (3) of 12.2.13. Notation 12.2.24. It is now convenient to reverse our convention in 11.1.2 that Ω is {K1 , K1 , K3 , K3 , . . .}; instead we take  , . . .}. Ω = {Kn , Kn−1 , Kn−3 , Kn−3

Subject to this change x = e1 is an element of Ff of weight 1 in WI contained in F f . By 12.2.23 and 12.2.13, Fx◦ ∼ = Ω2(n−1)+a [m] for a = 1 or 2, and with = −1 if a = 2. Set si = z(Ki ) for 1 ≤ i ≤ n. Then s1 = e1,2 so s1 x = e2 ∈ Mx and e2 ∈ V2 V2 . Much as in Notation 11.5.2, we can form the (2(n − 1) + a)-dimensional orthogonal space (U, q) (of sign −1 if a = 2) over a suitable field of order congruent to 1 modulo 4. Set L = Ω(U, q) and regard Fx◦ as FTx (L). This time we write U = U2 ⊥U3 ⊥ · · · ⊥Un ⊥U0 , where for 2 ≤ i ≤ n, Ui is a 2-dimensional subspace of U of sign +1, and U0 is of dimension a and sign −1 if a = 2. Further we choose notation so that Ui +Ui+1 = [U, si ] for 2 ≤ i < n. Set A = Un +U0 , A0 = U3 +· · ·+U0 , and A1 = Un−1 + Un + U0 . Then s1 x is the involution in Mx with [U, s1 x] = U2 . In particular C1 = CFx (s1 )◦ = CFx (s1 x)◦ = FTA0 (LA0 ) ∼ = Ω2(n−2)+a [m].   ⊥ Set WX = V2 V2 · · · Vn−2 Vn−2 . Then [U, WX ] = A , so X = O 2 (CFx◦ (WX )) = FX (LA ) ∼ = L2 [m], L2 [2m] for a = 1, 2, respectively, and for X Sylow in X . Further as s1 x = e2 ∈ V2 V2 , X ≤ C1 , so X centralizes K1 ≤ M . By construction, X centralizes K2 , . . . , Kn−2 ≤ M .

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We summarize our observations: Lemma 12.2.25. (1) X = O 2 (CFx◦ (WX )) ∼ = L2 [m], L2 [2m] for a = 1, 2 respectively. (2) The Sylow group X of X centralizes K1 , . . . , Kn−2 ≤ M . Notation 12.2.26. Suppose f = fξ with ξ = ZS . Then by 12.2.14, S is transitive on ξ and then by 12.2.18.3, n = 2|ξ| + 1 is odd. In this case replace f = fξ by f = fθ with |θ| = (n − 1)/4, and regard F as f, fZS , NF (W ) , a case that must be handled in case II of 12.2.17. Subject to this convention, we may choose notation so that f = eθ1 with θ1 ⊆ {1, . . . , n − 2}. Thus f centralizes LA1 , and hence also X1 = FX1 (LA1 ) ∼ = Ω4+a [m], ◦ where X1 is Sylow in LA1 . Then X1 = X1 ≤ F2 . As in 11.5.2, let Λ1 be an index set of order a. From 9.4.22, 9.4.27, and 9.4.28, there are E8 -subgroups B1,λ of X1 such that X1 = X1 Osn−1 , AutX (CX1 (B1,λ ) : λ ∈ Λ1 . If n is even, set Λ = Λ1 and Bλ = B1,λ . If n is odd, set Λ = {1} and let Bλ be a 4-subgroup of X. In either case set Rλ = CT (Bλ ) and Y = T Oτ , FT (T M ), AutX (Rλ ) : λ ∈ Λ . Lemma 12.2.27. (1) Tf = TMf X with |Tf : TMf | = 2a = |T : TM |. (2) If a = 1 then T M ∼ = ω(Cn , m). (3) If a = 2 then T M ∼ = 2ω(Cn , m). Proof. By 12.2.18.2, |Tf : TMf | = 2a , and then as T = TM Tf with TMf = TM ∩ Tf , (1) holds. Suppose a = 1. Then X ∼ = L2 [m] so X is dihedral of order m/2 and by induction on n, X ∩ TM is cyclic. Let r ∈ X − TM . From the structure of Ω2(n−1)+a [m], r Kn−1 = Kn , while r centralizes K1 , . . . , Kn−2 by 12.2.25.2, so T M = r M ∼ = ω(Cn m) by 5.9.2.5. Therefore (2) holds. So assume a = 2. Then X ∼ = Dm . By induction on n, the cyclic = L2 [2m], so X ∼ r = Kn . subgroup y of X of index 2 centralizes Vn−1 Vn , and we still have Kn−1 ∼ Thus T M = r, y M = 2ω(Cn , m) by 5.9.3 and an argument in the proof of 5.3.10, establishing (3).  Lemma 12.2.28. (1) For λ ∈ Λ, Rλ ≤ CT (f ). (2) AutX (Rλ ) is uniquely determined by T M and Ω. (3) Y ∼ = Ω2n+a [m]. Proof. By 12.2.18.5, either f ∈ Z(S) or 12.2.17.II holds and |S : CS (f )| = 2. In the first case (1) is trivial. In the second, X1 ≤ F2 so CS (Bλ ) acts on θ1 and θ2 , and hence CS (Bλ ) ≤ CS (f ) so (1) holds in this case too. By induction on n, Fx◦ is uniquely determined by Tx Mx and Ωx . From 12.2.24, we may regard Fx◦ as FTx (L), and use the representation of L on (U, q) described in 12.2.24 to calculate in Fx◦ . From the proof of 12.2.20, W = Wf Vi Vi , where i = k, k + 1 for n even, odd, respectively. Further by 12.2.18, n − k ≥ 2, 3 for n even, odd, respectively, so n ≥ i + 2. Therefore, using the representation of L on (U, q), we conclude that Yi = Ki Ki ∩ S centralizes LA , and hence also X . As Fx◦ is determined by Tx Mx and Ωx , the fact that Yi centralizes X is determined by T M and Ω.

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∼ Sn by 11.1.2, and simiNext μ0 is the core of the permutation module for μ ˜= ˜i , i = 1, 2. Therefore larly μi,0 = μi ∩ μ0 is the core of the permutation module for μ |μ0 | = 2n−1 , |μ1,0 | = 2k−1 , and |μ2,0 | = 2n−k−1 . Then as μ1,0 ∩ μ2,0 = 1, μ(τf )0 = μ(τf ) ∩ μ0 is of order 2k−1 · 2n−k−1 = |μ0 |/2, and indeed μ0 = μ(τf )0 Yi W/W . Fi ˜ nally C T (f ) = Tf , so CT (f ) = Yi Tf . Therefore Rλ = Yi T1 R2 where R2 = CT2 (Bλ ). As X1 ≤ F2 , AutX (Rλ ) centralizes T1 , and thus centralizes Yi T1 by earlier remarks. Further if n−k ≥ 5, then working in F2 by induction on n, AutX (R2 ) is determined by T2 M2 and Ω2 , so (2) holds in this case. So suppose n − k ≤ 4. Recall from above that n − k ≥ 2. If n − k = 2 then F2 = X1 and AutX (R2 ) is determined by the proofs of 9.4.30 and 9.4.31. If n − k = 3 then F2 ∼ = Ω7 [m] or Ω− 8 [m] and AutX (R2 ) is determined by the proofs of 9.5.29 and 9.5.32. Finally if n − k = 4 then F2 is Ω9 [m] or Ω− 10 [m] and AutX (R2 ) is determined by the proof of 12.1.25. Therefore (2) holds. ˜ Ω) ˜ the Lie packet of F˜ , η˜ ∈ η(˜ Let F˜ = Ω2n+a [m], τ˜ = (F, τ ), etc. By 5.3.9 and ˜ 5.3.10, a model for NF˜ (W ) is isomorphic to aω(Cn , m), so by 12.2.27 we may take ˜ M ˜ = M, Ω ˜ = Ω, etc. By 11.5.3, F˜ = T Oτ˜ , FT (T M ), Aut ˜ (Rλ ) : λ ∈ Λ . T = S, X By Theorem 2.7.3, T Oτ˜ = T Oτ , and by (2), AutX˜ (Rλ ) = AutX (Rλ ), so F˜ = Y. Thus (3) holds.  Notation 12.2.29. Let G1 = Ff and E1 = CT (f )Ff◦ . If case I of 12.2.17 holds, set G2 = NF (W ) and E2 = FT (T M ). If case II of 12.2.17 holds set G2 = Fc and E2 = T Fc◦ . Lemma 12.2.30. (1) F = G1 , G2 . (2) Ei  Gi for i = 1, 2. (3) Y = E1 , E2 . Proof. Suppose case I of 12.2.17 holds. Then (1) follows from 12.2.16. Also E1  G1 by 12.2.20.2. By 3.3.13, FTM (M )  NF (M ) = N , so to complete the proof of (2) it suffices by 1.5.2 in [Asc19] to show that T  NN (TM ). By 11.1.3.5, AutN (W0 ) = AutG (W0 ) = AutM (W0 ), so AutNN (TM ) (W0 ) = AutTM (W0 ), and hence NN (TM ) ≤ TM CN (W0 ). Finally CN (W0 ) ≤ CN (f ), and we just saw that CT (f )  CNN (TM ) (f ), so FT (T M )  NF (W ), completing the proof of (2) in this case. Next by 1.3.2 in [Asc19], E1 = CT (f )Ff◦ = CT (f )F1 , CT (f )F2 , and by 11.5.3, CT (f )Fi ≤ Y, so E1 ≤ Y. By definition, E2 = FT (T M ) ≤ Y, so E1 , E2 ≤ Y. Conversely T Oτ , AutX (Rλ ) : λ ∈ Λ ≤ E1 , so Y = E1 , E2 , establishing (3) in this case. So assume case II of 12.2.17 holds. Then ξ = ZS and by 12.2.16, F = Ff , Fy , NF (W ) , where y = fξ . If n is even, y = c generates Z(FS (M ))  NF (W ), so F = G1 , G2 and (1) holds. If n is odd then by 1.3.2 in [Asc19], Fy = SFy◦ , NF (Ty ) . By 11.5.3, SFy◦ = SOτ , FS (SMy ) , so as SOτ ≤ Fc ≥ NF (W ) ≥ FS (SM ), we have SFy◦ ≤ Fc . Thus to prove (1), it suffices to show that NFy (Wy ) ≤ NF (W ). Let L = CFy (Wy ) and L Sylow in L. Now Wy = V1 · · · Vn , so W = V1 Wy . By 1.3.2 in [Asc19], NFy (Wy ) = SL, NFy (L) , so as W is weakly closed in S with respect to F, if W  W L then NFy (Wy ) ≤ NF (W ), proving (1).  We may take K = Kn , so CFy (Wy ) ≤ Fz ≤ NF (C). Further WC = V1 · · · Vn−3 Vn−3 and WC  NNF (C) (Wy ), so indeed W  W L, completing the proof of (1).

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Recall E1  G1 by 12.2.20.2, so to prove (2) we must show that T  NFc (Tc ). But W ≤ TM = Tc , so NFc (Tc ) ≤ NF (W ). Then as we showed that T  NNF (W ) (TM ), the proof of (2) is complete. Let Y0 = E1 , E2 . First T Oτ , FT (T M ) ≤ T Fc◦ = E2 and AutX (Rλ ) : λ ∈ Λ ≤ E1 , so Y ≤ Y0 . By definition of Y, E1 ≤ Y. By 11.5.3 and 11.1.6,  E2 = T Fc◦ = T Oτ , FT (T M ) ≤ Y, so Y0 = Y, establishing (3). Lemma 12.2.31. F ◦ ∼ = Ω2n+a [m]. Proof. The proof is much like that of 12.2.22. We show that Y  F. Then F ◦ = Y, and 12.2.28.3 completes the proof. To prove Y  F, we verify that the tuple Ei , Gi , i = 1, 2, satisfies Hypothesis 1.4.3 in [Asc19], and then appeal to 1.4.5 in [Asc19]. By 12.2.30.1, F = G1 , G2 , so Hypothesis 1.4.1 in [Asc19] is satisfied. By 12.2.30.2, Ei  Gi for i = 1, 2, so 1.4.3.1 in [Asc19] holds. As T = TM and W and O(τ ) are normal in NF (TM ), AutF (T ) is contained in Aut(T Oτ ) and Aut(FT (M )), so 1.4.3.2 follows from 1.4.6 in [Asc19]. Except in case II with i = 1, T is Sylow in Ei , so the extension condition 1.4.3.3 is satisfied. Thus Hypothesis 1.4.3 in [Asc19]  is satisfied except possibly in case II. Finally Y is saturated with Y = O 2 (Y) by 12.2.28.3. Therefore Hypothesis 1.4.3 in [Asc19] is satisfied and the lemma holds, except in case II where we must verify the extension condition when i = 1. But if α ∈ AutFf (CT (f )) then as W ≤ CT (f ), α ∈ AutF2 (CT (f )). Further AutF2 (CT (f )) ≤ NF2 (T ), so the proof is complete. 

We close this section with a theorem summarizing the major lemmas in the section.

Theorem 12.2.32. Assume Hypothesis 11.1.1 with n ≥ 5. Define c = eI as in 11.1.2. If 1 = c ∈ Z(F) assume the Extended Inductive Hypothesis holds. Then one of the following holds: ˇ 2n or (1) 1 = c ∈ Z(F) and either F ◦ = FS∩M (M ) ∼ = AE = ω(Dn , m) or F ◦ ∼ + Ω2n [m]. (2) F ◦ = FS∩M (M ) ∼ = ω(Dn , m)/Z(ω(Dn , m)). (3) F ◦ ∼ = AE2n or AE2n+1 . (4) F ◦ ∼ = P Ω+ 2n [m]. ◦ ∼ (5) F = Ω2n+1 [m]. (6) F ◦ ∼ = Ω− 2n+2 [m]. Proof. If 1 = c ∈ Z(F) then (1) holds by 11.1.6.2, so we may assume Hypothˇ 2n−4 esis 12.2.7. If O = O then (2) holds by 12.2.8.2 and 12.2.10, while if C is AE or AE2n−3 then (3) holds by 12.2.11. Therefore by 12.2.9.3 we may assume Hypothesis 12.2.12. If 12.2.13.1 holds then (4) holds by 12.2.22, so we may assume Hypothesis 12.2.23. Now 12.2.31 completes the proof of the theorem. 

12.3. COMPLETING |Ω(z)| = 2

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12.3. Completing |Ω(z)| = 2 In this section we assume the following hypothesis: Hypothesis 12.3.1. (1) τ = (F, Ω) is a quaternion fusion packet such that A(τ ) = ∅ and for z ∈ ZS , |Ω(z)| = 2. (2) F ◦ is transitive on Ω. (3) For z ∈ ZS ∩ F f , D∗ (z) = D(z) and Z ∩ O(z) = {z}. (4) τ satisfies the Extended Inductive Hypothesis. Lemma 12.3.2. Hypothesis 9.1.8 is satisfied. Proof. Immediate from the definitions.



Notation 12.3.3. Adopt Notation 9.1.2 and 9.1.6, plus the notation in 9.1.9. Set O = O(z). By 3.3.14 there is a unique member η of η(τ ). From 9.1.6, W = η , G is a model for NF (CS (W )), M = K G ∈ M (τ ), and μ = μ(τ ). Lemma 12.3.4. Let ηi , 1 ≤ i ≤ r, be the orbits of M on η, and for 1 ≤ i ≤ r let Ωi = {J ∈ Ω : J ∩ W ∈ ηi }, Mi = JiM for Ji ∈ Ωi , Si = S ∩ Mi , ζi = (FSi (Mi ), Ωi ), and μi = μ(ζi ). Then (1) M = M1 ∗ · · · ∗ Mr is a central product. (2) μ = μ1 × · · · × μr with μi ∼ = W eyl(Dn ) for some n ≥ 3. (3) G is transitive on {M1 , . . . , Mr } and NG (S ∩ M ) is transitive on Ω. Proof. This follows from 9.1.9.



Notation 12.3.5. Adopt the notation of 12.3.4, and choose notation so that some z ∈ Z ∩ F f is in Z(ζ1 ) and let K ∈ Ω(z). If m > 8 set Oz = O(z). If m = 8 let Gz be a model for NFz (OCS (O)), Hz = W Gz , and Lz = O 2 (Hz ), except when n = 4 write L1 for O 2 (Hz ). Let x = z when n = 4, and set x = 1 when n = 4. When Lx = 1 set Ox = O and when Lx = 1 set Ox = FO2 (Lx ) (Lx ). Lemma 12.3.6. Fz is transitive on D∗ (z) − D(z). Proof. See 9.1.10.3.



Lemma 12.3.7. Suppose m = 8, let x = z when n = 4, and x = 1 when n = 4. Assume Lx = 1. Then: (1) Lx = O 2 (W1Gz ). (2) If n > 4 then Lz centralizes CS (O), O = O2 (Lz ), and Lz is of index 1 or 3 in SL2 (3) ∗ SL2 (3). (3) If n = 4 then L1 centralizes CS (O(ζ1 )), O(ζ1 ) = O2 (L1 ), and L1 is of index 3e in (SL2 (3) ∗ SL2 (3))2 for some 0 ≤ e ≤ 3. (4) If n = 3 then either (i) Lz centralizes CS (O), O = O2 (Lz ), and Lz is of index 1 or 3 in SL2 (3) ∗ SL2 (3), or (ii) O2 (Lz ) = O × Bz where Bz = bLz ∼ = E4 , b = za for a = Z(M1 ), and Lz = O2 (Lz )Xz , where Xz ∈ Syl3 (Lz ), Xz ∼ = E3e for some 1 ≤ e ≤ 3, and Z(Lz ) = z . Moreover CS (O) = Bz × CS (Lz ) and CS (Lz ) contains W2 · · · Wr and any Lz -invariant subgroup of CS (O) centralized by W1 .

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Proof. Set S0 = CS (O) and observe W = W1 ∗ U where U = W2 · · · Wr ≤ S0 and U = η − η1 = V ∈ η : V ≤ S0 . Thus U  Gz , so as W centralizes U , so does Hz . Thus, setting H1 = W1Gz , we have Hz = H1 U with U ≤ Z(Hz ), so Lx = O 2 (Hz ) = O 2 (H1 ), establishing (1). Parts (2) and (3) follow from 9.1.11.2, so we may take n = 3. Let A be the set of Lz -invariant subgroups of Gz with A ≤ CS0 (W1 ). Then H1 centralizes each A ∈ A, so Lz centralizes A by (1). In particular the last statement in (4ii) holds. For the remainder of (4), we appeal to (1) and argue as in the proof of 9.5.11.  Theorem 12.3.8. Assume Hypothesis 12.3.1 Then r = 1, so μ ∼ = Weyl(Dn ) for some n ≥ 3. Until the proof of Theorem 12.3.8 is complete, (near the end of the section) we assume that r > 1. Lemma 12.3.9. Fz◦ = Oz ∗ Cz ∗ C2 ∗ · · · ∗ Cr where (1) For 2 ≤ i ≤ r, ρi = (Ci , Ωi ) is a quaternion fusion packet, ηi ∈ η(ρi ), and Mi ∈ M (ρi ), so μi = μ(ρi ). (2) If n = 3 then Cz = 1, while if n > 3 then ρz = (Cz , Ωz ) is a quaternion fusion packet, where Ωz = Ω1 − Ω(z), ηz ∈ η(ρz ), and Mz ∈ M (ρz ), where ηz = C (z) {V ∈ η1 : V ≤ CS (O)} and Mz = Ωz M1 . Further μz = μ(ρz ) ∼ = W eyl(Dn−2 ). (3) (OKj , Kj ), j = 1, 2, ρz , and ρi , 2 ≤ i ≤ r, are the coconnected components of τz , unless n = 4 and Cz = O(t), where ZS (ζ1 ) = {z, t}. Proof. See the proof of 9.3.10.1.



Notation 12.3.10. Let I = {1, . . . , r} and I1 = I − {1}. For i ∈ I1 , let zi ∈ Z(ρi ) with zi ∈ Cif , and, when n > 3, set Di = CCi (O(zi ))◦ , Γi = Ωi − Ω(zi ), and δi = (Di , Γi ). Lemma 12.3.11. Assume m = 8, n = 4, and if n = 3 and Lz = 1 assume 12.3.7.4.i holds. (1) Oz  Fz . (2) For s ∈ ZS let φs ∈ AutF (O(τ )) with zφs = zs , and set Os = Oz φ∗s . Then Os is independent  of the choice of φs and NF (O(τ )) contains a normal central product Oτ = s∈Zs Os . Proof. Set S0 = CS (O). By 12.3.7, Lz centralizes S0 and O is Sylow in Lz , so by 2.2 in [Asc11], Fz contains a central product E = Oz ∗ CF (O). Then NE (S0 ) has model Lz CGz (O)  Gz as Lz  Gz , so NE (S0 )  NFz (S0 ). Therefore E  Fz by 1.5.1 in [Asc19]. Next E0 = CE (CF (O)) = Z(CF (O))Oz , so Oz = O 2 (E0 )  Fz , proving (1). By (1), Os is independent of the choice of φs in (2). As Lz centralizes S0 , N = NF (O(τ )) contains the central product Oτ by 2.2 in [Asc11]. Then as  AutF (O(τ )) permutes {O(s) : s ∈ ZS }, we have Oτ  N . Lemma 12.3.12. Assume m = 8 and n = 4. For i ∈ I set Oi = O(ζi ) and Zi = Z(Oi ). (1) O1  CF (Z1 ). (2) AutF (O(τ )) is transitive on {Oi : i ∈ I}.

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(3) For i ∈ I let φi ∈ AutF (O(τ )) with O1 φi = Oi , and set Oi = O1 φ∗i . Then Oi is independent  of the choice of φi and NF (O(τ )) contains a normal central product Oτ = i∈I Oi . Proof. The proof of 12.3.11 establishes (1) and (3). Next A = AutF (O(τ )) is transitive on {O(s) : s ∈ ZS } as F is transitive on Ω. By 12.3.9, {{O(zi ), O(ti )} : i ∈ I} is a system of imprimitivity for A, where ZS (ζi ) = {zi , ti }. Therefore (2) holds.  Lemma 12.3.13. If (m, n) = (8, 3) and Lz = 1 assume 12.3.7.4.i holds. Then for i ∈ I1 , one of the following holds: (1) Oz = O and Ci = FSi (Mi ). (2) m = 8, n > 4, O = O2 (Lz ), Lz is of index 3 in SL2 (3) ∗ SL2 (3), and Ci is ˇ 2n , or AE2n+1 . AE2n , AE (3) n > 4 if m = 8, Oz ∼ = SL2 [m] ∗ SL2 [m], O is Sylow in Oz , and Ci is − (P )Ω+ [m], Ω [m], or Ω 2n+1 2n 2n+2 [m]. (4) n = 4, m = 8, O(ζ1 ) = O2 (L1 ), L1 is of index 3e in (SL2 (3) ∗ SL2 (3))2 , and one of the following holds: ˇ 8 or AE9 and e = 2. (i) Ci is AE − (ii) Ci is Ω+ 8 [8], Ω9 [8], or Ω10 [8] and e = 0. 3+6 ˇ 3 (2)/2 and e = 3. (iii) Ci is L ˇ + (2) and e = 1. (iv) Ci is Ω 8 (5) n = 3, m = 8, 12.3.7.4.i holds, Lz is of index 3e in SL2 (3) ∗ SL2 (3), and either ˇ 6 and e = 1, or (a) Ci ∼ = AE − (b) Ci ∼ = Ω+ 6 [8] or Ω8 [8] and e = 0. Proof. First observe that Γ = Ω − Ω1 centralizes Oz . If m > 8 this follows from 2.6.12, while if m = 8 it follows from 12.3.7. Now for φ ∈ Aut(O(τ )W ), Γφ centralizes Ozφ , and by 12.3.11 we may choose φ so that zφ = zi and Oz φ∗ = Ozi , and hence z centralizes Ozi (or Oi if n = 4 and m = 8 by 12.3.12). Also W1 φ = Wi , so Ozi is contained in the normal closure Pi of Wi in CCi (zi ) (or in CCi (Zi ) if n = 4 and m = 8). Further for αi ∈ A(zi ) with zi α = z, Pi αi∗ ≤ Ozi . We conclude that Pi = Ozi . Then the lemma follows from 12.3.1.4 (together with Theorems 9.5.33, 12.1.25, and 12.2.32 for n = 3, n = 4, and n > 4, respectively), since by 12.3.9.1, ρi is a quaternion fusion packet with μi = μ(ρi ).  Lemma 12.3.14. Assume n > 3. Then (1) for all i, j ∈ I1 , Ci ∼ = Cj . (2) For all i ∈ I1 and αi ∈ A(zi ) with zi αi = z, we have Di αi∗ = Cz , so Di ∼ = Cz .
1 and the transitivity of F ◦ on Ω in 12.3.1.2.  Lemma 12.3.17. We can choose z, W1 , and O(ζ1 ) in F f . Proof. Let α ∈ A(W1 ). As Wi  W , W α = W , so α is a NF (W )-map. Then as NF (W ) permutes {Wi : i ∈ I}, it follows that W1 α = Wj for some j ∈ I. Further as S permutes {Wi : i ∈ I}, we have CS (z) ≤ NS (W1 ), so zα ∈ F f . Hence, replacing z by zα if necessary, we may assume W1 ∈ F f . Similarly let Qi = O(ζi ), Q = O(τ ), and β ∈ A(Q1 ). As in the previous paragraph, Q1 β = Qi for some i ∈ I and CS (z) ≤ NS (Q1 ) ≥ NS (W1 ), so zβ and W1 β ∈ F f . Hence we may assume z, W1 , Q1 are in F f .  Notation 12.3.18. Suppose first that n = 4. Then, using  12.3.12 when m = 8, for each j ∈ I, Oj is defined, as is a central product Oτ = j∈I Oj . So take n = 4. Then, using 12.3.16 and  12.3.11.2 when m = 8, for s ∈ZS , Os is defined, as is a central product Oτ = s∈ZS Os . For j ∈ I, let Oj = s∈ZS (ζj ) Os , and observe  we also have a central product Oτ = j∈I Oj . For j ∈ I, set Bj = Sj Oj , FSj (Mj ) ≤ F. For i ∈ I1 , by 12.3.17 there exists αi ∈ A(Wi ) with Wi αi = W1 . Let Ti be Sylow in Ci .  Lemma 12.3.19. (1) For i > 1 and (n, m) = (4, 8), Oi =2 Pi , where Pi = s∈ZS (ρi ) Ps , Ps is a NCi (O(ρi ))-conjugate of Pzi , and Pzi = O (Hzi ) where Hzi is the normal closure of Wi in CCi (zi ). (2) If (n, m) = (4, 8) and i > 1 then Oi = O 2 (Hi ) where Hi is the normal closure of Wi and CCi (Zi ). ˇ 2n , (P )Ω+ [m], L ˇ 3 (2)/23+6 , (3) For i > 1, Bi = Ci if Ci is FSi (Mi ), AE2n , AE 2n + ˇ or O8 (2). ˇ 2n ; Ω+ [m] if Ci is isomorphic to (4) For i > 1, Bi ≤ Ci is isomorphic to AE 2n − AE2n+1 ; Ω2n+1 [m] or Ω2n+2 [m], respectively.

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(5) (6) (7) (8)

∗ For αi ∈ A(Wi ) with Wi αi = W1 , we  have Bi αi = B1 . F contains a central product B0 = i∈I Bi . αi permutes {Sj : j ∈ I} and {Bj : j ∈ I}. Set D1 = Cz . Then αi permutes {Dj : j ∈ I}.

Proof. Parts (1) and (2) were established during the proof of 12.3.13. Then (3) and (4) follow from 12.3.13, (1), (2), and the following lemmas: 11.6.6 and ˇ k ; 11.5.3 when Ci is a Lie system; and 10.1.11.1 and 11.6.8 when Ci is AEk or AE 2.1.15 in the remaining cases. Choose αi as in (5). Then Si αi = S1 , so FSi (Mi )αi∗ = FS1 (M1 ). Also Si Oi αi∗ = S1 O1 from the definition of Oi in 12.3.18. Thus (5) follows.  Let Mi = FSi (Mi ). By 12.3.4.1, F contains a central product i∈I Mi . Observe also that for distinct i, j in I, Si centralizes Sj Oj , since Oi = O(ζi ) centralizes 8 this is 2.6.12, while if m = 8 it follows from 12.3.7 and 12.3.16. Oj . When m > Therefore S  = i∈I1 centralizes B1 and S1 centralizes Bi for i ∈ I1 , so (6) follows from 2.2 in [Asc11]. As in the proof of 12.3.15.6, αi permutes {Sj : j ∈ I}, {Mj : j ∈ I}, and {Sj Oj : j ∈ I}, so αi permutes {Bj : j ∈ I}, establishing (7). Part (8) is essentially established during the proof of 12.3.14.  Lemma 12.3.20. C2 is not AE2n+1 . Proof. Assume otherwise. By 12.3.16 and 12.3.13.5, n > 3. Let i ∈ I1 . By 11.6.7, Di ∼ = AE2n−3 , while by 11.6.8, |Ti : Si | = 2, and Ti = Si O2 (Di ). Set T1 = S1 O2 (D1 ). By 12.3.14.2, Di αi∗ = D1 , so Ti αi = T1 . Then by parts (7) and (8) of 12.3.19, αi permutes {Tj : j ∈ I}. Set Ui = O2 (Ci ) and U1 = J(T1 ). By 11.6.10, Ui = J(Ti ), so Ui αi = U1 and αi permutes {Uj : j ∈ I}. Next Ti centralizes D1 and Si centralizes S1 , so Si centralizes T1 . As αi permutes {Sj : j ∈ I}, {Dj : j ∈ I}, and {Tj : j ∈ I} for each i ∈ I1 , it follows that for all distinct j, k in I, Sj and Dj centralize Tk , so Tj = Sj O2 (Dj ) centralizes Tk . Similarly Bi centralizes S1 and O2 (D1 ), so Bi centralizes T1 . Then Ti Bi centralizes T1 , so Tj Bj centralizes Tk . Next T1 centralizes AutBi (Ui ), AutDi (Ui ) = AutCi (Ui ), so [T1 , Ci ] ≤ Ui . Therefore Cj : j ∈ I1 acts on U = k∈I Uk . Claim U  S, so that U ∈ F f . For S acts on {Sj : j ∈ I} and for  s ∈ S, s T1 = S1s O2 (D1 )s =  S1s O2 (D1s )  = T1s as z s ∈ F f . Therefore S acts on k∈1S Tk and hence on U  = k J(Tk ) = k Uk . Let J = I − 1S and u ∈ z S . Then u = z or u ∈ Cz or u ∈ Ci for some i ∈ 1S . In each case u centralize Cj for each j ∈ J. Then as u ∈ F f , ρj is the coconnected component of τu containing Ωj . Therefore S permutes {Cj : j ∈ J}, and hence also  U  = j∈J Uj . Thus S acts on U = U  U  , completing the proof of the claim. By the claim, τU = (Y, Ω) is a quaternion fusion packet, where Y = NF (U ). We’ve shown that Ci and αi are contained in Y for each i ∈ I1 , so Y is transitive on Ω. As z ∈ U ≤ O2 (Y), it follows from Theorem 2 that C1 = C2 α2∗ = Ci αi∗ for each i ∈ I1 , and C0 = C1 × · · · × Cr = Y ◦ . But now Hypothesis 8.1.8 is satisfied, so by 8.1.10, F ◦ = C0 , contrary to the assumption that r > 1 and to the transitivity  of F ◦ on Ω in 12.3.1.2.

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Lemma 12.3.21. Assume n = 3 and C2 is Ω7 [m] or Ω− 8 [m]. For i ∈ I1 , set Di = E(CCi (O(zi )). Then (1) D1 = Di αi∗ is a component of Fz independent of i and αi . (2) αi permutes {Dj : j ∈ I}. ∼ Proof. First if m = 8 then C2 is Ω− 8 [8] by 12.3.16 and 12.3.13.5, so Di = L2 [2m] is indeed quasisimple. Let Di be Sylow in Di and Ui = z, zi . Then Di is a component of CF (Ui ) = CFz (zi ), so D1 = Di αi∗ is a component of CF (Ui α) with Ui α = ui , z and ui = zαi . Therefore D1 is a component of CFz (ui ). By 12.3.19.4, ui ∈ Bk ≤ Ck for some k ∈ I1 , so ui = zk and either D1 is a component of Fz or D1 = Dk . Assume the latter. Now 1 = Vi = Si ∩ Di is cyclic of index 2 or 4 in Di , and Vi αi ≤ S1 ∩ Dk by 12.3.19.7. This is a contradiction as S1 ∩ Sk ≤ Z(Mk ). Therefore D1 is a component of Fz . Let j ∈ I1 and α ∈ A(zj ) with zj α = z, and suppose D1 = Dj αj∗ . Then z × D1 × Dj α ≤ CS (O(z1 )). Hence z × Vi αi × Vj α ≤ CS1 (O(z)), a contradiction as m(CS1 (O(z)) = 2. Therefore D1 = Dj αj∗ , completing the proof of (1). Then (1) implies (2).  Lemma 12.3.22. C2 is not Ω2n+1 [m] or Ω− 2n+2 [m]. Proof. Assume otherwise. If n = 3 define Dj , j ∈ I, as in 12.3.21, and observe that if also m = 8 then C2 is Ω− 8 [8] by 12.3.16 and 12.3.13.5. If n > 3 then Dj is defined in 12.3.10 when j ∈ I1 and D1 = Cz , as in 12.3.19.8. In any event let Dj be Sylow in Dj . Let i ∈ I1 . By 5.3.9 and 5.3.10, Ti = Di Si with |Ti : Si | = 2,4 for C2 equal to Ω2n+1 [m], Ω− 2n+2 [m], respectively. Set T1 = D1 S1 , Observe that Ti αi = T1 and αi permutes {Tj : j ∈ I}. For Di αi = D1 and αi permutes {Dj : j ∈ I} by 12.3.14.2 and 12.3.21.1, while Si αi = S1 and αi permutes {Sj : j ∈ I} by 12.3.19.7. Next Ti centralizes D1 and Si centralizes S1 , so Si centralizes T1 = D1 S1 . Then, from the previous paragraph, for all distinct j, k ∈ I, Sj centralizes Tk . Now as Ti centralizes D1 and S1 , Ti centralizes T1 = D1 S1 , so Tj centralizes Tk . Next we show that Ci = Bi , Σ , where Σ = O 2 (AutCi (Rλ ) :λ ∈ Λ for suitable Rλ ≤ Ti and Σ ⊆ CF (T1 ). Then Ci centralizes T1 . Set T  = i∈I1 Ti and  Q = j =i Tj ∈ NF (T1 )f . Then Ci = CF (Q)◦ as z ∈ Q, and T  ∈ F f as W1 ∈ F f , so there is β ∈ A(Q) with  Qβ = T  . Then C1 = Ci β ∗ = CF (T  )◦ . Similarly  ◦  C1 = CF (O ) , where O = i∈I1 O(ρi ), so Cj αj∗ = C1 for each j ∈ I1 . Therefore by 2.2 in [Asc11], there is a direct product C0 = C1 × · · · × Cr contained in F. But now we obtain a contradiction from 8.1.10, as in the proof of 12.3.20. It remains to establish the claim. If n is odd then from 11.5.2, Rλ = CTi (Bλ ), where Bλ is a 4-subgroup of Di and Σ ≤ AutDi (Rλ ), so indeed Σ ⊆ CF (T1 ). Thus we may take n even, so from 11.5.2, Rλ = CTi (Bλ ) for a suitable 8-subgroup of LA and A = Un/2 + U0 with Un/2 = [U, sn/2 ] for sn/2 ∈ ZTi ∩ Cif . If n ≡ 0 mod 4 then there is sn/2 = s ∈ ZTi ∩ Cif , so we may choose s = zi and hence zi centralizes LA , so that O 2 (AutCi (Rλ )) ≤ AutDi (Rλ ) ⊆ CF (T1 ). This leaves the case n ≡ 2 mod 4, where O(zi )  Ti , and we define O  = Ωi − Ω(zi ) and E = CCi (O  )◦ .  f Then E ∼ and = Ω5 [m] or Ω− 6 [m] and Σ is induced in E. Now Q = O αi ∈ F ∗ Eαi is a coconnected component of NF (Q). Also Q ≤ Cz , so Ck is a coconnected component of NF (Q). Therefore Eαi∗ centralizes Tk for each k ∈ I1 , so E centralizes  T1 , completing the proof of the claim and the lemma.

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Lemma 12.3.23. (1) For i ∈ I1 , Ci = Bi . ˇ 2n , or Ω+ [m], or (n, m) = (4, 8) and Ci is L ˇ 3 (2)/23+6 (2) Ci is FSi (Mi ), AE 2n + ˇ (2). or O 8  (3) Define C1 = B1 ; then F contains a central product C0 = j∈I Cj . (4) Z(Ci ) = ci ∼ = Z2 , where ci = Z(Mi ). Proof. First, by 12.3.16, if (n, m) = (3, 8) then either Lz = 1 or 12.3.7.4.i holds. Therefore the hypothesis of 12.3.13 is satisfied. Further by 12.3.22, Ci is neither Ω2n+1 [m] nor Ω− 2n+2 [m], while by 12.3.20, Ci is not AE2n+1 . Therefore by 12.3.13, either Ci appears in (2), or Z(Ci ) = 1 and Ci is AE2n or P Ω+ 2n [m]. Thus (4) implies (2). In any event by 12.3.19.3, Ci = Bi , so (1) holds and (3) follows from 12.3.19.6. By (3) and 8.1.10, Z(Ci ) = 1, establishing (4) and completing the proof of (2).  Notation 12.3.24. Set U = ci : i ∈ I . By 12.3.23.4, U = 1. Set U = NF (U ) and SM = S1 · · · Sr . Lemma 12.3.25. Z(F) = 1. Proof. Set F + = F/Z(F). Then, using 6.6.9, F + satisfies Hypothesis 12.3.1, so Theorem 12.3.8 holds in F + by induction on the order of F, contradicting r > 1.  Lemma 12.3.26. NF (W ) ≤ U. Proof. As U = ci : i ∈ I and ci = Z(Mi ), we have U = Z(FSM (M )). Then as FSM (M )  NF (W ), the lemma follows.  Lemma 12.3.27. (1) U controls fusion in CS (W ). (2) If X ≤ CS (W ) and α ∈ A(X) then Xα ≤ CS (W ) and α is a U-map. Proof. As W is weakly closed in S, NF (W ) controls fusion in CS (W ), and then (1) follows from 12.3.26. Similarly in the setup of (2), W α = W as W is weakly closed in S, so (2) follows from 12.3.26.   Lemma 12.3.28. Let ∅ = J ⊆ I and WJ = j∈J Wj . Then (1) if WJ ∈ F f then FSj (Mj ), j ∈ J, and Ci , i ∈ I − J are the coconnected components of ρJ = (NF (WJ ), Ω), and are subnormal in NF (WJ ). (2) NF (WJ ) ≤ U. Proof. Suppose WJ ∈ F f . By 3.3.9, FSj (Mj ) is a coconnected component of J = NF (WJ ) for each j ∈ J. Further for i ∈ I − J, Ci is contained in a coconnected component E of ρJ . Then as E centralizes zj for j ∈ J, Eαj∗ is contained in a coconnected component X of τz , so as Ci αj∗ ≤ Eαj∗ ≤ X we conclude from 12.3.9 that Ci = E, completing the proof of (1). In (2), conjugating in U and appealing to 12.3.27.2, we may assume WJ ∈ F f . By (1), J ◦ ≤ U. Now by 1.3.2 in [Asc19], J = NS (WJ )J ◦ , NJ (W ) ≤ U, using 12.3.26. That is (2) holds.  Lemma 12.3.29. Fz ≤ U. Proof. By 12.3.9, Fz◦ = Oz ∗ Cz ∗ C2 ∗ · · · ∗ Cr . Then as Oz ∗ Cz ≤ C1 , we have ≤ U. But by 1.3.2 in [Asc19] and 12.3.28.2, Fz = CS (z)Fz◦ , NFz (WI1 ) ≤ U. 

Fz◦

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Lemma 12.3.30. Assume ξ = (Y, Γ) is a proper subpacket of τ with Γ ⊆ Ω and Ci ≤ Y and Ωi ⊆ Γ for some i ∈ I. Then (1) Ci is a coconnected component of ξ. (2) Y ≤ U. Proof. We first prove (1). As Ci is coconnected, Ci is contained in a coconnected component ζ = (E, Σ) of Y. Arguing as in the proof of 9.3.10, Wi ∈ W (ζ) and μi ∼ = μ(ζ). Then by 12.3.1.4 and Theorem 12.3.50, and by 12.3.13, Ci = CE (ci ), ˇ 2n and E ∼ and then either Ci = E or Ci ∼ = AE2n+1 or Ci ∼ = AE = Ω+ 2n [m] and − E ∼ = Ω2n+1 [m] or Ω2n+2 [m]. Of course we may assume Ci = E, and, conjugating in U and appealing to 12.3.27, we may assume i = 1. Now if n > 3 then CE (z)◦ = Oz ∗ X where X is AE2n−3 or Ω2n−3 [m] or Ω− 2n−2 [m]. Then X is contained in a coconnected component Y of τz , so Y = Cj or Cz by 12.3.9. Then as Cz ≤ X , it follows that X = Cz , a contradiction as Cz = X . ˇ 6 or Ω+ [m] and E is AE7 , Ω7 [m], or Ω− [m]. Therefore n = 3, so C1 is AE 6 8 However in the first case, and in the second when m = 8, we have E = C1 , CE (z) , so E ≤ U by 12.3.29, so c1 ∈ O2 (E) and hence E is AE7 . Then O2 (E) = cE1 ≤ U ≤ CS (W ), whereas W1 does not centralize O2 (E). Therefore E is Ω7 [m] with m > 8 or Ω− 8 [m]. Let X = E(CE (O(z)). Then / O2 (c1 X ) contradicting c1 ∈ O2 (U). This finally X ≤ U by 12.3.29. But c1 ∈ completes the proof of (1). If C is a Y-conjugate of C1 distinct from C1 , then z centralizes C, so C = Ck for some k ∈ I by 12.3.9. Thus the product D of the Y-conjugates of C1 is contained in U. Now, arguing as usual using 1.3.2 in [Asc19] and 12.3.28.2, we obtain (2).  Lemma 12.3.31. (1) U = F. (2) {Ci : i ∈ I} is the set of coconnected components of τU = (U, Ω). (3) C0  U. Proof. If U = F then U  F, so X = CF (U )  F. By 12.3.25, X = F, so by 6.6.9 and induction on the order of F, C0 = X ◦  F. As usual this contradicts 12.3.1 and r > 1. Thus (1) holds. By (1) and 12.3.30.1, {Ci : i ∈ I} is contained in the set C of coconnected components of τU . Further if E ∈ C is not a Ci then E is contained in a coconnected component of τz , contrary to 12.3.9. Thus (2) holds, and of course (2) implies (3).  Lemma 12.3.32. U is tightly embedded in F. Proof. Let 1 = X ≤ U ; we will show NF (X) ≤ U. For α ∈ A(U ), α is a U-map by 12.3.27.2, so Xα ≤ U . Thus we may assume X ∈ F f . Then NF (X) ≤ U by 12.3.30. By 12.3.27, U controls fusion in U , so the lemma follows from the previous paragraph.  Lemma 12.3.33. Let P ≤ S such that for some i ∈ I, Wi ≤ P . (1) NF (P ) ≤ U. (2) For each φ ∈ homF (P, S), φ is a U-map. Proof. Set J = {j ∈ I : Wj ≤ P }. Then WJ is weakly closed in P , so (1) follows from 12.3.28.2.

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Assume the setup of (2). Then WJ φ = WA for some A ⊆ I. Let α ∈ A(WJ ); by 12.3.27.2, α is a U-map and WJ α = WB for some B ⊆ I. Let β ∈ A(WA ) with WA α = WB . By 12.3.27.2, β is a U-map. Also ζ = α−1 φβ is a NF (WB )-map, and hence also a U-map by 12.3.28.2. Therefore φ = αζβ −1 is a U-map, proving (2).  Lemma 12.3.34. Let i ∈ I and Ui = NS (Si )C0 . (1) Ci  Ui , so CS (Ci ) exists in Ui and is the largest subgroup of CS (Si ) centralizing Ci in Ui (2) Let 1 = X ≤ CS (Ci ) and α ∈ A(X). Then α is a U-map, Ci α∗ = Cj for some j ∈ I, and Xα ≤ CS (Cj ). (3) If X ∈ F f then Ci is a coconnected component of τX = (NF (X), NΩ (X)). (4) NF (X) ≤ U. Proof. As C0  U by 12.3.31, we can form Ui , and by construction, Ci  Ui . Then (1) follows from Theorem 4 in [Asc11] and a remark preceding that theorem. Assume the setup of (2). Applying 12.3.33 to α, NS (X) in the role of φ, P , we conclude that α is a U-map, and then (2) follows from 12.3.31. Part (3) follows from 12.3.30, which also says that NF (X) ≤ U when X ∈ F f . It remains to prove (4), where as α is a U-map, we may assume X ∈ F f . Then (4) follows from a remark in the previous paragraph.  ˇ 2n , or (n, m) = Lemma 12.3.35. Either C1 is isomorphic to FS1 (M1 ) or AE 3+6 ˇ (4, 8) and C1 is L3 (2)/2 . Proof. Assume otherwise; then by 12.3.23.2, C1 is Ω+ 2n [m] or (n, m) = (4, 8) ˇ + (2). In either case, C1 is quasisimple. Adopt the notation from [Asc19], and C1 is O 8 particularly Notation 6.1.1, 6.1.10, and 6.1.12 in [Asc19]; thus C1 ∈ C(F) the set of quasisimple subsystems C of F such that I(C) = ∅, where i ∈ I(C) if i is an involution centralizing C such that Cα∗ is a component of Fiα for α ∈ A(i). By 12.3.34, for each involution i ∈ CS (C1 ), we have i ∈ I(C). Therefore C1 is maximal in C(F) as defined in 6.1.12 of [Asc19]. As W1 ∈ F f , also S1 ∈ F f . By 12.3.34, for distinct j, k ∈ I, each involution in Sj is in I(Ck ), so {Ci : i ∈ I} ⊆ C1⊥ . Therefore by Theorem 2 in [Asc19], C1 is a component of F and C0  F. But now we obtain our usual contradiction to 12.3.1 and r > 1.  Notation 12.3.36. We now follow section 7 in [Asc19], beginning with Notation 7.3.14 in [Asc19]. By 12.3.22, U is tightly embedded in F, so we can adopt Notation 3.1.9 in [Asc19]. In particular P = {1 = P ≤ S : homF (P, U ) = ∅}, ∗

P is the set of maximal members of P under inclusion, and Pˆ = P ∗ − {U }. For P ∈ P and β ∈ A(P ) with P β ≤ U , set U (β) = (U ∩ NS (P )β)β −1 . From 7.3.15 in [Asc19] we have: Lemma 12.3.37. (1) Each member of P is contained in a unique member U (P ) of P ∗ . (2) U (P ) = U (α) for each α ∈ A(P ) with P α ≤ U . (3) P ∗ is AutF (S)-invariant. (4) For distinct P1 , P2 ∈ P ∗ , P1 ∩ P2 = 1.

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(5) Each member of P ∗ is a TI-subgroup of S.

Lemma 12.3.38. O2 (F) = 1. Proof. By 6.6.9.1 and induction on the order of τ , we may take F = F ◦ . By Theorem 2, z ∈ / O2 (F), so by 3.3.2, F = F ◦ centralizes O2 (F), contrary to 12.3.25.  Lemma 12.3.39. Pˆ =  ∅. Proof. If not, U is strongly closed in S, so U  F, contrary to 12.3.38.



ˆ Define ΣP to consist Notation 12.3.40. By 12.3.39 we may choose P ∈ P. of those 1 = X ≤ NS (P ) such that T(P ) = ∅, where T(P ) is the set of triples (P0 , ψ, i) such that 1 = P0 ≤ NP (X), ψ ∈ homF (XP0 , S) with P0 ψ ≤ U , and Xψ centralizes Ci . Write ΞP for the set of 4-subgroups A of NS (P ) with A# ⊆ ΣP and A ∩ P = 1. Lemma 12.3.41. Assume X ∈ ΣP and (P0 , ψ, i) ∈ T(X). Then (1) There is β ∈ A(X) and γ ∈ A(Xψ) such that Xψγ = Xβ, Ci γ ∗ = Cj for some j ∈ I, and NF (Xβ) ≤ U. (2) P0 ≤ U (β). (3) For each 1 = Y ≤ XP0 , Y ∈ ΣP with (CP (Y ), β|Y CP (Y ) , j) ∈ T(Y ). Proof. Part (1) follows from 12.3.34. Then the proof of 7.3.18 in [Asc19] establishes (2) and (3).  Lemma 12.3.42. Assume X ∈ ΣP . Then (1) NS (X) ≤ NS (P ). (2) NF (X) ≤ NF (U (β)) ≤ NF (P ) for β ∈ A(X) as in 12.3.41.1. (3) X centralizes no Ci . (4) U (β) ≤ P and U (β)  NS (X). Proof. See the proof of 7.3.19 in [Asc19].



Lemma 12.3.43. Assume 1 = X ≤ S such that Wi ≤ NS (X) for some i ∈ I, and let β ∈ A(X). Then (1) β is a U-map. (2) If Xβ ≤ CS (Ci ) for some i ∈ I then X ≤ CS (Ck ) for some k ∈ I. (3) X ∈ / ΣP . Proof. Part (1) follows from 12.3.33.2 applied to NS (X), β in the role of P, φ. Then (1) implies (2), and (2) and 12.3.42.3 imply (3).  Lemma 12.3.44. Let X ∈ ΣP . Then (1) X ≤ W . (2) P ∩ W = 1. (3) NW (P ) = CW (P ). Proof. Part (1) follows from 12.3.43.3. Each nontrivial subgroup of P is in  ΣP , so (1) implies (2). Then [NW (P ), P ] ≤ P ∩ W = 1 by (2), so (3) holds.

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Lemma 12.3.45. Assume A ∈ ΞP . Then (1) A acts on Wi for each i ∈ I. (2) U ≤ CS (P A). (3) ∅ = U F − {U } = P ! . Proof. Assume A does not act on Wi for some i ∈ I, say a ∈ A with Wia = Wj with i = j. If a = P ∩ A = PA and for each b ∈ A − P , b acts on Wi , then A = A − P also acts on Wi , a contradiction. Thus we may assume A = PA a . Let Y = Wi Wj , Y + = Y /ci , cj , and Ya = {wwa : w ∈ Wi }. Then Ya ≤ CY (a) and Ya+ is a full diagonal subgroup of Y + = Wi+ × Wj+ . By 12.3.44.3, P centralizes CY (a), so A = PA a centralizes Ya , and hence A acts on {Wi , Wj }, so B = NA (Wi ) is of index 2 in A. As B centralizes the full diagonal subgroup Ya+ , B centralizes Y + . Therefore [Wi , B] ≤ ci . But for V ∈ ηi , z(V ) is the involution in Φ(V ci ), so B centralizes z(V ) and hence acts on V and then centralizes V . Hence B centralizes Wi , so by 12.3.42.1, Wi ≤ NS (P ), contrary to 12.3.43.3. Thus (1) holds. By (1), A centralizes ci for each i ∈ I, so U ≤ CS (A) ≤ NS (P ), and then (2) ˆ so U α ∈ P ! = U F − {U }, follows from 12.3.44.3. Now for α ∈ A(P ), U α ∈ P, establishing (3).  Lemma 12.3.46. Let α ∈ A(P ). Then (1) P α ≤ U . ˆ (2) CU (P )α ≤ Q ∈ P. ! (3) If P ∈ P then CU (P )α = Q ∈ P ! and |P | = |CU (P ). (4) For each 1 = X ≤ P , X ∈ ΣP and (P, α, i) ∈ T(X) for each i ∈ I. (5) For each i ∈ I and 1 = X ≤ CS (Ci )∩NS (P ), Xα ∈ ΣQ and (NP (X)α, α−1 , i) ∈ T(X). Proof. By 12.3.37.2, there is β ∈ A(P ) with P β ≤ U . Let γ ∈ A(P β) with P βγ = P α. As W ≤ CS (P β), γ is a U-map by 12.3.33.2, so as P β ≤ U , also P α = P βγ ≤ U , proving (1). Let R = CU (P )α. Then Rα−1 ≤ U , so R ∈ P and then (2) holds. By 12.3.37, P and U are TI-subgroups of S with P ∩ U = 1, so |CU (P )| ≥ |P |; then (3) follows. Parts (4) and (5) are straightforward.   Lemma 12.3.47. For i ∈ I, set Zi = ZS (ρi ), ci and Z = i∈I Zi . Assume A ∈ ΞP and let α ∈ A(P ). Then (1) AP centralizes Z. (2) U α ∈ P ! with Zα ≤ NS (U α).  (3) For each i ∈ I, Xi α ∈ ΣUα , where Xi = i =j∈I Zi . (4) Let E4 ∼ = E ≤ Xi with 1 = E ∩ U . Then Eα ∈ ΞUα . (5) Xi α centralizes Z. (6) For K ∈ Ω, |NK (U α)| = 2. (7) m(U Xi ) ≤ 2. (8) n = 3 or 4 and r = 2. (9) c1 = c2 = c and U = c ∼ = Z2 . Proof. Set P  = U α; by 12.3.45.3, P  ∈ P ! . Suppose (6) fails. Then there is x of order 4 in NK (P  ). By 12.3.46.5, xα ∈ ΣP  . But xα ∈ D ∈ Δ, so xα ∈ Sj for some j ∈ I, contrary to 12.3.42.3. This proves (6). By 12.3.46, A acts on each Wi and AP centralizes U .

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Assume A does not centralize Z; then for some t ∈ ZS and a ∈ A, s = ta = t. As in the proof of 12.3.45, we may assume A = PA a . Let Y = Q(t)Q(s) and observe that Y = Q(t) × Q(s) and Ya = CY (a) is a full diagonal subgroup of Y . By 12.3.44.3, Wa = W ∩ Ya centralizes P , so as Wa ≤ Z(Y ) it follows that P acts on {Q(t), Q(s)}, so B = NA (Q(t)) is of index 2 in A. Also PA acts on Ya , and if [PA , Ya ] = 1 then Z(Ya ) ≤ [PA , YA ] ≤ P , contradicting Z(Ya ) ≤ W and 12.3.44.3. Thus A = PA a centralizes Ya , so B centralizes a full diagonal subgroup of Y and therefore centralizes Y . But this contradicts (6), showing that A centralizes Z. Then Z ≤ NS (P ), so [P, Z] ≤ P ∩ Z ≤ P ∩ W = 1, proving (1). Recall P  ∈ P ! and as Z ≤ NS (P ), we have Zα ≤ NS (P  ), proving (2). In (3), Xi centralizes Ci , so (3) follows from 12.3.46.5. Then (3) implies (4). By (4) and symmetry between P and P  , Eα centralizes Z, so (5) follows. Set B = (U Xi )α. By (5), B centralizes Z, so B acts on {K, K2 } = Ω(z), and then |B : NB (K)| ≤ 2. Let V be of order 4 in K ∩ W . Then |NB (K) : CB (V )| ≤ 2, so |B : CB (V )| ≤ 4. Hence to prove (7), it suffices to show that CB (V ) = 1. But if b ∈ CB (V )# then V ≤ CS (b) ≤ NS (P  ) by (4), contrary to (6). Hence (7) holds. Next m(Zi ) ≥ n/2, so n = 3 or 4 by (7). Similarly if r > 2 then m(Xi ) > 2, contrary to (7). This proves (8). Indeed if c1 = c2 then m(U Xi ) = 3, again contrary to (7), so (9) holds.  Before leaving the setup of this lemma, we derive a contradiction to the assumption that ΞP = ∅. Namely by (8) and (9), Z1 = z, c ∼ = E4 . Set b = cα, B = Z1 α, Y = KK2 , and Y + = Y /z . From the proof of (7), for some a ∈ B −b , K a = K2 , so setting Ya = CY (a), Ya+ is a full diagonal subgroup of Y + = K + × K2+ . Now Ya ≤ NS (P  ) = CS (b), so B = a, b centralizes Ya . Set d = NB (K). As d centralizes the full diagonal subgroup Ya+ and acts on K, it follows that d centralizes K + . But then |CK (d)| > 2, contrary to (6). We’ve shown: Lemma 12.3.48. ΞP = ∅. Lemma 12.3.49. (1) P = a ∼ = Z2 . (2) r = 2. (3) W1a = W2 . Proof. By 12.3.46.4, P ∈ ΣP , so (1) follows from 12.3.48.  By 12.3.46.3, P  = CU (P )α ∈ P ! . Suppose ∅ = J ⊂ I such that X = j∈J Wj is a-invariant. Then X ≤ CS (Ci ) for i ∈ I − J, so by 12.3.46.5, A = CX (a)α ∈ ΣP  . Then by 12.3.47, m(A) = 1, so as X is abelian, m(X) = 2. But if r > 1 we may choose J = {1, 2}, so m(X) ≥ 3, a contradiction. This establishes (2) and shows J = {1} and X = W1 . As m(X) = 2, n = 3 or 4. As m(A) = 1 and a centralizes c1 , z a = z. Thus Y = CO(z)O(za ) (z) is a full diagonal subgroup of O(z) × O(z a ), so Y ∼  = O(z), contradicting m(A) = 1. This completes the proof of (3). We are now in a position to obtain a contradiction, and hence establish Theorem 12.3.8. Adopt the notation of 12.3.49, and let v be of order 4 in K ∩ W and X = v, v a . By 12.3.49, W1a = W2 , so X ∼ = Z24 and Y = a X is wreathed. Therefore H = CX (a) ∼ = Z4 and the involution in H is b = zz a with ab ∈ aX . Let α ∈ A(P ), d = (ab)α, and Q = d α. As P ∈ P! is of order 2 and ab ∈ aX , we

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have Q ∈ P ! , so W1d = W2 by 12.3.49. But H ≤ CS (P ), so Hα ≤ NS (Q) and then d ∈ Φ(Hα) ≤ Φ(S), contradicting W1d = W2 . This contradiction completes the proof of Theorem 12.3.8. Theorem 12.3.50. Assume Hypothesis 12.3.1 and let M ∈ M (τ ) and SM = S ∩ M . Then μ(τ ) ∼ = Weyl(Dn ) for some n ≥ 3, and one of the following holds: (1) F ◦ = FSM (M ). ˇ 2n , or AE2n+1 . (2) m = 8 and F ◦ ∼ = AE2n , AE + ◦ ∼ (3) F = (P )Ω2n [m], Ω2n+1 [m], or Ω− 2n+2 [m]. ˇ + (2). ˇ 3 (2)/23+6 or Ω (4) (n, m) = (4, 8) and F ◦ ∼ =L 8 ◦ ∼ (5) (n, m) = (3, 8) and F = L3 (2)/E64 or Sp6 (2). Proof. By Theorem 12.3.8 and 12.3.4, μ(τ ) ∼ = Weyl(Dn ) for some n ≥ 3. If n = 3 then the theorem follows from Theorem 9.5.33. If n = 4 the theorem follows from Theorem 12.1.25. Thus we may assume n ≥ 5. Then the hypothesis of Theorem 12.2.32 is satisfied, and that theorem completes the proof.  12.4. The proof of Theorem 6 In this section we prove Theorem 6 under the additional hypothesis that τ satisfies the Extended Inductive Hypothesis. At the end of the paper, when we will finally be in a position to prove Theorem 1, we will remove the assumption that the Extended Inductive Hypothesis holds, and complete the proof of Theorem 6. So in this section we assume the hypothesis of Theorem 6, together with the Extended Inductive Hypothesis. Let z ∈ ZS ∩ F f . By the hypothesis of Theorem 6, |Ω(z)| > 1. Suppose first that |Ω(z)| > 2. Then Hypothesis 10.2.1 is satisfied. But then one of the conclusions of Theorem 6 holds by Theorem 10.2.13. Hence we may assume: Lemma 12.4.1. |Ω(z)| = 2. Lemma 12.4.2. Z ∩ O(z) = {z}. Proof. If not then one of the conclusions of Theorem 6 holds by Theorem 9.3.24.  Lemma 12.4.3. D∗ (z) = D(z). Proof. If not then one of the conclusions of Theorem 6 holds by Theorem 9.4.33.  Observe that by 12.4.1-12.4.3, Hypothesis 12.3.1 is satisfied. But now one of the conclusions of Theorem 6 holds by Theorem 12.3.50. This completes the proof that Theorem 6 holds under the additional assumption that the Extended Inductive Hypothesis is satisfied.

Theorems 7 and 8

CHAPTER 13

|Ω(z)| = 1 and μ abelian In Chapter 13 we prove Theorem 7, modulo the Extended Inductive Hypothesis. Thus our assumption throughout the chapter is that τ = (F, Ω) is a quaternion fusion packet such that for each z ∈ ZS (τ ) we have |Ω(z)| = 1, and such that μ = μ(τ ) is abelian. In section 13.1 we work under Hypothesis 13.1.1, which also assumes that either (i) F = SF ◦ and F ◦ is isomorphic to (P )Sp2ns [m], (P )Sp− 2nS [m], [m], or (ii) τ satisfies the Extended Inductive Hypothesis . Case (i) is or L− 2nS +1 included in order to establish various properties of such systems used in the generational results for the systems established in sections 13.3, 13.4, and 13.5. Section 13.1 proves some basic results and then in Theorem 13.1.15 reduces our treatment of systems satisfying Hypothesis 13.1.1 to the generic case of Hypothesis 13.2.1 where AutF (O(τ )) induces Sym(Ω) on Ω and some member of Z(τ ) moves some member of Ω. In section 13.2 we establish various results under this hypothesis, which set the stage for the generational results in the middle of the chapter. In the final section we use the theory developed in the earlier sections to determine in Theorem 13.6.6 the systems satisfying Hypothesis 13.1.1 such that F ◦ is transitive on Ω. Then in Remark 13.6.7 we observe that Theorem 13.6.7 and Theorem 1 imply Theorem 7.

13.1. Systems with μ abelian In this section we assume the following hypothesis: Hypothesis 13.1.1. τ = (F, Ω) is a quaternion fusion packet such that: (1) For each z ∈ ZS , |Ω(z)| = 1. (2) μ(τ ) is abelian. (3) Either − (i) F = SF ◦ and F ◦ = O 2 (F) ∼ = (P )Sp2nS [m], (P )SL− 2nS [m], or L2nS +1 [m], or (ii) τ satisfies the Extended Inductive Hypothesis Notation 13.1.2. Pick z ∈ F f ∩ Z and let K ∈ Ω(z). Set μ = μ(τ ). For ) = {t ∈ s ∈ Z, define Do (s) = CZ (s) − s⊥ . Set Z  = Z  (τ ) = Z − ZS , Z m = Z m (τ  Z : ZS ⊆ CS (t)}, and O = O(τ ). For λ ⊆ Ω set λ = Ω − λ and O(λ) = J∈λ J. For s ∈ S, write sΩ for the image in Sym(Ω) of s. Lemma 13.1.3. Suppose θ ⊆ Z such that for each t ∈ θ, θ ⊆ t⊥ . Set Q = θ . Then for α ∈ A(Q), θα ⊆ ZS . 341

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Proof. Replacing θ by θα, we may assume that Q ∈ F f . Let t ∈ θ, Γ = θ−{t}, and β ∈ A(Γ). By induction on |θ|, Γβ ⊆ ZS . By hypothesis, Γβ ⊆ (tβ)⊥ , so for each s ∈ Γ, tβ centralizes O(sβ) by 3.3.4. Let γ ∈ A(tβ); then O(sβ)γ ∈ Ω, so θβγ ⊆ ZS . Finally let δ ∈ A(Qβγ) with Qβγδ = Q. Then O = Oδ ≤ CS (Q), so  θ ⊆ ZS by 3.1.5.3. Lemma 13.1.4. Suppose t ∈ Do (z) and set E = z, t . Then (1) Kt ∼ = SD2n . (2) Do is a symmetric F-invariant relation. (3) Let v ∈ K be of order 4 with v  CS (z), α ∈ A(t), and u ∈ O(tα) of order 4 with u  CS (tα). Then GL(E) = cv , cu α−∗ = AutF (E). Proof. Part (1) follows from 3.3.6. As Do (s) = CZ (s) − s⊥ for s ∈ Z and D is a symmetric F-invariant relation by 3.1.13, (2) follows. Assume the setup of (3). By (1), t inverts v, so cv ∈ AutF (E) is the transvection on E with center z. By (2), zα ∈ Do (tα), so by symmetry cu is the transvection on Eα with center tα, and hence cu α−∗ is the transvection on E with center t. Thus (3) holds.  Lemma 13.1.5. Assume 13.1.1.3.ii holds and ρ = (Y, Γ) is a subpacket of τ with Γ ⊆ Ω. (1) ρ satisfies conditions (1) and (2) of Hypothesis 13.1.1 and the Inductive Hypothesis. (2) ρ◦ = ρ1 ∗· · ·∗ρr is a central product of quaternion fusion packets ρi = (Yi , Γi ) such that ρi satisfies conditions (1) and (2) of Hypothesis 13.1.1 and the Inductive Hypothesis, and Yi = Yi◦ is transitive on Γi . (3) Assume ρ is proper in τ and let 1 ≤ i ≤ r, Zi = Z(ρi ), Ki ∈ Γi , zi = z(Ki ), and μi = μ(ρ)i . Then one of the following holds: (i) Γi = {Ki } and Yi = OKi . (ii) μi ∼ = Z2 , Zi − {zi } = D o (zi ), and Yi ∼ = L2 [2m](1) . ⊥ ˆ 5. (iii) m = 8, μi ∼ = E4 , CZi (zi ) = zi , and Yi ∼ = AE (iv) μi ∼ = E2ni , CZi (zi ) = zi⊥ , and Yi ∼ = (P )Sp2ni [m] for some integer ni ≥ 2. (v) μi ∼ = E2ni , Do (z(J)) = ∅ for J ∈ Γi , and Yi ∼ = (P )SL− 2ni +a [m] for some ni and a ∈ {0, 1} with 2ni + a ≥ 3. Proof. Part (1) is trivial. Part (2) follows from (1), 13.1.1.3.ii, and 6.6.6. Assume ρ is proper. Then for 1 ≤ i ≤ r, ρi satisfies the hypothesis of Theorem 1 and the Inductive Hypothesis by (2), so ρi satisfies one of the conclusions of Theorem 1. We conclude that one of the following holds: (a) Zi ⊆ ZS , and (cf. 13.1.11) Yi = O(ρi ). (b) Zi = ∅ = Zim and (cf. 13.1.12) Yi ∼ = L2 [2m](1) or L− 3 [m]. m ˆ (c) Z1 = ∅ and (cf. 13.1.15 and 13.6.6) Yi is AE5 , (P )Sp2n [m], (P )SL− 2n [m], or L− 2n+1 [m]. In case (a), as Yi = Yi◦ is transitive on Γi , (3i) holds. In case (b), (3ii) or (3v) holds. Finally in case (c), (3iii), (3iv), or (3v) holds. This completes the proof.  Lemma 13.1.6. τz◦ = τ1 ∗· · ·∗τr where τi = (Fi , Ωi ) is a quaternion fusion packet satisfying conditions (1) and (2) of Hypothesis 13.1.1 and the Inductive Hypothesis, and τi is described in 13.1.5.3. Proof. In 13.1.1.3.i the lemma follows from the discussion in section 5.4, so assume that 13.1.1.3.ii holds. Then the lemma is essentially the special case of

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13.1.5 with ρ = τz . If z ∈ / Z(F) the application is immediate. If z ∈ Z(F) then  OK  F and we apply 13.1.5 to ρ = (F, Ω − {K}). Lemma 13.1.7. Let t ∈ D(z). (1) For α ∈ AFz (t), tα ∈ ZS . (2) Let T be Sylow in Fz◦ . Then t ∈ T and tα = tϕφ for some ϕ ∈ AutFz (T ) and φ ∈ homFz◦ (tϕ, T ). Indeed tϕ and tα are in Ti Sylow in Fi for some i, and tϕ is fused to tα in Fi . (3) Let θ ⊆ Ω with O(θ) ∈ F f and t ∈ D(z(J)) for each J ∈ θ. Let ρ = (CF (O(θ)), θ ). Then t ∈ Z(ρ). Proof. Let θ = {z, t} and β ∈ A(θ ). By 13.1.3, θβ ⊆ ZS . Let γ ∈ A(zβ) with zζ = z where ζ = βγ. Then Oγ = O, so tζ ∈ ZS . Let δ ∈ AFz (tζ) with tζδ = tα. Then Oδ = O, so tα ∈ ZS , establishing (1). By 13.1.6, τz◦ = τ1 ∗ · · · ∗ τr , and by (1), tα ∈ ZS (τi ) for some i. As Fz◦  Fz and tα ∈ T , also t ∈ T and ϕ and φ exist as in (2). Then as Fz◦ is the central product of the Fi , tϕ is fused to tα in Fi . Thus (2) holds. Finally (3) follows from (2) by induction on |Ω|.  Lemma 13.1.8. Assume Ki ∈ Ω and zi = z(Ki ) for i = 1, 2 with 1 = u = z1 z2 ∈ Z(F). Assume t ∈ Z such that either (a) K1t = K2 , or (b) t ∈ D o (zi ) for i = 1, 2. Set Y = [K1 ]F and Γ = K1F . Then (1) ρ = (Y, Γ) is a quaternion fusion packet and ρ = ρ1 ∗ · · · ∗ ρr is a central product where ρj = (Yj , Γj ), |Γj | = 2, Γ1 = {K1 , K2 }, t is fused to z1 and z2 in ˆ 5 , Sp4 [m], or SL− [m]. Y1 , and Yj ∼ = AE 4 ◦ (2) τ = ρ ∗ ξ1 ∗ · · · ∗ ξk , where the ξj = (Xj , Σj ) are coconnected components of τ ◦ and Xj is described in 13.1.5.3. Proof. In 13.1.1.3.i, as u ∈ Z(F) we conclude that F ◦ is Sp4 [m] or SL− 4 [m], and then the lemma follows. Therefore we may assume that 13.1.1.3.ii holds. By 13.1.5.2, τ ◦ = τ1 ∗ · · · ∗ τn is a central product, where τi = τi◦ = (Ti , Ωi ) and Ti is transitive on Ωi . By 2.5.2, ρ = (Y, Γ) is a subpacket of τ , and by construction, Y  F. Thus we may choose notation so that τj = ρj = (Yj , Γj ) for 1 ≤ j ≤ r are the conjugates of Y1 , and for 1 ≤ j ≤ k, τr+j = ξj = (Xj , Σj ). Further we may choose K1 ∈ Γ1 . Now in cases (a) and (b), z1 , t, z2 is a path in Dc , so it follows that as K1 ∈ Γ1 , also z1 , t, z2 ∈ Z1 = Z(ρ1 ). Set F + = F/u and τ + = (F + , Ω+ ). Then τ + is a quaternion fusion packet by 3.3.2.2. Moreover (τ ◦ )+ = τ1+ ∗· · ·∗τn+ is a central product with τj+ = (Tj+ , Ω+ j ) and + + + Tj is transitive on Ωj . By 6.6.9.2, τ satisfies the Extended Inductive Hypothesis. Further Ω(z1+ ) = {K1+ , K2+ } is of order 2, and from 13.1.1.2, μ(τ + ) is abelian. It + + + + + follows from Theorem 9.4.33 that Γ+ 1 = {K1 , K2 }, t is fused to z1 in Y1 , and + ∼ − Yj = AE5 , Ω5 [m], or Ω6 [m] for each 1 ≤ j ≤ r. Then (1) follows from 5.1.19 and 3.3.16. Part (2) follows from the discussion above together with 13.1.5.3.  Lemma 13.1.9. Assume Ki ∈ Ω and zi = z(Ki ) for i = 1, 2 with 1 = u = z1 z2 ∈ F f . Assume t ∈ Z such that either (a) K1t = K2 , or (b) t ∈ D o (zi ) for i = 1, 2. Then

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(1) τu◦ has a coconnected component τ1 = (Y, Γ) where Γ = {K1 , K2 }, t is fused ˆ 5 , Sp4 [m], or SL− [m]. to z1 and z2 in Y, and Y ∼ = AE 4 (2) Suppose s ∈ Z centralizes K1 and K2 . Then s ∈ Z(τu ) and either s centralizes Y or s ∈ {z1 , z2 }. Proof. From 2.5.2, τu = (Fu , Ω) is a quaternion fusion packet. Suppose first that (a) holds. Then tu ∈ tK1 , so θ = {t, tu} ⊆ Z. Claim tu ∈ D(t). If not then tu ∈ Do (t), so by 13.1.4.3, u ∈ tAutF (t,u ) ⊆ Z and u ∈ / D(t). Let α ∈ A(t) with tα = z. Observe X = CK1 K2 (t) ∼ = K1 with u = z(X), so uα induces an inner automorphism on K, and hence uα ∈ D(z), so u ∈ D(t), contradicting and earlier remark. This proves the claim. Set Q = θ and let α ∈ A(Q); by 13.1.3, θα ⊆ ZS . Let β ∈ A(uα) with uζ = u, where ζ = αβ. Then O(θα)β = O(θζ) so θζ ⊆ ZS . Therefore as ζ ∈ homFu (Q, Qζ), it follows that t ∈ Z(τu ). Hence (1) follows from 13.1.8 if u ∈ Z(F), while if u∈ / Z(F) then τu is proper in τ , and (1) follows from 13.1.5 and the structure of the coconnected components of τu in 13.1.5.3; cf. section 5.4. Next assume that (b) holds. Let α ∈ A(t) with tα = z. By 13.1.4.2, zi α ∈ Do (z) for i = 1, 2, so zi α inverts the CS (z)-invariant cyclic subgroup V of K of index 2. Therefore uα = z1 α · z2 α centralizes V . Let β ∈ A(uα) with uζ = u, where ζ = αβ. Then ζ ∈ homFu (CS (z1 , u, t ), S). Now V β ≤ J ∈ Δ(tζ), and by 13.1.1.2, J ∈ Ω, so tζ ∈ ZS . Hence again t ∈ Z(τu ), so again (1) follows from 13.1.8 if u ∈ Z(F) and from 13.1.5 and the structure of the coconnected components of τu in 13.1.5 when u ∈ / Z(F). Assume the hypothesis of (2) with s ∈ / {z1 , z2 }, and let α ∈ A(s) with sα = z and β ∈ A(uα) with uζ = u, where ζ = αβ. Then Ki α centralizes K, so Kβ centralizes (K1 K2 )ζ and thus as u ∈ (K1 K2 )ζ, we have sζ ∈ ZS centralizes u. Therefore as ζ ∈ homFu (K1 K2 s , S), s ∈ Z(τu ) so sζ ∈ Z(ρ ) for some coconnected compo/ {z1 , z2 } centralizes K1 K2 , ρ = ρ, so s centralizes Y, proving nent ρ of τu◦ . As s ∈ (2).  Lemma 13.1.10. Let t ∈ Z. Then (1) t ∈ ZS iff t ∈ s⊥ for each s ∈ ZS . (2) Suppose Ki ∈ Ω and zi = z(Ki ) for i = 1, 2 with 1 = u = z1 z2 and t ∈ Z such that either (a) K1t = K2 , or (b) t ∈ D o (zi ) for i = 1, 2. ◦ Let α ∈ A(u). Then there exists a coconnected component ρ = (Y, Γ) of τuα such ∼ ˆ that Γ = {K1 α, K2 α}, tα is fused to z1 α and z2 α in Y, and Y = AE5 , Sp4 [m], or SL− 4 [m]. (3) If t ∈ Z m then t induces a transposition (K1 , K2 ) on Ω and centralizes each member of Ω − {K1 , K2 }. (4) If Do (t) ∩ ZS = ∅ then |D o (t) ∩ ZS | ≤ 2 and ZS − D o (t) ⊆ D(t). (5) Assume the hypothesis of (2) with t ∈ Z m . Then Z centralizes u and acts on {K1 , K2 }. (6) Let ti ∈ Z  and Γi = {J ∈ Ω : [ti , J] = 1}, for i = 1, 2. Assume Γ1 ∩Γ2 = ∅. Then [t1 , t2 ] = 1. (7) If t, a ∈ Z m with tΩ = aΩ then [t, a] = 1, Proof. Part (1) follows from 13.1.3.

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Assume the hypothesis of (2); replacing u by uα, we may assume u ∈ F f and α = 1. Then (2) follows from 13.1.9. Suppose t moves some K1 ∈ Ω or is nontrivial on two members K1 and K2 of Ω; then we may assume the setup of (2) holds with u ∈ F f and α = 1. Therefore for each J ∈ Ω − Γ, J ∈ Γ where ρ = (Y  , Γ ) is a coconnected component of τu◦ distinct from ρ, and hence as t ∈ Z(ρ), t centralizes J. It follows that (3) and (4) hold. Suppose t ∈ Z m . As distinct transpositions in S Ω commute, it follows that all members of Z act on {K1 , K2 } and hence centralize u, establishing (5). Assume the hypothesis of (6). If Γ1 = {K1 , K2 } is of order 2, then, replacing u = z1 z2 by uα for α ∈ A(u), we may assume u ∈ F f . Then (6) follows from 13.1.9.2. Thus we may assume Γi = {Ki } is of order 1. Let θ = Ω − {K1 , K2 }. If n = |Ω| > 2 then appealing to 13.1.7.3 and induction on n, (6) holds, so we may take n = 2. Then by 13.1.6 and 13.1.7.2, Fz◦i = O(zi ) ∗ Yi with Yi ∼ = L2 [2m](1) or − L3 [m] and t3−i ∈ Ti Sylow in Yi with Ti ∼ = SD2m . Then [T1 , T2 ] ≤ CT1 (O(z1 )) ∩ CT2 (O(z2 )) = 1, completing the proof of (6). Assume the hypothesis of (7). Then t = (K1 , K2 ) and a = (J1 , J2 ) are distinct cycles on Ω, so (7) follows from (3) and (6).  Lemma 13.1.11. If Z = ZS and F is transitive on Ω then F ◦ = O(τ ). Proof. By 13.1.1.1, OK is tightly embedded in F. Thus the lemma follows from Theorem 3.6.8 in [Asc19].  Lemma 13.1.12. Assume Z  = ∅ = Z m then F ◦ is the central product of conjugates of L2 [2m](1) or L− 3 [m]. ∼ L− [m], so the Proof. If 13.1.1.3.i holds then as Z  = ∅ = Z m , we have F ◦ = 3 lemma holds. Thus we may assume that 13.1.1.3.ii holds. Let n = |Ω|. If n = 1 the lemma is a consequence of Theorem 7.1.6, so assume n > 1. (a) For each t ∈ Z  , there exists a unique J ∈ Ω with [J, t] = 1. As Z m = ∅, t acts on each member of Ω, while as t ∈ Z  , [J, t] = 1 for some J ∈ Ω by 13.1.10.1. If t is nontrivial on two members of Ω, then by 13.1.10.2, there ˆ 5 , Sp4 [m], or SL− [m]. But is a subpacket ρ = (Y, Γ) of τ with Y isomorphic to AE 4 m m then there is s ∈ Z (ρ), contradicting Z = ∅. From 3.1.5.4 and the transitivity of F on Ω: (b) AutF (O) is transitive on Ω. Let I = {1, . . . , n}, Ω = {Ki : i ∈ I}, and set zi = z(Ki ). Set Zi = {t ∈ Z  : [Ki , t] = 1} and Ti = Zi , Ki . As Z m = ∅, T = T1 · · · Tm is in the kernel of the action of S on Ω, so by (b) and a Frattini argument: (c) AutF (T ) is transitive on Ω and {Ti : i ∈ I}. By (a), Zi = ∅ for some i ∈ I, and hence Zi = ∅ for each i ∈ I by (c). Take z = z1 . (d) τz◦ = O(z) ∗ τ2 ∗ · · · ∗ τn , where for 1 < i ∈ I, τi = (Yi , Ki ) is a quaternion ∼ fusion packet, Yi ∼ = L2 [2m](1) or L− 3 [m], Ti is Sylow in Yi , and Ti = SD2m . ◦ By 13.1.6, τz = τ1 ∗ · · · ∗ τr with τi = (Yi , Γi ) and Yi is described in 13.1.5.3. Let t ∈ Zi for i > 1. By (a), t centralizes K, so by 13.1.7.2, t ∈ Z(τj ) for some j ∈ I. Then t centralizes Yk for k = j, so Ki ∈ Γj . As Z m = ∅, Z m (τl ) = ∅ for

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each l, so from 13.1.5.3, Yl ∼ = L2 [2m](1) or L− 3 [m]. Thus Γj = {Ki } and a Sylow group Qj of Yj is isomorphic to SD2m . Therefore Qj = Ti , establishing (d). (e) T1 centralizes Yi for 1 < i ∈ I. From (d), K centralizes Yi , so it remains to show t ∈ Z1 centralizes Yi . By 13.1.10.6, T1 centralizes Ti . Hence by 2.6.12, T1 centralizes Oi = CYi (zi ). Next Yi = Oi , AutYi (R) where R is a 4-subgroup of Ti . Let G be a model for t NYi (R); then G = a × H where H ∼ = S4 is a model for NYi (R) and a = t or tzi . If a = t then (e) holds, so assume a = tzi . Let α ∈ AutF (T ) with zi α = z. Then T1 α = Tl for some 1 < l ∈ I, so (tα)z f is fused to (zα)z in Fz . Therefore a = tzi ∈ uF , where  u = zj zk ∈ F , so there ◦ is β ∈ A(a) with aβ = u. Now Fu = O(zj ) ∗ O(zk ) ∗ r =j,k Yr by 13.1.5, and as Z m = ∅. As u = tβ · zi β with Ki β ∈ Ω, it follows that zi β = zj and tβ = zk . This is a contradiction as Yi β ∗ ≤ Fu◦ and zi ∈ / Z(Yi ). Hence the proof of (e) is complete. By (c), (e), and 2.2 in [Asc11]: (f) There is Y1 ≤ F with Sylow group T1 such that Y = Y1 ∗ · · · ∗ Yn ≤ F is a central product and AutF (T ) is transitive on {Yi : i ∈ I}. (g) Let ρi = (Yi , Ki ), i ∈ I, and set Zi = Zi ∪ {zi }. Then Zi = Z(ρi ) and {Zi : i ∈ I} is the set of connected components of D(τ )c . From (a), Z  = i∈I Zi , and as Yi is L2 [2m](1) or L− 3 [m] with Ti Sylow in Yi , and as Ti ∼ SD , it follows that Z is the set of involutions in Ti , so Zi = Z(ρi ). = 2m i Then from (f), {Zi : i ∈ I} is the set of connected components of D(τ )c . By (f) and (g) and 8.1.11, {ρi : i ∈ I} is the set of coconnected components of  τ ◦ , completing the proof of the lemma. In light of 13.1.11 and 13.1.12, during the remainder of the section we assume the following hypothesis: Hypothesis 13.1.13. Hypothesis 13.1.1 holds and (1) F is transitive on Ω, and (2) Z m = ∅. Lemma 13.1.14. Let A = AutF (O) and write AΩ for the image of A in Sym(Ω). For t ∈ Z m let ct be conjugation by t. Set B = ct : t ∈ Z m ≤ A. (1) A is transitive on Ω. (2)  There is an A-invariant partition Λ of Ω such that for λ ∈ Λ, |λ| ≥ 2, and B = λ∈Λ Bλ is a direct product, where Bλ = ct : t ∈ Zλ , Zλ = {t ∈ Z m : λ ⊆ F ixΩ (ct )}, and BλΩ acts faithfully as Sym(λ) on λ.

(3) For each λ ∈ Λ, Bλ centralizes O(λ ) and Zλ centralizes Zλ = σ⊆λ Zσ . Proof. By 13.1.13.1, F is transitive on Ω, so (1) follows from 3.1.5.4. Let t ∈ Z m . By 13.1.10.3, cΩ t is a transposition. Then (2) follows from this fact and (1). In particular {Zλ : λ ∈ Λ} is a partition of Z m with λ ⊆ F ixΩ (ct ) for t ∈ Zλ . Then by 13.1.10.7, Zλ centralizes O(λ ) and Zλ . Finally as B acts on  O(λ ) and Bλ = ZλB , Bλ centralizes O(λ ), completing the proof of (3). Theorem 13.1.15. Assume AutF (O) does not induce Sym(Ω) on Ω. Then ˆ 5 , (P )Spn [m], or (P )SL− F ◦ is the central product of the conjugates of AE n [m] for some n ≥ 4.

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During the remainder of the section assume that F is a counter example to Theorem 13.1.15. As AutF (O) does not induce Sym(Ω) on Ω, 13.1.1.3.i does not hold, so 13.1.1.3.ii holds. Adopt the notation in 13.1.14. As AΩ = Sym(Ω), the partition Λ of Ω in 13.1.14 is nontrivial. Set n = |λ| for λ ∈ Λ and k = |Λ|. Thus n, k ≥ 2 and |Ω| = nk. Pick λ ∈ Λ with O(λ) ∈ F f . As NS (O(λ)) = NS (O(λ )), also O(λ ) ∈ F f . Set Y = Yλ = CF (O(λ ))◦ and ρ = ρλ = (Y, λ), and let Tλ be Sylow in Y. By 2.5.2, ρ is a quaternion fusion packet. Indeed: Lemma 13.1.16. (1) n = 2 and Y (2) n ≥ 3 and Y

Either ∼ ˆ 5 , Sp4 [m], or SL− [m], or = AE 4 ∼ = (P )Sp2n [m] or (P )SL2n+a [m] with a = 0 or 1.

Proof. By 13.1.5, ρ = ρ1 ∗ · · ·  ∗ ρr with ρi = (Yi , Γi ) described in 13.1.5.3. As B acts on O(λ ), B acts on O(λ) = i O(Γi ), and permutes {Γ1 , . . . , Γr }. Hence as ˆ 5 then n = 2, B λ = Sym(λ) is primitive on λ, we conclude that r = 1. If Y ∼ = AE while if Y ∼ = (P )SL2n+1 [m]; = (P )Spl [m] or (P )SLl [m] then either l = 2n or Y ∼ thus the lemma follows from 13.1.5.3.  Lemma 13.1.17. For σ ∈ Λ, set Zσ = {t ∈ Z  : σ  ⊆ CS (t)}. Then (1) {Zσ : σ ∈ Λ} is a partition of Z  . (2) Z  (ρ) = Zλ . Proof. We first prove (1). Let t ∈ Z  . If t ∈ Z m then t ∈ Zσ for a unique σ ∈ Λ by 13.1.10.3, so we may assume ZS ⊆ CS (t). By 13.1.10.1, t is nontrivial on some J ∈ Ω. Let J ∈ σ ∈ Λ. If t centralizes Ω−{J} then σ is the unique ζ ∈ Λ with t ∈ Zζ , so assume L ∈ Ω − {J} with [L, t] = 1. Set u = z(J)z(L) and let α ∈ A(u). By 13.1.10.2, there exists s ∈ Z m with cycle (Jα, Lα) on Ω, so Jα, Lα ∈ ζ ∈ Λ. As O centralizes u, α ∈ AutF (O), so ζ = σα, and hence L ∈ σ, completing the proof of (1). Choose notation so that K ∈ λ. By definition of ρ, Z(ρ) = z Y and Z  (ρ) = Y z − ZTλ . As z centralizes O(λ ), Z  (ρ) ⊆ Zλ . Conversely if t ∈ Zλ then for each  J ∈ Ω with [J, t] = 1, J ∈ λ, so by 13.1.7.3, t ∈ Z  (ρ). Set S0 = CS (O) and let G be a model for NF (OS0 ). For σ ∈ Λ let Hσ = (Zσ )NG (σ) O(σ). 

Lemma 13.1.18. (1) Hλ = (Zλ )Gλ O(λ) = O 2 (Gλ ), where Gλ is model for NY (O(λ)). (2) Tλ ∈ Syl2 (Hλ ). (3) H = Hσ : σ ∈ Λ is a central product of the Hσ , and G acts transitively on {Hσ : σ ∈ Λ} via Hσg = Hσg . Proof. Let X = NF (O(λ )). By construction Y  X and NG (λ) is a model for NX (OS0 ). By 2.6.12, S0 centralizes O(τ ), so we conclude from the structure of Y in 13.1.16 that OS0  OS0 NY (O). Then NY (O)  NX (OS0 ), so we may take Gλ  NG (λ). Then as Zλ ⊆ Gλ by 13.1.17, Hλ ≤ Gλ . Indeed from the description

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of Y in 13.1.16 and 5.4.8-5.4.10, O 2 (Gλ ) = (Zλ )Gλ O(λ), so (1) holds. Then (2) follows from (1). As Hλ  NG (λ), Hλg = Hλg for g ∈ G. In particular as G is transitive on Λ, G is transitive on {Hσ : σ ∈ Λ}. Let GΛ be the kernel  of the action of G on Λ. As H ≤ GΛ ≤ NG (σ), Hσ  H. Set Hλ = σ =λ Hσ . Then Hλ ∩ Hλ ≤ CHλ (O(λ)) ∩ CHλ (O(λ ) ≤ Z(O(λ)) ∩ Z(O(λ )) ≤ Z(H). Therefore [Hλ , Hλ  ] ≤ Z(H), so [Hλ , O 2 (Hλ )] = 1. Then as Hλ = O 2 (Hλ )Zλ  , where Zλ  = σ∈Λ−{λ} Zσ , and as Zλ centralizes Zσ for each σ ∈ Λ − {λ} by 13.1.10.6,  we have [Hλ , Hλ ] = 1. This completes the proof of (3).   For σ ∈ Λ, let Tσ = S ∩Hσ and set T = σ∈Λ Tσ . For θ ⊆ Λ set Tθ = σ∈θ Tσ . Define Yσ = Yc∗g for g ∈ NG (T ) with λg = σ. By 13.1.18, NG (T ) ∩ NG (λ) ≤ NG (Tλ ), so Yσ is independent of g. Lemma 13.1.19. (1) F contains a central product Y0 = (2) AutF (T ) acts on Y0 and on Z = Z(Y0 ).

 σ∈Λ

Yσ .

Proof. By 13.1.18.3, Q = Tλ centralizes Hλ ; therefore Q centralizes NY (O(λ)) using 13.1.18.1, and observing from 13.1.16 that CGλ (Tλ ) = Z(Gλ ). Take K ∈ λ and t ∈ Zλ with K t = K. Set u = zz t . By 13.1.9.2, Zλ  centralizes CTλ (u)Xu , where Xu is the coconnected component of (CY (u), λ) described in 13.1.9.1. Therefore by 1.3.2 in [Asc19], U = Zλ  O(λ ) centralizes Y = CTλ (u)Xu , NY (O(λ)) . Let X = NF (O(λ )), X a model for NX (OS0 ), and C = CX (Y). Then [U ]X ≤ C. But Hλ ≤ X with Tλ ≤ Hλ = U Hλ by 13.1.18, so Tλ ≤ C. Therefore Tλ centralizes Y, so Tλg centralizes Yλg for g ∈ NG (T ). Then (1) follows from 2.2 in [Asc11]. As AutF (T ) permutes Λ, it permutes the Tσ and Yσ by construction. Thus  AutF (T ) acts on Y0 , and hence also on Z(Y0 ), proving (2). Lemma 13.1.20. Z(F) = 1. Proof. Assume otherwise and let y ∈ Z(F) be an involution and set F + = F/y . Suppose first that y = zz t for some z ∈ ZS and t ∈ Z. Then by 13.1.8, F ◦ ˆ 5 , Sp4 [m], or SL− [m], contrary is the central product of the conjugates of Y ∼ = AE 4 to the choice of F. Hence y is not of that type, so, using 6.6.9, τ + = (F + , Ω+ ) satisfies Hypothesis 13.1.1. Then by induction on the order of τ , (F + )◦ is the central product of copies  of Y + , again contrary to 5.1.19, 3.3.16, and the choice of F. Lemma 13.1.21. (1) Suppose θ = (X , Γ) is a proper subpacket of τ with Γ ⊆ Ω. Then each coconnected component of θ ◦ is contained in τσ = (Yσ , σ) for some σ ∈ Λ, so if τσ is a subpacket of θ then it is a component of θ ◦ . (2) For some σ ∈ Λ − {λ}, Tλ ∩ Tσ = 1. (3) Z = Z(Y0 ) = 1. (4) Z is not normal in F. ˆ 5 , Sp2n [m], or SL− [m]. (5) Yλ ∼ = AE 2n Proof. Let ξ = (U, Σ) be a coconnected component of θ ◦ . Then ξ is described in 13.1.5.3. In particular taking Gξ to be a model for NU (O(ξ)) and setting H =

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Z(ξ)Gξ , H is transitive on Σ, so Σ ⊆ σ for some σ ∈ Λ. Therefore (1) follows from 13.1.18. Next by 13.1.20, Z(F) = 1. Hence if Tλ ∩ Tσ = 1 for each σ ∈ Λ − {λ}, then we can verify the hypotheses of 8.1.8 using (1) and arguments in the proof of 8.1.11. Hence F ◦ = Y0 by 8.1.10, contrary to the choice of F. This proves (2). As Tλ ∩ Tσ ≤ Z(Y), (2) implies (3). Suppose Z  F. By 13.1.20, CF (Z) = F so by induction on the order of τ , Y0 = F ◦ , contrary to the choice of F. Thus (4) holds. Part (5) follows from (3) and 13.1.16.  Set B = NF (Z), where Z = Z(Y0 ), and set τ0 = (B, Ω). From 13.1.21.3, Z = 1 and B = F by 13.1.21.4, so τ0 is a proper subpacket of τ . Then from 13.1.5 and  13.1.21.1, τ0◦ = σ∈Λ τσ is a  central product where τσ = (Yσ , σ). In particular Y0  B. For θ ⊆ Λ, set Yθ = σ∈θ Yσ . Lemma 13.1.22. (1) NF (T ) ≤ B. (2) For each ∅ = θ ⊆ Λ with Tθ ∈ F f , NF (Tθ ) ≤ B. (3) Let ∅ = θ ⊂ Λ with Tθ ∈ F f , 1 = P ∈ Yθf with P ≤ Z, and α ∈ A(P ). Then Yθ α∗  NF (P α) and P B ∩ F f = ∅. If P ∈ F f then NF (P ) ≤ B. Proof. By 13.1.19.2, AutF (T ) acts on Z, so AutF (T ) = AutB (T ). Next CF (T ) ≤ CF (Z) ≤ B. Let R = T CS (T ) and N = NF (T ). By 1.3.2 in [Asc19], N = SCF (T ), NN (R) , and as AutF (T ) = AutB (T ), NN (R) ≤ B, establishing (1). Pick θ as in (2) and let Q = Tθ and ξ = (CF (Q), Γ), where Γ consists of those J ∈ Ω contained in a member of Λ − θ. Then ξ is a quaternion fusion packet, and from 13.1.21.1, Yθ = CF (Q)◦  M = NF (Q). By 1.3.2 in [Asc19], M = NS (Q)Yθ , NM (T ) , so (2) follows from (1). Pick P and α as in (3) and let P = NF (P α). As usual by 13.1.21.1, X = Yθ α∗ is a product of coconnected components of P ◦ . As P ≤ Z, all other coconnected components are of smaller order, using 13.1.21.1, so X  P. Let β ∈ A(Tθ α) with Tθ ζ = Tθ , where ζ = αβ. By (2), ζ ∈ homB (NS (P ), S). Also NS (P α)β ≤ NS (P ζ), so P ζ ∈ F f . Finally if P ∈ F f we may take α = 1 to obtain Yθ  P = NF (P ). Then by 1.3.2 in [Asc19], P = NS (P )Yθ , NP (Tθ ) , so P ≤ B by (2).  Lemma 13.1.23. (1) B is transitive on Z and Ft ≤ B for each t ∈ Z. (2) Suppose P ≤ S with Z ∩ Z(P ) = ∅. Then homF (P, S) = homB (P, S). (3) Suppose Q ∈ F f and φ ∈ homF (Q, S) such that CZ (Qφ) = ∅. Then Qφ ∈ QB . Proof. As AutF (T ) is transitive on Λ and Y is transitive on Z(ρ), B is transitive on Z. By 13.1.22.3, Fz ≤ B, completing the proof of (1). Assume u ∈ Z(P )∩Z and let ψ ∈ homF (P, S). Let β ∈ AB (u). By (1), uB = Z, so there is α ∈ AB (uψ) with uψα = uβ = v. Now ζ = β −1 ψα ∈ homF (P β, S) with vζ = v, so as Fv ≤ B by (1), ζ ∈ homB (P β, S). Then as α, β, and ζ are B-maps, so is ψ, proving (2). Assume the hypothesis of (3) and let u ∈ CZ (Qφ) and set P = u, Qφ and let α ∈ A(Qφ) with Qφα = Q. Then u ∈ Z(P ), so by (2), α|P ∈ homB (P, S). Hence α|Qφ ∈ homB (Qφ, S), so Qφ ∈ QB , proving (3). 

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Lemma 13.1.24. (1) Let 1 = P ≤ Z and α ∈ A(P ). Then P α ∈ P B and P α ≤ Z. If P ∈ F f is of order 2 then NF (P ) ≤ B. (2) Z is not strongly closed in S with respect to F. Proof. As Z ⊂ T ≤ CS (P ), it follows from 13.1.23.3 that P α ∈ P B . Then as Z  B, P α ≤ Z. Suppose P ∈ F f is of order 2. Then as P centralizes Y0 , 13.1.20 and our usual argument using 13.1.21.1 shows that Y0  N = NF (P ). By 1.3.2 in [Asc19], N = SY0 , NN (T ) and NF (T ) ≤ B by 13.1.22.1, so N ≤ B, completing the proof of (1). Suppose (2) fails. Then Z is a strongly closed abelian subgroup of F, so by 14.1 in [Asc11], Z  F, contrary to 13.1.21.4.  By 13.1.24.2 there is an involution a ∈ Z with aF not contained in Z, and by 13.1.24.1 we may take a ∈ F f . Let b ∈ aF − Z and α ∈ A(b) with bα = a. By 13.1.23.3, CZ (b) = ∅, so: Lemma 13.1.25. b is has no fixed points on Ω. Suppose for the moment that b acts on some nonempty proper subset θ of Λ and set P = CTθ (b). Lemma 13.1.26. For each involution x ∈ P , CZ (xα) = ∅. Proof. Assume CZ (xα) = ∅. By 13.1.22.3, there is β ∈ AB (x) with y = xβ ∈ F f , and Fy ≤ B by 13.1.22.3. Now φ = β −1 α ∈ homF (y, S), so by 13.1.23.3, xα = yφ ∈ y B . Thus there is γ ∈ AB (xα) with xαγ = y. Therefore ζ = φγ ∈ homFy (bβ, y , S). As Fy ≤ B, ζ, β, and γ are B-maps, so α is also a B-map, contradicting b ∈ / aB .  By 13.1.26, P α is semiregular on Ω, so nk = |Ω| ≥ |P |. Enlarging θ if necessary, we may assume |θ| = k − , where = 1 or 2, and = 1 if k is odd. Thus the set Γ of J ∈ Ω contained in a member of θ is of order n|θ| = n(k − ). For J ∈ Γ, J b = J by 13.1.25, so |CJJ b (b)| = m. Thus |P ∩O(θ)| = mn(k−)/2 /e, where e = |Z(Yθ ) : CZ(Yθ ) (b)| ≤ 2k−−1 . Therefore as nk ≥ |P | ≥ |P ∩ O(θ)|, 2k−−1 nk ≥ enk ≥ mn(k−)/2 . In particular if = 1 then 2k−2 nk ≥ mn(k−1)/2 , which is impossible as k ≥ 2, n ≥ 2, and m ≥ 8. Thus = 2, so k is even and hence k ≥ 4 as θ = ∅. But then 2k−3 nk ≥ mn(k−2)/2 , impossible as k ≥ 4, n ≥ 2, and m ≥ 8. We’ve shown that b acts on no nonempty proper subset of Λ, so: Lemma 13.1.27. k = 2 and λb = λ. We are now in a position to derive a contradiction, and hence establish Theorem 13.1.15. As |Z(Y)| = 2 by 13.1.16 and 13.1.21.3, and as k = 2, it follows from 13.1.21.2 that Z = a = Z(Y). Set X = O 2 (CY0 (b)) and let X be Sylow in X . As λb = λ, it follows from 10.11 in [Asc11] that X ∼ = Y/a ∼ = AE5 , P Sp2n [m], or ∗ 2 ∗ L− [m]. Then X α ≤ B and as X = O (X ), X α acts on Y. 2n

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Next X = {ttb : t ∈ Tλ } and for t an involution in Tλ distinct from a, b = bt = btt ∈ bX. Therefore a = (bt )α ∈ a(Xα) ⊆ NS (λ). As c = (bt )α acts on λ, we have c ∈ Z by 13.1.27. Thus as Z = a , we conclude that a = c, contradicting a = (bt )α. This contradiction completes the proof of Theorem 13.1.15. b

13.2. Generic systems with μ abelian In this section we assume the following hypothesis: Hypothesis 13.2.1. Hypothesis 13.1.1 holds and (1) AutF (O(τ )) induces Sym(Ω) on Ω, and (2) Z m = {t ∈ Z : ZS ⊆ CS (t)} = ∅. Notation 13.2.2. Adopt Notation 13.1.2, set S0 = CS (O), let G be a model ≤ G. Write GΩ for the image of G in Sym(Ω). for NF (OS0 ), and set H0 = (Z m )G  Set n = |Ω| and W = ZS . Let c = s∈ZS s. Lemma 13.2.3. (1) GΩ = H0Ω = Sym(Ω). (2) {tΩ : t ∈ Z m } is the set of transpositions in S Ω . (3) The map J → z(J) is a G-equivariant bijection between Ω and ZS . (4) Either c = 1 and W is the permutation module for GΩ , or c = 1 and W is the image of the permutation module modulo its center. Proof. Recall that AutF (O) = AutG (O), so GΩ = Sym(Ω) by 13.2.1.1. By 13.2.1.2 there is t ∈ Z m , and by 13.1.10.3, tΩ is a transposition. Further as GΩ = Sym(Ω), all transpositions in S Ω are fused in GΩ , so (2) holds. Also GΩ is generated by its transpositions, so as t ∈ H0  G, GΩ = H0Ω , completing the proof of (1). Part (3) is a consequence of 13.1.1.1. By (1) and (3), W is an image of the  permutation module for GΩ , and CW (G) is generated by c, so (4) holds. Notation 13.2.4. Let I = {1, . . . , n}. By 13.2.3.4, if c = 1 then we can regard W as the power set on I under symmetric difference, while if c = 1 then W is the image of that module modulo eI . Thus for θ ⊆ I, we write eθ for θ regarded as a member of W when c = 1, or its  image modulo eI when c = 1. In particular ZS = {ei : i ∈ I}. Write O(θ) for i∈θ O(ei ). Lemma 13.2.5. Let ∅ = θ ⊆ I and 1 = x = eθ ∈ F f . Let θ1 = θ, θ2 = I − θ, τx = (Fx , Ω), nj = |θj |, Ωj = i∈θj O(ei ), Fj = Fj (x) = [Ωj ]Fx◦ , and τj = (Fj , Ωj ) for j = 1, 2. (1) τx is a quaternion fusion packet and if 13.1.1.3.ii holds then τx satisfies Hypothesis 13.1.1. (2) τx◦ = τ1 ∗ τ2 , where τ2 = ∅ if x = c. (3) For j ∈ {1, 2}, one of the following holds: (i) Fj ∼ = Sp2nj [m] or SL− 2nj [m] and eθj = Z(Fj ). (ii) nj = 1, m = 8, and Fj = O(θj ). ˆ 5. (iii) nj = 2, m = 8, and Fj ∼ = AE (1) ∼ . (iv) j = 2, c = 1, and either F2 ∼ = L− 2n2 +1 [m] or n2 = 1 and F2 = L2 [2m] m m   (4) Let Zj = {t ∈ Z : θ3−j ⊆ F ix(t)} and Zj = {t ∈ Z : Ω3−j ⊆ CS (t)}. Then Zjm = Z m (τj ) and Zj = Z  (τj ).

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Proof. By 2.5.2, τx is a quaternion fusion packet, and then (1) follows from 13.1.5.1. If 13.1.1.3.i holds then the lemma follows from the discussion in section 5.4, so assume that 13.1.1.3.ii holds. Ω Ω Ω Let Gx = CG (x). From 13.2.3.4, GΩ x = G1 × G2 , where Gj is the pointwise stabilizer in GΩ of θ3−j , and GΩ j acts faithfully as Sym(θj ) on θj . If x = c assume x ∈ / Z(F). Then by 13.1.5.2, τx◦ = ρ1 ∗ · · · ∗ ρr is the central product of its coconnected components ρi = (Yi , Γi ). Let K ∈ Ωj ; then K ∈ Γi for some i. By 13.1.5.3, AutYi (O(ρi ))Γi = Sym(Γi ), so we have Γi ⊆ Ωj . On the other hand as Gj is primitive on Ωj , it follows that Ωj = Γi . We conclude that r = 2 (or x = c and r = 1) and Ωj = Γi , and then that ρi = τj and (2) holds. Further if nj > 1 then by 13.1.5.3, either Fj ∼ = (P )Sp2nj [m] or (P )SL− 2nj +a [m] with a ∈ {0, 1}, or ∼ ˆ nj = 2, m = 8, and Fj = AE5 . As eθ1 = x ∈ Z(F1 ), we have F1 ∼ = Sp2n1 [m], ˆ 5 , and eθ = Z(F1 ). If n = 1 then F1 = O(θ1 ). Similarly either [m], or AE SL− 1 2n1 ˆ 5 , and eθ = Z(F2 ), or F2 ∼ [m] (or F2 ∼ = Sp2n [m], SL− [m], or AE = L− 2

2n2

2

2n2 +1

/ Z(F2 ), so that eθ2 = x, and hence c = 1. possibly L2 [2m](1) if n2 = 1) and eθ2 ∈ Thus (3) holds. By 13.1.10.3, Zjm centralizes O(θ3−j ), and then by 13.1.7.3, Zjm ⊆ Z m (τj ). Similarly Zj ⊆ Z  (τj ) by 13.1.7.3. As Zjm ⊆ Z m (τj ) and Zj ⊆ Z  (τj ) and as (2) holds, (4) also holds. Finally suppose x = c ∈ Z(F) and set F + = F/c . If n = 2 the lemma holds by 13.1.8, so take n > 2. Then, using 6.6.9.2, τ + = (F + , Ω+ ) satisfies Hypothesis 13.2.1, so by induction on the order of τ , F ◦+ ∼ = P Sp2n [m] or P SL− 2n [m], and then − ◦ ∼ F = Sp2n [m] or SL2n [m] by 3.3.16, so again the lemma holds.  Lemma 13.2.6. E(τ ) = SO(τ ), NF (O), NF (O(θ)) : θ ∈ Ξ , where Ξ consists of those θ ⊆ I such that NS (θ) is transitive on θ, and O(θ) ∈ F f . Proof. This follows from 3.4.12.1.



Definition 13.2.7. Define B(τ ) = SO(τ ), NF (O), Feθ : θ ⊆ I, |θ| is a power of 2, and eθ ∈ F f . Lemma 13.2.8. (1) NF (W ) = SO(τ ), NF (O) ≤ B(τ ). (2) For each θ ⊆ I with |θ| a power of 2, Feθ ≤ B(τ ). Proof. By 3.3.7.2, O(τ )  NF (W ). Then (1) follows from 1.3.2 in [Asc19]. Set x = eθ . There is g ∈ G such that α = c∗g ∈ A(x). By definition of B = B(τ ), Fxg ≤ B. Therefore as NF (O) ≤ B by (1), we can complete the proof as in 11.3.4.  Theorem 13.2.9. F = B(τ ). Proof. Observe that if θ ⊆ I then NF (O(θ)) ≤ CF (eθ ). Therefore E(τ ) ≤ B(τ ) by 13.2.6. Observe also that Hypothesis 11.2.1 is satisfied by 13.2.3.4. Also 11.3.8.1 holds in 13.1.1.3.ii by 13.1.5.4. Hence by Theorem 11.3.10, either F = B(τ ) or conclusion (2) of that theorem holds. However the latter case is contrary to 13.2.1.2. 

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Lemma 13.2.10. Let 1 = x = eθ for some θ ⊆ I with x ∈ F f . Then one of the following holds: (1) Fx ≤ E(τ ). (2) x = c. (3) n is even, c = 1, and |θ| = n/2. Proof. See the proof of 11.1.8 and use 13.2.5.



Lemma 13.2.11. One of the following holds: (1) F = E(τ ). (2) n = 2k , c = 1, and F = E(τ ), Fc . (3) n = 2k , c = 1, and F = E(τ ), Fx , where x ∈ F f , x = eθ , and |θ| = n/2. Proof. By 13.2.6 and Theorem 13.2.9, F = E(τ ), Feθ : θ ⊆ I and |θ| is a power of 2 . Then the lemma follows from 13.2.10.



Lemma 13.2.12. Let θ ⊆ I and x = eθ with 1 = x ∈ F f . (1) NF (O(θ)) ≤ Fx . (2) O(θ) ∈ F f iff x ∈ F f . (3) Either (a) c ∈ Z(Fx ), or (1) ∼ (b) F2 (x) ∼ . = L− 2n2 +1 [m] or n2 = 1 and F2 (x) = L2 [2m] Proof. Set P = O(θ). As NF (P ) permutes {O(i) : i ∈ θ}, (1) follows. As W is the permutation module for GΩ = Sym(I), or the image of that module modulo eI , it follows from 13.2.3.3 that either CS (x) = NS (P ) or n is even, c = 1, and |θ| = n/2 with |S : NS (P )| = 2. In either case (2) holds. Next Fx◦  Fx , and by 13.2.5, c = eθ1 · eθ2 . Moreover by 13.2.5.3, either Fx permutes {eθ1 , eθ2 }, or (3b) holds so (3) follows.  Lemma 13.2.13. (1) Either (a) c ∈ Z(F), or (b) For some 1 = x = eθ ∈ F f with |θ| a power of 2, F2 (x) ∼ = L− 2n2 +1 [m] or (1) n2 = 1 and F2 (x) ∼ = L2 [2m] . (2) If 1 = c ∈ Z(F) then either F ◦ ∼ = Sp2n [m] or SL− 2n [m], or n = 2 and ◦ ∼ ˆ F = AE5 . Proof. First c ∈ Z(SO(τ )). By 13.2.12.1, NF (O) centralizes c, while by 13.2.12.3, either Fx centralizes c for each 1 = x ∈ W with x = eθ ∈ F f and |θ| a power of 2, or (1b) holds. Therefore (1) follows from Theorem 13.2.9. Then (2) follows from 13.2.5.3.  Given 13.2.13, during the remainder of the section we assume: Hypothesis 13.2.14. Hypothesis 13.2.1 holds and c = 1 if c ∈ Z(F). Lemma 13.2.15. Let x = eθ ∈ F f . (1) If n1 = |θ| = 2 then F1 (x) ∼ = Sp4 [m] or SL− 4 [m]. (2) If n1 = n − 1 then either F2 (x) ∼ = SL2 [m] or L− 3 [m], or m = 8 and F2 (x) = O(n).

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∼ AE ˆ 5 , so Proof. Set Fi = Fi (x). Assume (1) fails; then by 13.2.5.3, F1 = m = 8. Let z = e1 and set Gi = Fi (z). By 13.2.5.3, G2 ∼ = Sp2n−2 [8], SL− 2n−2 [8], − (1) ∼ ∼ ˆ or L2n−1 [8], or n = 3 and G2 = AE5 , or n = 2 and G2 = L2 [16] . In the first case there is y ∈ G of order 3 centralizing K with O(2) = [O(2), y]. Let g ∈ G with eg1,2 = x. Now y g centralizes x so y g acts on L the model in G for O 2 (NF1 (O(θ))). Further y g acts on O2 (F1 ) and hence on O(θ) ∩ O2 (F1 ) = E. ˜ ∼ ˜ This Set F˜1 = F1 /x . Then E = E8 , so as y g centralizes K g , y g centralizes E. g g g contradicts O(2) = [O(2) , y ]. Therefore case two or three holds. Suppose n = 3; then we may take θ = {2, 3}, and there is h ∈ G of order 3 centralizing K with O(θ) = [O(θ), h]. Then for a ∈ G with (1, 2, 3)a = (3, 1, 2), ha centralizing O(3) and O(2) = [O(2), ha ], a contradiction as in the previous paragraph. Finally suppose n = 2. There is l ∈ G of order 3 with O = [O, l]. But l centralizes z, so l acts on F2 , a contradiction as O(2) = [O(2), l] and Z4 ∼ = O(2) ∩ O 2 (F2 ). This completes the proof of (1). Moreover the argument in case three establishes (2).  Lemma 13.2.16. Assume m = 8. Then there exists O  Fz such that K is Sylow in O and O ∼ = SL2 [8]. Proof. Set T = CS (K). We first show there exists O ≤ CFz (T ) such that K is Sylow in O and O ∼ = SL2 [8]. Choose notation so that there is t ∈ Z m such that tΩ = (1, 2) and x = e1,2 ∈ F f . Adopt the notation of 13.2.5. By 13.2.15, F1 ∼ = Sp4 [8] or SL− 4 [8], so there is ∼ O1  CF1 (e1 ) with K1 = O(1) Sylow in O1 = SL2 [8]. Then by 2.3.5 in [Asc19], S0 O(I − {1}) centralizes O1 . As G is transitive on Ω, it follows from 2.2 in [Asc11]  that there is a central product L = i∈I Oi ≤ NF (O) with Ki = O(i) Sylow in Oi ∼ = SL2 [8] for each i ∈ I. As NF (O) permutes Ω, AutL (O)  AutF (O), so L  NF (O). Therefore there is a model L for L normal in G, and of course L = i∈I Li is a central product of groups Li ∼ = SL2 (3) with Ki = O2 (Li ). Let z = ej , so that K = Kj , and as S permutes {Li : i ∈ I}, T acts on Lj . Then [Lj , T ] ≤ CLj (K) = z , so Lj = O 2 (Lj ) centralizes T , and therefore O = Oj centralizes T . Next let E = CF (K) and C = CFz (E). As K  Fz , it follows from Theorem 4 in [Asc11] that C  Fz and E ∗ C ≤ Fz . Let R be Sylow in C and set Z = Z(E). Then CR (K) ≤ R ∩ T = Z, so KZ ∈ C f c is normal in C, and hence C is constrained. Let Gc be a model for C. Next AutO (KZ) ≤ O 2 (AutCFz (T ) (KZ)), so by Theorem 4 in [Asc11], AutO (KZ) ≤ AutC (KZ). Therefore we may take O ≤ C. Then as O is irreducible on K/z , we conclude KZ = O2 (Gc ), so that O = O 2 (C)  Fz , completing the proof of the lemma.  Notation 13.2.17. Suppose for the moment that t ∈ ZS . If m > 8 set Ot = O(t) and observe that O(t) ∼ = SL2 [m] by 13.2.15. If m = 8 set Ot = Oα−∗ for α ∈ A(t) with tα = z and O the  system in 13.2.16. As O  Fz , Ot is independent Oτ  NF (W ) using 13.2.16 of α. In either case set Oτ = t∈ZS Ot . Observe that  when m = 8. Write Oi for Oei and for θ ⊆ I set Oθ = i∈θ Oi . Let L0 be a model for NOτ (OS0 ) in G and H = Z G L0 ≤ G. Let M be the kernel of the action of H on Ω.

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Definition 13.2.18. Suppose for the moment that x = eθ ∈ F f with |θ| = 2. Define F to be of type P Sp2n [m] if F1 (x) ∼ = Sp4 [m], and define F to be of − − − ∼ type L− [m], L [m] if F (x) SL [m] and F2 (z) ∼ = = SL− 1 2n 2n+1 4 2n−2 [m], L2n−1 [m], respectively. Lemma 13.2.19. Let 1 = x = eθ ∈ F f with |θ| = n1 and set Fi = Fi (x) for i = 1, 2. − (1) F is of type P Sp2n [m], L− 2n [m], or L2n+1 [m]. (2) If F is of type P Sp2n [m] or L− 2n [m] then c = 1, while if F is of type − L2n+1 [m] then c = 1. (3) If F is of type P Sp2n [m] then Z  = Z m and Fi ∼ = Sp2ni [m] for i = 1, 2, unless ni = 1 and m = 8 where Fi ∼ = K.  m o  m (4) If F is of type L− 2n [m] then Z = Z , |D (t)∩ZS | = 2 for each t ∈ Z −Z , − ∼ ∼ and Fi = SL2ni [m] for i = 1, 2, unless ni = 1 and m = 8 where Fi = K.  m  o (5) If F is of type L− 2n+1 [m] then Z = Z , there exists t ∈ Z with |D (t) ∩ − ZS | = 1, and F1 ∼ = SL2n1 [m] (unless n1 = 1 and m = 8 where F1 ∼ = K, and − ∼ F2 = L2n2 +1 [m]). (6) If n = 2 then F is of type L− 5 [m]. Proof. By 13.2.1.2, n ≥ 2. Thus there is u = eξ ∈ F f with |ξ| = 2. Set Gi = Fi (u) for i = 1, 2. Set F2 (z) = C. By 13.2.15.1, G1 ∼ = Sp4 [m] or SL− 4 [m]. Assume first that n = 2. Then c = u = 1 by 13.1.1.1. Now by 13.2.14, (1) , and then the c ∈ / Z(F), so by 13.2.13.1, C is isomorphic to L− 3 [m] or L2 [2m] − former holds by 13.2.15.2. We will show that G1 ∼ = SL4 [m], so F is of type L− 5 [m] [m] there is and (1) and (6) hold. Also (2)-(4) are vacuously true and, as C ∼ = L− 3 t ∈ Do (z2 ) ∩ z2C with Do (t) ∩ ZS = {z2 }, so (5) holds. Therefore when n = 2 it remains only to show that G1 ∼ = SL− 4 [m]. Hence by 13.2.15.1 we may assume: (a) G1 ∼ = Sp4 [m]. Let Ω = {K1 , K2 } with K = K1 and set zi = z(Ki ). Let Tu , T2 be Sylow in G1 , C, respectively, r ∈ Z m ∩ Tu , T1 = T2r , and T = T1 , T2 . By (a), Tu = KK2 r , r ∼ while as C ∼ = L− 3 [m], Ti = SD2m . Next T2  NS (K), so T1 = T2  NS (K) and then T0 = T1 ∩ T2  S. Therefore T0 ≤ Z(T1 ) ∩ Z(T2 ) = z ∩ z2 = 1, so (b) T = T1 × T2 . But Tu  S, so [T, r] ≤ Tu ∩ T = KK2 , a contradiction as for t ∈ T − K, [t, r] ∈ / KK2 by (b). Thus we may assume that n > 2. As n > 2 there is b ∈ uF with b ∈ C f , and a coconnected component X of (CC (b), Ω − {K}) with X β ∗ = G1 for β ∈ A(b) with bβ = u. It follows that − − ∼ C ∼ = Sp2n−2 [m] if G1 ∼ = Sp4 [m] and C ∼ = SL− 2n−2 [m] or L2n−1 [m] if G1 = SL4 [m]. Therefore (1) holds. Similarly if ni ≥ 2 then there is a ∈ uF with a ∈ Fif and a coconnected component Y of (CFi (a), Ωi ) such that Yα∗ = G1 for α ∈ A(a) with aα = u. Therefore Fi ∼ = Sp2n2 [m] iff F is of type P Sp2n [m], while Fi ∼ = SL− 2n2 [m] or − L2n2 +1 [m] otherwise. f Suppose F2 ∼ = L− 2n2 +1 [m]. Let j ∈ θ with y = ej ∈ F1 and γ ∈ A(y) with yγ = ∼ − z. Then F2 = CC (O(θ)γ)◦ , so as F2 ∼ = L− 2n2 +1 [m], it follows that C = L2n−1 [m], − and therefore F is of type L2n+1 [m]. Similarly the converse is true: if F is of type ∼ − L− 2n2 +1 [m] then F2 = L2n1 +1 [m]. Together with the previous paragraph, these observations establish the statements in (3)-(5) about Fi .

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Suppose F is of type P Sp2n [m] or L− 2n [m]. From the previous paragraph, [m], so c ∈ Z(F) by 13.2.13.1. Therefore c = 1 by 13.2.14. F2 ∼ = Sp2n2 [m] or SL− 2n2 − / Z(F2 ), so c = 1, completing On the other hand if F is of type L2n+1 [m] then x ∈ the proof of (2).  If G1 is SL− 4 [m] then there is t ∈ Z (ρ1 ) (where ρ1 is the Lie packet of G1 ) with o  m ei ∈ D (t) for i ∈ ξ, so Z = Z . Conversely if t ∈ Z  with Do (t) ∩ ZS = ζ of order 2, then by 13.1.10.2, G1 ∼ = SL− 4 [m].  Next if t ∈ Z with Do (t) ∩ ZS = {ei } of order 1, then conjugating in F we may take θ2 = {i}, and then by 13.1.7.3 and 13.2.15.2, F2 ∼ = L− 3 [m], so F is of type − L2n+1 [m]. Together with the previous paragraph, this shows Z  = Z m when F is of type P Sp2n [m], completing the proof of (3). Similarly if F is of type L− 2n [m] then |Do (t) ∩ ZS | = 2 for each t ∈ Z  − Z m , completing the proof of (4). Finally  ∼ − if F is of type L− 2n+1 [m] then, taking n2 = 1, F2 = L3 [m], so for t ∈ Z (τ2 ), o  |D (t) ∩ ZS | = 1, completing the proof of (5). Lemma 13.2.20. Assume x = e1,2 ∈ F f and let T1 be Sylow in F1 = F1 (x). Then (1) CS (T1 ) = CS (F1 ). (2) Let t ∈ Z m ∩ T1 and set T¯1 = O(1, 2)t . Then [T1 , CG (O(1, 2))] ≤ e1 , e2 , CS (O(1, 2)) = e1 × CS (T¯1 ), and CS (T¯1 ) centralizes a subsystem F¯1 of F1 isomorphic to Sp4 [m]. In particular if F is of type P Sp2n [m] then F1 = F¯1 . (3) NF1 (O(1, 2)) = T1 if m > 8 and T1 O1,2 if m = 8. (4) For each r ∈ Dm , r ∈ T1g for some g ∈ G. (5) Let Do = D o (τ ) = {s ∈ Z : Do (s) ∩ ZS = ∅}. Then CZ  (W ) = Do . (6) For s ∈ Do , |Do (s) ∩ ZS | ≤ 2 with s ∈ T1g for some g ∈ G in case of equality. If |Do (s) ∩ ZS | = 1 then F is of type L− 2n+1 [m] and [s, CG (O)] ≤ W . Proof. By 13.2.15, F1 ∼ = Sp4 [m] or SL− 4 [m]. Hence by 9.4.18, 9.4.22, and 9.4.28, together with 2.1.15, F1 = G1 , O 2 (AutF1 (Ri )) : i ∈ I  , where G1 = T1 O1,2 and Ri = CT1 (Qi /x ) for suitable Qi ≤ T1 isomorphic to Z4 ∗ Q8 . Let G2 be a model for NF1 (Qi ) normal in a model for NF (Qi CS (Qi )), and set P = CS (T1 ). Then [P, G2 ] ≤ CG2 (Ri ) ≤ CRi (O 2 (G2 )), so O 2 (G2 ) centralizes P , and therefore P centralizes F1 , proving (1). Part (3) follows from 5.4.8-5.4.10. Set Q = O(1, 2) and Y = CG (Q). If m > 8 set L1,2 = 1, while if m = 8 let L1,2 be a model for O1,2 in G. By (3), [T1 , Y ] ≤ CT1 L1,2 (Q) = e1 , e2 , so Y acts on D = t, e1 ∼ = D8 and on e1 , e2 . Thus the first statement in (2) holds and [Y, D] ≤ Z(D) = x , so |Y : CY (D)| = 2. Then as CY (D) = CY (T¯1 ), the second statement in (2) follows. The third statement follows from the second and (1) applied to the subsystem F¯1 of F1 on T¯1 isomorphic to Sp4 [m]. Thus (2) holds. Part (4) follows from 13.1.10.2 and 13.2.3, while (5) follows from 13.1.1.2. Suppose s ∈ Do . By 13.1.10.4, Γ = Do (s) ∩ ZS is of order at most 2, and in case of equality, s ∈ T1g for some g ∈ G by 13.1.10.2. So assume Γ = {ei } is of order 1. We may assume ei = z = e1 , so by 13.1.10.4, s ∈ D(z). Then by 13.1.7, s ∈ Z(τz ). Hence as Γ is of order 1 it follows from 13.2.19.1 that F is of  type L− 2n+1 [m]. Let Hz  CG (z) be a model for NF2 (z) (O(1 )). Then s ∈ Hz so  [s, CG (O)] ≤ CHz (O(1 )) ≤ W , completing the proof of (6).  Lemma 13.2.21. (1) H Ω = Sym(Ω), [H, S0 ] ≤ W , and if F is of type P Sp2n [m] then S0 = CS (H)W .

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 (2) L0  G, and if m > 8 then L0 = O, while if m = 8 then L0 = i∈I Li where Li ∼ = SL2 (3) is the model for Oi . (3) If F is of type P Sp2n [m] then M = L0 . (4) Suppose F is not of type P Sp2n [m] and set Z 1 = CZ  (W ) and Z 2 = ˜ = M/L0 is the permutation {t ∈ Z 1 : |D o (t) ∩ ZS | = 2. Then M = Z 1 L0 and M module, core of the permutation module for H/M = Sym(I) for F of type L− 2n+1 [m], − 1 2 2 ˜ ˜ L2n [m], respectively, with t of weight 1,2 in M for t ∈ Z −Z , t ∈ Z , respectively. ¯ (n, m) is the wreath (5) If F is of type P Sp2n [m] then H ∼ = σ(n, m), where σ product L1 wr Sn with L1 = K, SL2 (3) for m > 8, m = 8, respectively, and σ(n, m) = σ ¯ (n, m)/Z(¯ σ(n, m)). ∼ (6) If F is of type L− 2n+1 [m] then H = 2λ(n, m), the wreath product M1 wr Sn with M1 = L1 t1 , where L1 = O(1) if m > 8 and t1 ∈ Z with Do (t1 ) ∩ ZS = {e1 }. ∼ If F is of type L− 2n [m] then H = λ(n, m), the subgroup of 2λ(n, m) of index 2 ∼ containing L0 with λ(n, m)/L0 = W eyl(Cn ). (7) Some complement to M in H contains a member of Z m . Proof. By 13.2.3, m(W/c ) = n − 1. Assume the setup of 13.2.20. For Ω 1 ≤ i < n, set T¯i = T¯1gi and Ti = T1gi for gi ∈ G with (1, 2)gi = (i, i + 1). Then T¯iΩ = (i, i + 1) , so H Ω = Sym(I) = Y Ω , where Y = T¯i : 1 ≤ i < n . By 13.2.20, S0 = CS0 (T¯1 ) × e1 . Therefore |S0 : CS0 (Y )| ≤ 2n−1 = |W : c |. Then as CW (H) = c , S0 = W CS0 (Y ). From 13.2.17, Oτ  NF (W ), so the model L0 for NOτ (OS0 ) is normal in G. Then (2) follows as Oi ∼ = SL2 [m]. Let M0 = Z 1 L0 , N the kernel of the action of G on Ω, and SN = S ∩ N . Claim M0  G. By a Frattini argument, G = N NG (SN ), and NG (SN ) acts on Z 1 , and hence also on M0 , so it remains to show that M0  N . But N = SN L0 CG (O), and by 13.2.20.6, [Z 1 , CG (O)] ≤ W , so we conclude from (2) that M0  G. By 13.2.20.3, T1  CG (e1,2 ) if m > 8, while T1 L1 L2  CG (e1,2 ) if m = 8. Therefore as Y is 2-transitive on I, as M0  G, and as Ti ≤ T¯i Z 1 , it follows that T1G ⊆ Y M0 . By parts (4)-(6) of 13.2.20, Z G ⊆ T1G ≤ Y M0 unless F is of G type L− ⊆ (Z 1 )G , T1G ≤ Y M0 as M0  G. So in any event, 2n+1 [m], where Z G Z ⊆ Y M0 , and hence H = Z G L0 ≤ Y M0 . As T1 ≤ Z O and H  G, we have Y M0 ≤ H, so H = Y M0 . By 2.3.5 in [Asc19], L0 centralizes S0 , and for t ∈ Z  , [S0 , t] ≤ W by parts (2) and (6) of 13.2.20, so [H, S0 ] ≤ W . Further if F is of type P Sp2n [m] then M0 = L0 by 13.2.19.3, so H = Y L0 . Then as S0 = W CS0 (Y ) and L0 centralizes S0 by 2.3.5 in [Asc19], the proof of (1) is complete. G Ω Suppose t1 ∈ Z m with tΩ 1 = (1, 2) and t2 ∈ t1 with t2 = (2, 3). Set y = t1 t2 3 Ω and u = y . Then y = (1, 3, 2) is of order 3, so u fixes O(i) for 1 ≤ i ≤ 3. Suppose y u is of even order and let v be the involution in u . Then t = t1 v ∈ ti for i = 1 m t t1 or 2, so t ∈ Z . Then as O(1) = O(1) = O(2), t ∈ T1 by 13.1.10.2, so v ∈ T1 . But then y acts on O(1, 2) by 13.1.10.4, a contradiction. Therefore |u| is odd. Set G+ = G/L0 . As u+ is of odd order and acts on O(i), u+ centralizes t+ 1 , so as t1 inverts u we have u+ = 1. Thus u ∈ L0 , so as ti inverts u for i = 1, 2 and / F ixΩ (ti ), we have u ∈ L1 L2 ∩ L2 L3 = L2 . But now as y centralizes Lj for j ∈ centralizes u, y acts on L2 , a contradiction. We’ve shown that |t1 t2 | = 3. Similarly if ti ∈ Z m with tΩ i = (i, i + 1) and i > 2 then |t1 ti | = 2 by 13.1.10.7. Set X = tj : 1 ≤ j < n . It follows that

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∼ Sn is a complement (X; tj : 1 ≤ j < n) is a Coxeter system of type An−1 , so X = to M0 in H. Therefore as H = M0 Y , also H = M0 X, so M = M0 , completing the proof of (3), establishing (7), and proving the first statement in (4). Suppose F is of type P Sp2n [m]. Then X is a complement to L0 in H and t1 centralizes OI−{1,2} by 13.1.10.3, so CX (e1 ) centralizes O(1), and hence also O1 , so − (5) follows. Thus we may assume F is of type L− 2n [m] or L2n+1 [m]. In the latter case for each i ∈ I there is ti ∈ Z  with D o (ti ) ∩ ZS = {ei }. Then Pi = O(i)ti ∼ = SD2m and by 13.1.10.4 and 13.1.10.6, P = Pi : i ∈ I is the direct product of the Pi . Further from 13.2.19.5, Pi Li  CG (ei ), so the first half of (6) follows. m Therefore we may assume F is of type L− ∩ Tn−1 2n [m]. There exists tn ∈ Z + + + Ω Ω with [tn−1 , tn ] = 1 and Tn−1 = tn−1 , tn . As tn = tn−1 , an earlier observation shows that |tn−2 tn | = 3 and tn centralizes ti for i < n−2. Set Y = X, tn ; it follows that (Y, ti : 1 ≤ i ≤ n) is a Coxeter system of type Dn , so Y ∼ = Weyl(Dn ). Further + Tn−1 ≤ Y + , so as W is the image of the permutation module for X, Ti ≤ Y L0 for each 1 ≤ i < n, and hence H = L0 Y , so Y is a complement to L0 in H. Hence the second half of (6) holds, completing the proof of the lemma.  Notation 13.2.22. Let T = S ∩ H, so that T is Sylow in H. Let F˜ be the − fusion system such that F is of type F˜ ; thus F˜ is P Sp2n [m], L− 2n [m], or L2n+1 [m]. ˜ ˜ ˜ ˜ ˜ Let τ˜ = (F , Ω) be the Lie packet for F, S the Sylow group of F , etc. We see in the ˜∼ ˜ = Ω, and G ˜ = H. next lemma that G = H, so we may take S˜ = T , Ω ˜∼ ˜ = Ω, and G ˜ = H. Lemma 13.2.23. (1) G = H, so we may take S˜ = T , Ω ˜ (2) SOτ˜ = T Oτ . ˜ = W. (3) Set W = T Oτ , FT (H) . Then W ˜ ) = W. ˜ (4) NF˜ (W (5) W  NF (W ), FT (H)  NF (O), and T  T CF (O). Proof. By 5.4.6-5.4.10, F˜ satisfies Hypothesis 13.2.14, so as H is determined ˜ ∼ up to isomorphism by 13.2.21, we conclude that H = H. But by 5.4.8-5.4.10, ∼ ˜ ˜ ˜ H = G, so G = H. Then following the convention in 13.2.22, we may take S˜ = T , etc, so that (1) holds. Next by Theorem 2.7.3, T Oτ is determined by T and Ω, so T Oτ = T Oτ˜ , establishing (2). Then (1) and (2) imply (3). As W is normal in T Oτ and FT (H), W ≤ NF (W ). As Oτ  NF (W ) by ˜ = W ˜ , so 13.2.17, NF (W ) = SOτ , NF (O) by 1.3.2 in [Asc19]. Further CS˜ (O) ˜ = F ˜ (G) ˜ and hence (4) holds. NF˜ (O) S Set G = NF (W ), G1 = SOτ , G2 = NF (O), and C = CF (O). We just saw that G = G1 , G2 . Set E1 = T Oτ and E2 = FT (H). We will show E2  G2 . Then Hypothesis 1.4.7 in [Asc19] is satisfied by the tuple Gi , Ei , i = 1, 2, so W = E1 , E2  G1 , G2 = G by 1.4.8 in [Asc19]. Similarly G2 = SCF (O), NG2 (S0 ) by 1.3.2 in [Asc19], and E2  NG2 (S0 ) by construction, so if T  SC then E2 = E2 , T  G2 by 1.4.8 in [Asc19]. Hence to complete the proof of (5), it remains to show that T  SC. Suppose T is strongly closed in S with respect to SC. Then as [T, S0 ] ≤ CT (O) = W ≤ Z(C), it follows from 9.5.2 in [Asc11] that T  T C, so the lemma holds. Therefore it suffices to show that T is strongly closed in S with respect to SC.

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Let x = eθ be an involution in CW (S) and Tx Sylow in Fx◦ . As Fx◦  Fx and SC ≤ Fx , Tx is strongly closed in S with respect to SC. Thus it suffices to show we can choose x with Tx = T . If F is of type L− 2n+1 [m] we choose x = c, so that T = Tc . Thus we may assume F is of type P Sp2n [m] or L− 2n [m]. If F is of symplectic type then (cf. section 5.4) either Tx = T or n is a power of 2 with |θ| = n/2, and |T : Tx | = 2, while if F is of linear type then either |T : Tx | = 2 and T = Tx t for / θ, or n is a power of 2 with t ∈ Z  with D o (t) ∩ ZS = {ei , ej } with i ∈ θ and j ∈ |θ| = n/2 and |T : Tx | = 4. Suppose F is linear but |θ| = n/2. Let u = eξ ∈ F f with |ξ| = 2, b = ei,j , and g ∈ G with bg = u and CS (b)g ≤ CS (u). Then a Sylow group Tu of Fu◦ is strongly closed in CS (u) with respect to CS (u)C and tg ∈ Tu . Let β ∈ homCS ((b)C (t, S); then α = β cg ∈ homCS (u)C (tg , S) with (tβ)g = tg α, so as Tu −1 is strongly closed we have tg α ∈ T . Therefore tβ ∈ T g ∩ S ≤ T as T is strongly SC closed in S with respect to G. Thus t ⊆ T as S acts on T , so as Tx is strongly closed of index 2 in T , also T is strongly closed. Hence the lemma follows unless the exceptional case |θ| = n/2 holds, which we now assume. Set G2+ = G2 /W . As C  G2 , also C +  G2+ by 8.9.2 in [Asc08a]. Now E2 = [Tx ]E2 ≤ CG2 (C + ) with T Sylow in E2 , so here too T  T C with T Sylow in E2 .  Notation 13.2.24. Let Ξ = Ξ(τ ) be the set of nonempty subsets ξ of I such one of the following holds: (a) ξ is S-invariant and ξ = I, or f (b) n is a power of 2, F is of type P Sp2n [m] or L− 2m [m], |ξ| = n/2, and eξ ∈ F , or (c) n is a power of 2, F is of type L− 2n+1 [m], and ξ = I. For ξ ∈ Ξ and F of type P Sp2n [m] or L− 2n [m], define Bξ (τ ) = Feξ , NF (W ) . In case (c) set Bξ (τ ) = Fc , Feσ , where |σ| = n/2 and eσ ∈ F f . Finally if F is of type L− 2n+1 [m] and case (a) holds, set Bξ (τ ) = Feξ , Fceξ , NF (W ) . Lemma 13.2.25. Let ξ ∈ Ξ, f = fξ , and B = Bξ (τ ). (1) f ∈ Z(S). (2) NF (O) ≤ B. (3) SOτ ≤ B. (4) Let θ ⊆ I with Q = O(θ) ∈ F f , and let TQ be Sylow in NF (Q). Let Σ be a nontrivial orbit of TQ on θ and g ∈ H such that α = c∗g ∈ A(O(Σ)). Then if TQ αCF (O(Σ)α)◦ ≤ B, we have NF (Q) ≤ B. Proof. In cases (a) and (c) of 13.2.24, ξ is S-invariant, so f ∈ Z(S). In case (b), |S : NS (ξ)| = 2 with f S = {f, f c}, so as c = 1 by 13.2.19.2, again f ∈ Z(S). Therefore (1) holds. As W  NF (O), and as NF (W ) ≤ Fc in case (c) of 13.2.24, (2) holds. As SO(τ ) ≤ Ff ≤ B, (3) holds. Assume the hypothesis of (4). By 1.3.2 in [Asc19], NF (Q) ≤ NF (O), TQ CF (Q)◦ and NF (O) ≤ B by (2), so to prove (4) it remains to show that TQ CF (Q)◦ ≤ B. But as α is a B-map and (TQ CF (Q)◦ )α∗ ≤ TQ αCF (O(Σ)α)◦ , TQ CF (Q)◦ ≤ B if  TQ αCF (O(Σ)α)◦ ≤ B. Lemma 13.2.26. Let ξ ∈ Ξ and B = Bξ (τ ). Then F = B.

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Proof. Set f = fξ and assume F = B. Claim NF (O(θ)) ≤ B for some θ ⊆ ZS with O(θ) ∈ F f . For if not, then by 13.2.6 and parts (2) and (3) of 13.2.25, E(τ ) ≤ B, so by 13.2.11, n is a power of 2 and either c = 1 and F = E(τ ), Fc , or c = 1 and F = E(τ ), Fx , where x = eθ ∈ F f and |θ| = n/2. By 13.2.19.2, in the first case F is of type L− 2n+1 [m] and in the second, F is of type P Sp2n [m] − or L2n [m]. Thus in the first case, case (c) of 13.2.24 holds, so Fc ≤ B, and hence B = F, contrary to assumption. Similarly in the second case, case (b) of 13.2.24 holds, so Fx ≤ B, for the same contradiction. This establishes the claim. By the claim and 13.2.25.4, there is θ ⊆ I such that Q = O(θ) ∈ F f , θ is an orbit of a Sylow group TQ of NF (Q), and Y = CF (Q)◦ ≤ B. As NF (O) ≤ NF (W ) ≤ B, θ = I. As TQ ≤ S, θ is contained in an orbit Σ of S. Let ξ  = I − ξ. In case (a) of 13.2.24, S acts on ξ, so as TQ ≤ S is transitive on θ, we have θ ⊆ Σ ⊆ ζ, for some ζ ∈ {ξ, ξ  }. If F is of type P Sp2n [m] or L− 2n [m] then by 13.2.19.2, c = 1 so f = f c, so interchanging the roles of ξ and ξ  if necessary, we may assume ζ = ξ. On the other hand if F is of type L− 2n+1 [m] then as both Ff and Fcf are contained in B, interchanging ξ and ξ  if necessary, we may again assume ζ = ξ. Further if θ = ξ then Y ≤ Ff ≤ B, a contradiction. So θ ⊂ ξ in case (a). Similarly in cases (b), (c), conjugating in H we may take θ ⊂ δ, where δ = ξ, σ, respectively and TQ acts on δ. ¯ and τ¯ = (Y, Ω). ¯ = {J ∈ Ω : z(J) = ei with i ∈ θ}, ¯ Then Set θ¯ = I − θ, Ω ¯ a = |θ| = n − |θ| and τ¯ is a quaternion fusion packet. In case (a) of 13.2.24, as Q ∈ F f , Σ is the orbit of S on I of smallest length subject to |Σ| ≥ |θ|, the TQ -orbits and S-orbits on I − Σ are the same, and |Σ| = |S : TQ ||θ|. In cases (b) and (c), as Q ∈ F f , |δ| = |NS (δ) : TQ ||θ|. Let α = ξ in (a) and (b) and α = σ in (c). In (a), S acts on ξ so TQ acts on α. In (b) and (c), TQ acts on δ = α. So in any event, TQ acts on α. ¯ Then as TQ acts on α and α − θ with θ = α, Suppose TQ is transitive on θ.  we conclude α = ∅, so α = I, contrary to 13.2.24 where α is a proper subset of I. ¯ Therefore TQ is not transitive on θ. ¯ Let α ¯ = α ∩ θ. As TQ is Sylow in NF (Q) it is also Sylow in TQ Y. We’ve seen that TQ acts on α, so as TQ acts on θ it also acts on α ¯ . Thus α ¯ ∈ Ξ(¯ τ) ¯ < n. As TQ is not transitive on θ, ¯ we have a > 1. appears in case (a), with a = |θ| τ ). However Bα¯ (¯ τ ) ≤ Fy◦ , NF (W ) ≤ B, Therefore by induction on n, TQ Y = Bα¯ (¯ − (or Fy , Fcy , NF (W ) if F is of type L2n+1 [m] and case (a) holds) where y = f or eσ , contradicting TQ Y ≤ B. This completes the proof. 

13.3. Symplectic groups and systems In this section we assume Hypothesis 5.4.1.III, except that we write L for G. Thus q is an odd prime power, F = Fq is the field of order q, and m = (q 2 − 1)2 is the 2-share of q 2 − 1. Let n be a positive integer, U a 2n-dimensional vector space over F , f a symplectic form on U , and L = O(U, f) = Sp(U ) be the isometry group of f. Thus L ∼ = Sp2n (q) is the n-dimensional symplectic group over F . We also adopt Notation 5.4.2. Thus U = U1 ⊥ · · · ⊥Un , where Ui , i ∈ I = {1, . . . , n}, is a nondegenerate line, and Li = CL (Ui ⊥ ) acts naturally as Sp(Ui ) on Ui . Pick Ki ∈ Syl2 (Li ) and set Ω = {Ki : i ∈ I}. Let S ∈ Syl2 (L) with O = K1 · · · Kn ≤ S and form the fusion system F = FS (L). By 5.4.6.3, τ = (F, Ω)

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is a Lie packet of L. Set zi = z(Ki ). By 5.4.6.4, F is transitive on Ω and for i ∈ I, Ω(zi ) = {Ki }. Set c = z1 · · · zn . Then c = Z(L) and we set L+ = L/c and F + = F/c = FS + (L+ ). Hence L+ ∼ = P Sp2n (q). By definition, F = Sp2n [m] and F + = P Sp2n [m]. Let z = z1 and K = K1 , and choose notation so that z ∈ F f . As in Notation zZ.7, Z = Z(τ ) and ZS = {z1 , . . . , zn }. Set Z  = Z − ZS . Lemma 13.3.1. (1) {K} = Ω(z) and if n > 2 then Ω(z + ) = {K + }. (2) CZ (z) = z ⊥ and if n > 2 CZ + (z + ) = (z + )⊥ .  (3) Let Σ = {Ui : i ∈ I}, W = ZS , and L0 = i Li . Then NL (W ) = NL (Σ) = L0 Y where Y acts faithfully as Sym(Σ) on Σ. Further NL+ (W + ) = NL (W )+ . Proof. We already observed that {K} = Ω(z), while if n > 2 then z + = zi+ for i > 1, so also Ω(z + ) = {K + } and CL+ (z + ) = CL (z)+ . In particular (1) holds. By 5.4.9.1, CZ (z) = z ⊥ and then as CL+ (z + ) = CL (z)+ when n > 2, (2) holds. As c ∈ W , NL+ (W + ) = NL (W )+ . By 3.1.5.3, NL (W ) = NL (ZS ) and as zi is the unique involution s with [U, s] = Ui , it follows that NL (W ) = NL (Σ). As Li = O(Ui , f), L0 is the kernel of the action of NL (Σ) on Σ, so NL (W )/L0 ≤ Sym(Σ). Finally let B = {xi , yi : i ∈ I} a basis for U with {xi , yi } a basis for Ui and f(xi , yi ) = 1 for each i ∈ I. Represent Sym(I) on U via xgi = xig and yig = yig . Then the image Y of Sym(I) in GL(U ) preserves f, so Y ≤ L and by construction Y acts faithfully as Sym(Σ) on Σ, completing the proof of (3).  Notation 13.3.2. Choose notation so that OA = Kn−1 Kn ∈ F f and set A = Un−1 + Un and LA = CL (A⊥ ). Then LA = Sp(A) ∼ = Sp4 (q). Further t = zn . T = S ∩ LA is Sylow in LA with T = OA t where t ∈ Z  with zn−1 Set LA = FT (LA ) and τA = (LA , {Kn−1 , Kn }), so that τA is a quaternion fusion packet and LA ∼ = Sp4 [m]. Set u = zn−1 zn ; then u = Z(LA ) = Z(LA ), and we set ˜ ˜ n−1 , K ˜ n }). LA = LA /u , L˜A = LA /u , and τ˜A = (L˜A , {K ∼ Observe that T = Qm wr Z2 , so T is transitive on the involutions in T −OA , and this set of involutions is Z  (τA ). Further CLA (zn ) is transitive on the set P(τA ) of subgroups P = t, v, zn−1 of T such that t ∈ Z  (τA ) and v is a normal subgroup of COA (t) of order 4. Set P(˜ τA ) = {P˜ : P ∈ P(τA )}. Observe that P = v ∗ Q, where Q = vt, zn t is the unique Q8 -subgroup of P . If ξ = (X , Ξ) is a quaternion fusion packet with Sylow group X such that X = XX ◦ with X ◦ ∼ = (P )Sp4 [m], write P(ξ) for the set of P ∈ P(ξ ◦ ) such that v, zn (or ˜ v , z˜n ) is normal in X. When X ◦ is Sp4 [m] and R = CX (P˜ ), define AutP (R) = φˆ : φ ∈ O 2 (Aut(P )) , where φˆ ∈ Aut(R) is defined by (cx)φˆ = c · xφ for c ∈ CS (Q) and x ∈ Q. Lemma 13.3.3. Let ξ = (X , Ξ) be a quaternion fusion packet with Sylow group X such that X = XX ◦ and X ◦ ∼ = P Sp4 [m]. Let P ∈ P(ξ) and set R = CX (P ). Then X = XOξ , O 2 (AutX (R)) , where Oξ = O 2 (CX (Z(O(ξ)))). Proof. This follows from 9.4.22, 9.4.27, and 9.4.34.2.



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Lemma 13.3.4. Let ξ = (X , Ξ) be a quaternion fusion packet with Sylow group X such that X = XX ◦ and X ◦ ∼ = Sp4 [m]. Set u = Z(X ◦ ) and X˜ = X /u . Let ◦ ◦ X be Sylow in X and P ∈ P(ξ), and set R = CX (P˜ ). Then (1) X = XOξ , O 2 (AutX (R)) . (2) R = CX (Q)Q, where Q is the Q8 -subgroup of P . (3) CX ◦ (Q) ∼ = Dm . (4) If m > 8 then A4 ∼ = O 2 (AutX (R)) = AutP (R). (5) Suppose m = 8. Then Q◦ = QCX ◦ (Q) ∼ = D8 Q8 , 2 2 ∼ O (AutX (R) = O (Aut(Q◦ )) via the restriction map, and is Inn(Q◦ ) extended by A5 . Further O 2 (AutX (R)) = O 2 (AutNX (RX ◦ ) (R)), AutP (R) . (6) X = XOξ , AutP (R) . Proof. As X˜ satisfies the hypotheses of 13.3.3, we conclude from that lemma ˜ ˜, O 2 (Aut ˜ (R)) . ˜ that X˜ = XO Therefore by 2.1.15, X is generated by the preimX ξ 2 ages XOξ and O (AutX (R)) of the two subcategories generating X˜ , so that (1) holds. Recall P = v ∗ Q with Q = vt, zn t ∼ = Q8 and R = CX (P˜ ). As Inn(Q) = ˜ we have R = CR (Q)Q. Observe that CX (Q) centralizes vzn and hence CAut(Q) (Q), also v˜, so CX (Q) ≤ CX (P˜ ) = R, so R = CX (Q)Q, proving (2). Set y = zn t. As zn ∈ Z(OA ), COA (y) = COA (t) ∼ = K, so COA (Q) = COA (y) ∩ COA (v) = V is cyclic of order m/2. Let l ∈ COA (y) − V . Then i = yl is an involution centralizing Q and inverting V , so Y = CX ◦ (Q) = V i ∼ = Dm , proving (3). Let G be a model for NX (R) and H the normal subgroup of G which is a model for NX ◦ (R). Observe G = NX (R)H, so O 2 (G) = O 2 (H). Also Y Q = Q◦  G. Suppose m > 8 and set Q0 = Φ(Q◦ ). Then Q0 = Φ(Y ) ∼ = Zm/4 and Q1 = CQ◦ (Q0 ) = V Q. Hence Q1 and V = Z(Q1 ) are normal in G, as is Ω1 (Q1 ) = P . ˜ ∼ Thus O 2 (G) acts on P and centralizes Q◦ /P , so O 2 (G) = O 2 (H) and O 2 (H) = 2 2 2 ∼ ∼ ∼ O (Aut(Q)) = A4 , so O (H) = SL2 (3). Hence A4 = O (AutX (R)). Let h ∈ H be of order 3. Then [CX (Q), h] ≤ CO2 (H) (Q) = u ≤ CH (h), so [CX (Q), h] = 1. Thus completes the proof of (4). So assume m = 8. Then Q◦ ∼ = Q8 D8 , so Σ = O 2 (Aut(Q◦ ) is Inn(Q◦ ) extended ◦ ˜ ), Σ1 , where Σ1 /Inn(Q◦ ∼ by A5 . Then Σ = NΣ (X = S3 , and O 2 (Σ1 ) is the image of AutP (R) under the restriction map. Thus (5) holds. Finally (6) follows from (1), (4), and (5).  Definition 13.3.5. Assume γ = (G, Γ) is a quaternion fusion packet with Sylow group Sγ , such that G = Sγ G ◦ with G ◦ = F. Let O∗ = K1 · · · Kn−2 ∈ G f and set P1 = Sξ CG (O∗ )◦ , where Sξ = NSγ (O∗ ) and ξ = (P1 , {Kn−1 , Kn }). Write P(γ) for P(ξ). Pick P1 ∈ P(γ) and set R1 = CSξ (P/zn−1 zn ) ). By 13.3.4.6, P1 = Sξ Oξ , AutP (R1 ) .

13.4. Linear and unitary groups and systems In this section we assume case I or II of Hypothesis 5.4.1, except that we write L for G. Thus q is an odd prime power, F = Fq or Fq2 in case I or II, respectively,

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and m = (q 2 − 1)2 is the 2-share of q 2 − 1. Set π = ±1 where q ≡ π mod 4 and let a ∈ {0, 1}. Let n be a positive integer, U a (2n + a)-dimensional vector space over F , and take L = SL−π (U ). Thus U is a linear or unitary space over F and L is the subgroup of the isometry group of U contained in SL(U ). We also adopt Notation 5.4.2. Thus U = U0 ⊥U1 ⊥ · · · ⊥Un , where Ui , i ∈ I = {1, . . . , n}, is a line with Ui nondegenerate in case II, and U0 is of dimension a. Further for i ∈ I, Li = CL (Ui ⊥ ) acts naturally as SL−π (Ui ) on Ui . Here, in case / J . Pick I, if A is the sum of the Uj with j ∈ J ⊆ I ∪ {0} then A⊥ = Uk : k ∈ Ki ∈ Syl2 (Li ) and set Ω = {Ki : i ∈ I}. Let S ∈ Syl2 (L) with O = K1 · · · Kn ≤ S and form the fusion system F = FS (L). By 5.4.6.3, τ = (F, Ω) is a Lie packet of L. Set zi = z(Ki ). By 5.4.6.4, F is transitive on Ω and for i ∈ I, Ω(zi ) = {Ki }. Let z = z1 and K = K1 , and choose notation so that z ∈ F f . As in Notation zZ.7, Z = Z(τ ) and ZS = {z1 , . . . , zn }. Set Z  = Z − ZS . − By definition F = SL− 2n+a [m] and we write L2n+a [m] for F/Z(F). Of course Z(F) is cyclic of order (q+π, 2n+a)2 , so if a = 1 then Z(F) = 1 and F ∼ = L− 2n+1 [m]. Suppose on the other hand that a = 0. Then Z(F) = c is of order 2, and we set L+ = L/c and F + = F/c = FS + (L+ ). Thus F + = L− 2n [m]. Lemma 13.4.1. (1) K = Ω(z) and if n > 2 then Ω(z + ) = {K + }. (2) CZ (z) = {z} ∪ D(z) ∪ Do (z), where D(z) = {t ∈ Z : [U, t] ≤ U⊥ 1 } and o D (z) = {t ∈ Z : dim([U1 , t]) = dim([U⊥ , t] = 1}. 1 (3) Let Σ = {Uj : j ∈ I ∪ {0}}, W = ZS , and M the kernel of the action of NL (Σ) on Σ. Then NL (W ) = NL (Σ) = M Y where Y acts faithfully as Sym(Σ) on Σ. Further if a = 0 then NL+ (W0+ ) = NL (W )+ . (4) Set L0 = i∈I Li . Then M = O(M )L0 M0 where M0 is Y -invariant and generated by members of Z  , and either (a) a = 0 and M0 is the core of the n-dimensional permutation module for Y over F2 , or (b) a = 1 and M0 = ti : i ∈ I is the n-dimensional permutation module with ti ∈ Z  satisfying [U, ti ] = [Ui , ti ] + U0 . Proof. The proof of (1) is the same as that of 13.3.1. Suppose t ∈ Z − {z} centralizes z. Then t acts on U1 = [U, z] and U⊥ 1 = CU (z), and hence t centralizes L so t ∈ D(z), or [U , t] = X and so either [U, t] ≤ U⊥ 1 1 1  o , t] = X are 1-dimensional. In the latter case [K, t] =  1, so t ∈ D (z). This [U⊥ 1 proves (2). As c ∈ W , NL+ (W0+ ) = NL (W )+ when a = 0. By 3.1.5.3, NL (W ) = NL (ZS ), so as zi is the unique involution s with [U, s] = Ui , and U⊥ i = CU (s), and as U0 = CU (W ), it follows that NL (W ) = NL (Σ). As M is the kernel of the action of NL (Σ) on Σ, NL (W )/M ≤ Sym(Σ). For i ∈ I, let {xi , yi } be a basis for Ui , with the basis hyperbolic in case II. Represent Sym(I) on U via xgi = xig and yig = yig for i ∈ I, and have Sym(I) centralize U0 . Then the image Y of Sym(I) in GL(U ) is contained in L and by construction Y acts faithfully as the stabilizer in Sym(Σ) of U0 on Σ, completing the proof of (3). ¯ = N/O(N ) Let N be the kernel of the action of NGL−π (U) (Σ) on Σ. Set N ¯ = and for i ∈ I, choose a basis {xi , yi } for Ui as in the proof of (3). Then N ∼ ¯0 × N ¯1 × · · · × N ¯n , where Nj = CN (U ⊥ ) for j ∈ J = I ∪ {0}, and N ¯i = ¯ N s Z2 = i j for i ∈ I, and also for i = 0 if a = 1. Moreover for i ∈ I, si induces an outer automorphism on Li and we may choose notation so that [Ui , si ] = xi . Now if

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a = 0 we may take M0 = si sj : i, j ∈ I , so that M0 ∼ = E2n−1 , and, from the construction of Y in the proof of (3), M0 is the core of the permutation module for Y . Also for i = j, dim([U, si sj ]) = 2, so si sj ∈ Z  . Therefore (4a) holds in this case. Similarly if a = 1 we may take M0 = ti : i ∈ I , where ti = si s0 , so that (4b) holds.  Notation 13.4.2. Choose notation so that OA = Kn−1 Kn ∈ F f and set A = Un−1 +Un +U0 and LA = CL (A⊥ ). Then LA = SL−π (A) ∼ = SL−π 4+a (q). Further TA = S ∩ LA is Sylow in LA . Set LA = FTA (LA ) and τA = (LA , {Kn−1 , Kn }), so that τA is a quaternion fusion packet and LA ∼ = SL− 4+a [m]. Set u = zn−1 zn ⊥ ∼ and B = Un−1 + Un . Thus B = [U, u], LB = CL (B ) = SL−π 4 (q), TB = S ∩ LB is Sylow in LB , LB = FTB (LB ) ∼ = SL− 4 [m], and τB = (LB , {Kn−1 , Kn }) is a ˜ B = LB /u , quaternion fusion packet. Also u = Z(LB ) = Z(LB ), and we set L ˜ ˜ ˜ ˜ LB = LB /u , and τ˜B = (LB , {Kn−1 , Kn }). For i ∈ I let {xi , yi } be a basis for Ui chosen as in the proof of 13.4.1. We may t choose our bases so that there is t ∈ Z m (τB ) with xtn−1 = xn and yn−1 = yn , and  s ∈ Z (τB ) so that [U, s] = xn−1 , xn and CB (s) = yn−1 , yn . Then from 13.4.1.4, TB = OA t, s and ts ∈ Z m (τB ). Similarly if a = 1 we may choose t to centralize U0 , and, as in 13.4.1.4, ti ∈ Z  (τA ) with [U, ti ] = yi , U0 and CA (ti ) = yn−1 , yn . Again from 13.4.1.4, TA = t = Tn . (Tn−1 × Tn )t ∼ = SD2m wr Z2 , where Ti = Ki ti and Tn−1 Notation 13.4.3. Write Pm (τB ) for the set of subgroups P = t, v, zn−1 of T such that t ∈ Z m (τB ) and v is a normal subgroup of COA (t) of order 4. Set Pm (˜ τB ) = {P˜ : P ∈ Pm (τB )}. As in 13.3.2, P = v ∗ Q, where Q = vt, zn t is the unique Q8 -subgroup of P . Write Po (τB ) for the set of subgroups P = s, zn , v such that s ∈ Do (τB ). Set ˜∼ τB ) = {P˜ : P ∈ Po (τB )}. From 9.4.16, CT˜B (P˜ ) = Q Po (˜ = E16 and AutL˜ B (P˜ ) is of + index 2 in O4 (2), so it follows that CTB (P˜ ) = Q ∼ = Q28 . Suppose ξ = (X , Ξ) is a quaternion fusion packet with Sylow group X such ! that X = XX ◦ with X ◦ ∼ = (P )SL− 4 [m]. For ! ∈ {m, o}, write P (ξ) for the set of P ∈ P! (ξ ◦ ) such that when ! = m, v, zn (or ˜ v , z˜n ) is normal in X, and when ˜ Write R(P ) for CX (P˜ ). ! = o, CX (P˜ ) = CX (Q). Lemma 13.4.4. Assume ξ = (X , Ξ) is a quaternion fusion packet with Sylow group X such that X = XX ◦ and X ◦ ∼ = L− 4 [m]. Then either o (1) P (ξ) = ∅, X has two orbits on Pm (ξ) with representatives P1 and P2 , and X = XOξ , O 2 (AutX (R(Pi ))) : i = 1, 2 , or (2) Po (ξ) = ∅, X is transitive on Pm (ξ) and Po (ξ) with representatives P1 and P2 , and X = XOξ , O 2 (AutX (R(Pi ))) : i = 1, 2 , Proof. This follows from 9.4.22, 9.4.27, and 9.4.34.2.



Lemma 13.4.5. Let ξ = (X , Ξ) be a quaternion fusion packet with Sylow group ◦ ˜ X such that X = XX ◦ and X ◦ ∼ = SL− 4 [m]. Set u = Z(X ) and X = X /u . Let ◦ ◦ X be Sylow in X . Then (1) X has two orbits on Pm (ξ) ∪ Po (ξ) with representatives Pi , i = 1, 2, and with P1 ∈ Pm (ξ). Further, setting Ri = R(Pi ) = CX (P˜i ) for i = 1, 2, X = XOξ , O 2 (AutX (Ri )) : i = 1, 2 . (2) R1 = CX (Q)Q, where Q is the Q8 -subgroup of P1 , and CX ◦ (Q) ∼ = SD2m .

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(3) A4 ∼ = O 2 (AutX (R1 )) = {φ ∈ Aut(R1 ) : φ|Q ∈ O 2 (Aut(Q)) and [φ, CX (Q)] = 1}.

∼ Q2 , and Z3 /E16 = ∼ O 2 (AutX (R2 )) (4) Assume P2 ∈ Po (ξ). Then Q = R2 ∩X ◦ = 8 2 consists of those φ ∈ Aut(R2 ) such that φ|Q ∈ O (Aut(Q)), φ acts on P2 , and for φ of order 3, CR2 (φ) is the preimage in R2 of O2 (CAut(Q) (φ|Q )) under the conjugation map R2 → Aut(Q). Proof. As X˜ satisfies the hypotheses of 13.4.4, we conclude from that lemma ˜ ∪ Po (ξ) ˜ with representatives P˜i , i = 1, 2, and that ˜ has two orbits on Pm (ξ) that X ˜ ˜, O 2 (Aut ˜ (R ˜ i )) : i = 1, 2 . Therefore by 2.1.15, X is generated by the X˜ = XO X ξ preimages XOξ and O 2 (AutX (Ri )), i = 1, 2, of the three subcategories generating X˜ , so that (1) holds. Adopt the notation in the proof of 13.3.4; in particular P1 = v ∗ Q with Q = vt, zn t ∼ = Q8 and R1 = CX (P˜1 ). Then the proof of 13.3.4.2 establishes the first statement in (2), and setting y = zn t, the proof of 13.3.4.3 shows COA t (Q) = V, i ∼ = Dm , where V = COA (Q) is cyclic of order m/2, l ∈ COA (y) − V , and i = yl is an involution inverting V . Next sy centralizes Q with (sy)2 = u, and s COA (t) ∼ = Ks ∼ = SD2m , so CX ◦ (Q) = i, sy ∼ = SD2m , completing the proof of (2). Then (2) and the proof of 13.3.4.4 establish (3). Assume the hypothesis of (4), let N = NX (R2 ), let G be a model for N , and let H be the normal subgroup of G which is a model for NX ◦ (R2 ). Then Q = O2 (H) and G = NX (R2 )H, so O 2 (G) = O 2 (H). From the discussion in 13.4.3, ˜ H ∼ = SL2 (3) ∗ SL2 (3) and NH (P ) is of index 3 in H. As Inn(Q) = CAut(Q) (Q), ˜ ˜ CX (Q) = CR2 (Q) = CX (Q)Q. Let h ∈ H be of order 3. Then [CX (Q), h] ≤ CO2 (H) (Q) = u = Z(H), so [CX (Q), h] = 1. Then from the discussion in 13.4.5, ˜ = 2 with C ˜ (h) a complement to Q ˜ in R ˜ 2 , and CR (h) is the |R2 : CX (Q)| 2 R2 preimage in R2 of O2 (CAut(Q) (ch|Q )) under the conjugation map R2 → Aut(Q). This completes the proof of (4).  Definition 13.4.6. Assume a = 0. Assume that γ = (G, Γ) is a quaternion fusion packet with Sylow group Sγ , such that G = Sγ G ◦ with G ◦ = F. Let O∗ = K1 · · · Kn−2 ∈ G f and for i = 1, 2, set P = NSγ (O∗ )CG (O∗ )◦ and ξ = (P, {Kn−1 , Kn }). For i = 1, 2, write Pi (γ) for Pi (ξ), pick Pi = Pi (γ) ∈ P(γ), and set Ri = Ri (γ) = R(Pi ). By 13.4.5.3, O 2 (AutP (R1 ) = {φ ∈ Aut(R1 ) : φ|Q1 ∈ O 2 (Aut(Q1 )) and [φ, CSγ (Q1 )] = 1}, where Q1 is the Q8 -subgroup of P1 . Further if P2 ∈ Po (ξ), then by 13.4.5.4, Q2 = R2 ∩ X ◦ ∼ = Q28 , where X ◦ is Sylow in CG (O∗ )◦ , and O 2 (AutP (R2 )) consists of those φ ∈ Aut(R2 ) such that φ|Q2 ∈ O 2 (Aut(Q2 )), φ acts on P2 , and for φ of order 3, CR2 (φ) is the preimage in R2 of O2 (CAut(Q2 ) (φ|Q2 )) under the conjugation map R2 → Aut(Q2 ). Notation 13.4.7. Assume a = 1. Choose notation so that Kn ∈ F f and set C = Un + U0 and LC = CL (C ⊥ ). Then LC = SL(C) ∼ = SL−π 3 (q) with TC = S ∩ LC Sylow in LC . Set LC = FTC (LC ) and τC = (LC , {Kn }), so that τC is a quaternion fusion packet and LC ∼ = L− 3 [m]. Let {xn , yn } be a basis for Un chosen as in Rr.1, and choose the basis so that there is t ∈ Z  (τC ) with [U, t] = xn + U0 and CC (s) = yn . Then TC = Kn t ∼ = SD2m .

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Write P(τC ) for the set of subgroups P = t , zn of TC such that t ∈ Z  (τC ). Then Kn is transitive on P(τC ) and for P ∈ P(τC ), P = CTC (P ) and LC = TC Ozn , O 2 (AutLC (P )) . Definition 13.4.8. Assume a = 1. Assume that γ = (G, Γ) is a quaternion fusion packet with Sylow group Sγ , such that G = Sγ G ◦ with G ◦ = F. Let O∗ = K1 · · · Kn−1 ∈ G f , set P3 = P3 (γ) = NSγ (O∗ )CG (O∗ )◦ . Observe that CG (O∗ )◦ = LC , pick P3 = P3 (γ) ∈ P(τC ), and set R3 = R3 (γ) = R(P3 ) = CSγ (P ). Next U0 = CU (c) and [U, c] = U⊥ 0 , so CL (c) = CL (U0 )t where P = t, zn . Further c ∈ Z(Sγ ) and Gc = CG (c) = Sγ Gc◦ , where Gc◦ = FCS (U0 ) (Lc ) ∼ = SL− 2n [m]. o Of course γc = (Gc , Ω) is a quaternion fusion packet. As t ∈ Sγ , P (Gc ) = ∅. For i = 1, 2, define Pi (γ) = Pi (γc ), Ri (γ) = Ri (γc ), and Pi (γ) = P(γc ). Lemma 13.4.9. Assume a = 1. For i ∈ I, let {xi , yi } be a basis for Ui chosen as in the proof of 13.4.1, and more generally adopt the notation of that proof. Let H be a model for NF (O(τ )).  ¯ ¯0 = ¯ ¯ 0 where L (1) H = LY i∈I Li and either m > 8 and Li = Ki or m = 8, ∼ ¯ ¯ Ki = O2 (Li ), and Li = SL2 (3). Further Y0 = M0 Y where M0 = ti : i ∈ I with ti ∈ Z  such that [U, ti ] = xi + U0 and CUi (ti ) = yi . Also Y = Sym(I) is represented on U so that Y centralizes U0 and for g ∈ Y and i ∈ I, xgi = xig and yig = yig . In particular M0 is the n-dimensional F2 -permutation module for Y . (τ ) as in Rr.8, and choose (2) Define P3 = P3 (τ ), R3 = R3 (τ ), and P3 = P3 notation so that t = tn . Then R3 = P3 XT where X = 1≤i 1 then Sf F2 = Sf W2 , AutP2,j (R2,j ) : j ∈ I2 . But Sf Wi ≤ SW ≤ Y and we will show that AutPi,j (Ri,j ) ≤ Y, so Ff ≤ Y by 1.3.2 in [Asc19]. So assume d2 ≤ 1. By 13.5.4.2, d2 = 0, so d2 = 1. Then F2 is empty, so the lemma holds. Thus we may assume that d2 > 1 and it remains to show that AutPi,j (Ri,j ) ≤ Y. If di = 2 then SL2 [m] ∼ = Fi ≤ Oτ , so we may take di > 2. Suppose first that di is even, so that di ≥ 4. Then Kni Kni −1 is conjugate in W to Kn Kn−1 , so AutPi,j (Ri,j ) ≤ Y. So take di odd. Then Kni is conjugate in W to Kn , and hence again AutPi,j (Ri,j ) ≤ Y. So assume case (II) holds. Arguing as above, Fe ≤ Y0 if CS (e) acts on F1 , so CS (e) does not act on F1 . Then by definition of Y, |θ| = n/2, completing the proof of the lemma.  Lemma 13.5.6. Assume (I) holds. Then (1) n is a power of 2. (2) L = P Sp2n (q) or P SL− 2n (q). (3) Let θ ⊆ ZS be of order n/2 and f = eθ ∈ F f . Then Ff ≤ Y. Proof. Observe NF (W ) = W, so NF (W ) ≤ Y. Let ξ ∈ Ξ and B = Bξ (τ ). By 13.2.26, F = B, so B ≤ Y. But from 13.2.24, one of (a), (b), or (c) of 13.2.24 holds and B = Fxj , NF (W ) : j ∈ J for some set of involutions xj = eθj , so Fxj ≤ Y for some j ∈ J. Thus by 13.5.5, (2) holds and |θj | = n/2, so also (3) holds. Thus it remains to show that n is a power of 2. In case (b) and (c) this holds by definition, so we may assume (a) holds. Therefore S is not transitive on I and we may choose  ξ to be an orbit of S, contradicting |θj | = n/2. Lemma 13.5.7. Case (II) holds. Proof. Assume instead that (I) holds and choose θ and f as in 13.5.6.3. Set U1 = Uj : j ∈ θ and U2 = U⊥ 1 . By 13.5.6.1, n is a power of 2 so S is transitive on I and centralizes f . Set Li = LUi , Ti = S ∩ Li and Fi = FTi (Li ). Then O 2 (Ff ) = F1 ∗ F2 and S is transitive on {F1 , F2 }. Therefore Ff satisfies Hypothesis 13.5.1.II. By 13.5.5, for each x ∈ X, CF (x) ≤ Y, so by 13.5.6.3, Ff is a counterexample to Theorem 13.5.3. Thus (II) holds by minimality of τ .  Notation 13.5.8. Recall that we abuse notation and work in SL(U ) rather than in the projective group P SL(U ). Let θ1 = {z ∈ ZS : [U, z] ≤ U1 }. Now Π1 = {θ1 , ZS − θ1 } is an S-invariant partition of ZS . Proceeding recursively, assume Πk is an S-invariant partition of ZS into 2k blocks of size n/2k . Pick θk ∈ Πk and let θk+1 ⊆ θk be of order n/2k+1 such that S , so that Πk+1 is an S-invariant |NS (θk ) : NS (θk+1 )| = 2, and set Πk+1 = θk+1

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partition of ZS into 2k+1 blocks of size n/2k+1 . Continuing this process, we arrive at θr = {z} for some z ∈ ZS ; that is n = 2r . Let Πk = {θk,1 , . . . , θk,2k } and Uk,i = [U, fθk,i ]. Thus  U = Uk,1 ⊥ · · · ⊥Uk,2k and we set Uk = {Uk,1 , . . . , Uk,2k }, Lk,i = LUk,i and Lk = i Lk,i the direct product of 2k copies of L2r−k+1 (q), where L = Sp or SL− . Let U k = {U1 , . . . , Uk } and Gk the stabilizer in SL of U k . Set Dk = FS (Gk ). Lemma 13.5.9. Gk = SLk O(Gk ). Proof. By construction, SLk ≤ Gk and Lk is contained in the kernel X of the k action of Gk on Uk . By 2.1.16, GU k is contained in a Sylow 2-group of Sym(Uk ). For x ∈ X of odd order, x|Uk,i ∈ Lk,i O(Gk ) as Lk,i acts faithfully as L(Uk,i ) on Uk,i . Therefore X/Lk O(Gk ) is a 2-group, completing the proof.  ∼ SL2 (q) has Sylow 2-group O(si ) for that si ∈ ZS Lemma 13.5.10. (1) Lr,i = such that si ∈ Z(Lr,i ) and Ur,i = [U, si ]. (2) Oτ = FO(τ ) (Lr ). (3) Dr = FS (Gr ) = SW ≤ Y. Proof. By construction, Ur,i = [U, si ] for some si ∈ ZS , and then (1) follows. In particular by (1), O(τ ) is Sylow in Lr and (2) holds. By 13.5.9, Gr = SLr , so  FS (Gr ) = FS (SLr ) = SOτ , establishing (3). Lemma 13.5.11. E(τ ) ≤ Y. Proof. From Qq.6, E = E(τ ) = NF (O(ξ)), SOτ , NF (O) : ξ ∈ Ξ , where Ξ consists of those ξ ⊆ ZS with O(ξ) ∈ F f and NS (O(ξ)) transitive on ξ. Now NF (O) ≤ SOτ , FS (SH) = SW ≤ Y. Thus it remains to show that N = NF (O(ξ)) ≤ Y for ξ ∈ Ξ. But N ≤ Fx , where x = eξ , and by definition of Y, Fx ≤ Y unless ξ = θ, so we may the latter holds. Now N = FS (X), where X = NL1,1 (O(ξ))L1,2 S. Therefore N = FS (SL1,2 ), NN (S2 ) , where S2 is Sylow in L1,2 . By minimality of τ , FS (SL1,2 ) ≤ Y. Further NN (S2 ) ≤  NN (O), which is contained in Y by the previous case. Notation 13.5.12. Observe that D1 = F, so D1 ≤ Y. On the other hand by 13.5.10.3, Dr ≤ Y. Thus there exists a least k such that Dk ≤ Y. Set D = Dk , . . . , 2k } and ρ = (D, Ω), let S0 = S ∩ Lk , and write D0 for FS0 (Lk ). Let I = {1, for i ∈I set Si = S ∩ Lk,i and Li = FSi (Lk,i ). From 13.5.8, Lk = i∈I Lk,i , so D0 = i∈I Li is a direct product. Set Ωi = Ω ∩ Si and ρi = (Li , Ωi ). Observe: Lemma 13.5.13. The pair D, D0 satisfies Hypotheses 11.4.1 and 11.4.6. Notation 13.5.14. As in 11.4.2 and 11.4.6, for X ≤ S and 0 ≤ i ≤ 2k , write Xi for X ∩ Si . As D ≤ Y, it follows from the Alperin-Goldschmidt Fusion Theorem that the set R of subgroups R ∈ D e such that AutD (R) ≤ AutY (R) is nonempty. Pick R ∈ R and let G be a model for ND (R) and set G∗ = G/R. By 13.5.13 and 11.4.4.1, R0 ∈ F f rc , so there is a model G0 for ND0 (R0 ). By 11.4.4, we can embed G and G0 in a certain group Gc with G0  Gc and G = NGc (R). Thus we can form H0 = NG0 (R) and by 11.4.4, H0  G with R0 = O2 (H0 ) and H0∗ ∼ = H0 /R0 .

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Set Q = NR (Lk,1 ) and X = FQS1 (QLk,1 ). Let 1 ∈ Σ be a set of coset representatives for Q in R. Write R∗ for the set of those R ∈ R of maximal order. Lemma 13.5.15. For each K ∈ Ω, |K ∩ R| ≤ 2. Proof. Assume otherwise; then from 3.4.5, AutD (R) = AutE(ρ) (R). But E(ρ) ≤ E(τ ) and by 13.5.11, E(τ ) ≤ Y, contrary to the choice of k.  Lemma 13.5.16. (1) For each i ∈ I, Ri = Si . (2) R is transitive on {Lk,i : i ∈ I}. (3) Q ∈ X e . Let GQ be a model for NX (Q). (4) GQ /Q ∼ = H0∗ .  (5) σLet W1 = (ZS (ρ1 )) and define α : W1 ∩ Z(Q) → W ∩ R by α : x → k σ∈Σ x . Then xα ∈ Z(R) and if x is of weight p in W1 then xα is of weight 2 p in W . (6) If p = |ZS (ρ1 )| then CD (xα) ≤ Y. Proof. If Ri = Si then Ωi ⊆ R, contrary to 13.5.15. Thus (1) holds. By 13.5.13, we can appeal to results in section 11.4. In particular (2) follows from (1) and 11.4.8.1, (3) follows from 11.4.8.4, and (4) follows from 11.4.8.5. From 13.5.14, 1 ∈ Σ is a set ofcoset representatives for Q in R. If σi ∈ Σ with R1 σi = Ri then R0 = i∈I Ri = i∈I R1σi , so for x ∈ W1 ∩ Z(Q) of weight p, xα centralizes Q, Σ = R and xα is of weight |Σ|p = |I|p = 2k p in W . Thus (5) holds. Assume the hypothesis of (6) and set y = xα. Observe that |ZS (ρ1 )| = n/2k = 2r−k . As p = |ZS (ρ1 )|, y = 1 so y is an involution. Then by definition of Y, either Fy ≤ Y or 2k p = n/2 = 2r−1 , and we may assume the latter, so that p = 2r−k−1 . R ∪ (θk − θk+1 )R . Then conjugating in Y, we may take x = fθk+1 , so Πk+1 = θk+1 But then CSLk (y) = CS (y)Lk+1 , so as Dk+1 ≤ Y, again CD (y) ≤ Y.  Lemma 13.5.17. ρ1 = (L1 , Ω1 ) satisfies Hypothesis iI.1 and if R ∈ R∗ then ρ1 , Q satisfies Hypothesis iI.7. Proof. Visibly ρ1 satisfies Hypothesis 11.2.1 with |Ω(z)| = 1. So assume R ∈ R∗ . By 13.5.16.4, Q ∈ Le1 . From 11.2.6, W(Q) consists of those involutions x ∈ Z(Q) ∩ W1 such that x = eα for some α ⊆ I(ρ1 ) of weight a power of 2. Passing to L1 /Z(L1 ) if necessary we may assume eI(ρ1 ) = 1. Hence by 13.5.16.6, for each x ∈ W(Q), CD (xα) ≤ Y. Let x ∈ W(Q), y = xα and Z = y G . As R ∈ R∗ and CD (y) ≤ Y, the proof of  11.3.5.1 shows that O 2 (CG (Z)) = R. Set Zx = xGQ . Then Z is a full diagonal   subgroup of σ∈Σ Zxσ and the isomorphism of 13.5.16.4 maps O 2 (CGQ (Zx ))/Q to   O 2 (CG (Z))/R = 1, so O 2 (CGQ (Zx )) = Q. Hence ρ1 , Q satisfies 11.2.7.1. Similarly if ρ, R satisfies the setup of 11.2.7.2 and U = ∅ then the proof of 11.3.5.2 shows that m2 (G∗ ) = 1 and U is not normal in G∗ . Let B ∗ be the subgroup of G∗ generated by its involutions. By 11.4.5.2, B ∗ is the subgroup of H ∗ ∗ generated by its involutions. Thus isomorphism of 13.5.16.4 maps the subgroup BQ ∗ ∗ ∗ ∗ of GQ = GQ /Q generated by involutions to B , so m2 (GQ ) = 1 and UQ = BQ . Therefore ρ1 , Q satisfies 11.2.7.2. This completes the proof of the lemma.  We are now in a position to obtain a contradiction, completing the proof of Theorem 13.5.3. Namely choose R ∈ R∗ . By 13.5.17, the pair ρ1 , Q satisfies Hypotheses 11.2.1 and 11.2.7. Further condition (a) of Theorem 11.2.33 is satisfied,

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as is condition (b) by 13.5.15. Condition (c1) of 11.2.33 hold as X = FQS1 (SLk,1 ). But then by Theorem 11.2.33, |ZS (ρ1 )| = 1, a contradiction. This completes the proof of Theorem 13.5.3. 13.6. Finishing μ abelian In this section we continue to assume Hypotheses 13.2.1 and 13.2.14, and adopt the various notational conventions from section 13.2. In addition we adopt the following notation: Notation 13.6.1. Pick ξ ∈ Ξ and in cases (a) and (b) of 13.2.24 set f = eξ and θ1 = ξ. In case (c) of 13.2.24, set f = eσ and θ1 = σ. For i = 1, 2, set Fi = Fi (f ), θ2 = I − θ1 , and ni = |θi |. Let Ti be Sylow in Fi . If F is of type P Sp2n [m] or L− 2n [m] then c = 1 by 13.2.19.2. Thus n > 2 in this case by 13.1.1.1, so ni0 > 1 for some i0 ∈ {1, 2}, and, interchanging the roles of 1 and 2 if necessary, we may assume i0 = 2. On the other hand if F is of type ∼ L− 2n+1 [m] then n ≥ 2 by 13.2.1.2, and (a) or (c) holds, so F2 = L2n2 +1 [m] with n2 ≥ 1. Set ρ2 = (T F2 , Ω2 ).   If F is of type P Sp2n [m] or L− 2n [m] set I (τ ) = I (ρ2 ), so that by 13.5.2,  I (τ ) = {1} or {1, 2}, respectively. Further set Y = W, AutPj (Rj ) : j ∈ I  (τ ) , where Rj = Rj (τ ) = Rj (ρ2 ) and Pj = Pj (τ ) = Pj (ρ2 ). If F is of type L− 2n+1 [m] then by 13.5.2, I  (τc ) = {1, 2} is of order 2 and I  (ρ2 ) = {3}, {1, 2, 3} for n2 = 1, n2 > 1, respectively. In this case set I  (τ ) = {1, 2, 3} and Y = W, AutPj (Rj ) : j ∈ I  (τ ) , where for j = 1, 2, Rj = Rj (τ ) = Rj (τc ) and Pj = Pj (τ ) = Pj (τc ), while R3 = R3 (τ ) = R3 (ρ2 ) and P3 = P3 (τ ) = P3 (ρ2 ). Lemma 13.6.2. Y = F˜ . ˜ = Ω, G ˜ = H, and W ˜ = W. By 13.5.3 applied Proof. By 13.2.23, S˜ = T , Ω  ˜ ˜ ˜ ˜ to F, we have F = W, AutP˜j (Rj ) : j ∈ I (˜ τ ) . Let j ∈ I  (τ ). If j = 3, let θ = {n − 1, n} ⊆ I with O(θ) ∈ F f . If j = 3 ¯ ◦ , where let θ = {n} with O(θ) ∈ F f . Set θ¯ = I − θ and consider X = CH (O(θ)) H = Fc if F is of type L− [m] and j = 1 or 2, while H = F otherwise. 2 2n+1 Suppose j = 3. Then X ∼ = Sp4 [m] if F is of type Sp2n [m] and X ∼ = L− 4 [m] m otherwise. A Sylow group X of X is O(θ)ti : i ∈ I(τ ) , where ti ∈ Z (τX ) with i = en and τX = (X , {Kn−1 , Kn }). The subgroup X is determined by T , Ω, etn−1 and θ by 13.1.10. Further from sections 13.3 and 13.4, Pj ∈ P(H) is determined by X and {Kn−1 , Kn } up to conjugation in X, so we may take P˜j = Pj . Then ¯ = R/u and u = zn−1 zn . Thus Rj is determined by T and Rj = CT (P¯j ), where R ˜ j = Rj . Pj , so R Consider the case j = 1. By parts (4) and (5) of 13.3.5 and by 13.4.5.3, AutP1 (R1 ) = {φ ∈ Aut(R1 ) : φ|Q ∈ O 2 (Aut(Q)) and [φ, CR1 (Q)] = 1}, where Q is the Q8 -subgroup of P1 . Then visibly AutP1 (R1 ) is determined by P1 and ˜ 1 ) = AutP (R1 ). T , so AutP1 (R1 ) is determined by T and Ω. Therefore AutP˜1 (R 1 In particular F˜ = Y if F is of type P Sp2n [m]. So suppose j = 2. By 13.4.5.4, R ∩ X has a unique Q28 -subgroup Q, and AutP2 (R2 ) consists of those φ ∈ Aut(R) such that φ|Q ∈ O 2 (Aut(Q)), φ acts on

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P2 , and for φ of order 3, CR (φ) is the preimage in R of O2 (CAut(Q) (φ|Q )) under the conjugation map R → Aut(Q). Thus AutP2 (R2 ) is determined by T and Ω, so ˜ 2 ) = AutP (R2 ), and in particular F˜ = Y if F is of type L− [m]. AutP˜2 (R 2 2n Hence we may assume F is of type L− [m] and it remains to consider the 2n+1 ˜ 3 and AutP (R3 ) = Aut ˜ (R ˜ 3 ). Then case j = 3, where we must show R3 = R 3 P3 − o ∼ ∼ X = L3 [m], and X = Kn tn = SD2m for tn ∈ D (zn ). Further P3 = tn , zn is determined up to conjugation in X and R3 = CT (P3 ). In particular, using 13.1.10, ˜ = H, P3 and R3 are determined P3 and R3 are determined by H and Ω, so as G ˜ ˜ by T and Ω. Thus we may take P3 = P3 and R3 = R3 . Next from the proof of 13.2.21, a Sylow group Q of the kernel M of the action of H on Ω is of the form Q = X1 × · · · × Xn , where Xi = Ki ti and ti ∈ Do (zi ). Further there is C ≤ M acting faithfully as Sym(I) on I with Xih = Xih for h ∈ C ˜ = P3 Xn B, where Xn =  and QC ∼ = X wr Sn . From 13.4.9.2, R i 2. Now [Cn , L] ≤ CL (P3 ), so O 2 (Cn ) centralizes L. Then as Cn = O 2 (Cn )s for s ∈ Z m ∩ C, it remains to show that s centralizes L. Take z1s = z2 , u = z1 z2 , and α ∈ A(u). By 13.1.10.2, there is ◦ a coconnected component ρ = (G, Γ) of τuα containing sα with Γ = {K1 α, K2 α}. ∗ Then X α centralizes G, so X centralizes s, so indeed Cn centralizes L. Next by construction, Ki centralizes X , so it remains to show t = ti zn centralizes L. As ti centralizes X, either t or ti centralizes L, and we may assume the latter. If n > 2 there is i = k < n and α ∈ A(zk ). Then working in F2 (z), tα centralizes X α∗ , contradicting [t, L] = 1. Therefore n = 2. Let T2 = X, s ∈ Z m , and T1 = T2s . By construction, T1 , T2 = T1 × T2 with Ti = ti Ki . Now Ki centralizes X and t1 centralizes FP3 (L), T2 Oz2 = X , so T1 centralizes X . Set C2 = X and C1 = C2 c∗s , so that T2 = T1s centralizes C1 . Therefore by 2.2 in [Asc11], we have C1 × C2 ≤ F. Observe that Hypothesis 8.1.17 is satisfied. Further C2 = X ∼ = L− 3 [m], so C2 has one class of involutions. It follows from 8.1.21 that F ◦ = C1 × C2 . This ◦  contradicts K2 ∈ K1F by 13.2.3.1. Lemma 13.6.3. NY (W ) = W  NF (W ). ˜ ) = W  NF (W ). ˜ so by 13.2.23, NY (W ) = N ˜ (W Proof. By 13.6.2, Y = F, F  Lemma 13.6.4. Let g = f , or if F is of type L− 2n+1 [m], let g = c or cf . Then (1) CT (g)NFg◦ (W )  NFg (W ). (2) Yg  Fg . Proof. Let Hg be the model for NFg◦ (W ) contained in CG (g). To prove (1), it suffices to show that CT (g)Hg  CG (g). Then as L0 ≤ Hg , it suffices to show that CT (g)L0  CG (g). We use the description of H in 13.2.21. By parts (5) and (6) of that lemma, if F is of type P Sp2n [m] or L− 2n+1 [m], then H is M1 wr Sn ,

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where M1 = L1 or L1 t1 , respectively, with L1 = O(1) if m > 8 and L1 ∼ = SL2 (3) if m = 8. Thus M1 is O(1), SL2 (3), SD2m , or SL2 (3)(1) . If F is of type P SL− 2n [m] [m]. then H is of index 2 in the group for L− 2n+1 Next CG (g) is the stabilizer in G of a partition Λ = {θ1 , θ2 } of I. Thus, setting ni = |θi |, CG (g)I ∼ = Sn1 × Sn2 , unless n1 = n2 = n/2 and c = 1, where wr Z . Moreover g ∈ Z(S), except in (c) where |S : Sf | = 2. S CG (g)I ∼ = n/2 2 Next if ni > 1 then the factor Sni is generated by transpositions that are induced by members of Tg , so SgI = TgI , except in the case ni = n/2 and c = 1. It follows that CT (g) = NCH (g) (Tg )  CG (g), except when F is of symplectic type and m = 8, where at least CT (g)L0  CG (g). Therefore (1) holds. Next by 13.6.2, Yg = CT (g)Fg◦ , so by 1.5.1 in [Asc19], to prove (2) it suffices to show that NYg (Tg )  NFg (Tg ). But NYg (Tg ) = CT (g)NFg◦ (Tg ). Also O(τ ) ≤ Tg , so NFg (Tg ) ≤ NFg (W ). Then as Oτ ≤ NFg◦ (Tg ), (2) follows from (1).  Lemma 13.6.5. F ◦ = Y ∼ = F˜ . Proof. We show Y  F. Then as Ω ⊆ T and T is Sylow in Y, F ◦ = Y, so the lemma follows from 13.6.2. If F is of type P Sp2n [m] or P SL− 2m [m], set J = {1, 2}, G1 = Ff , E1 = Yf , G2 = NF (W ), and E2 = W. In case (c), set J = {1, 2}, G1 = Ff , E1 = Yf , G2 = Fc , and E2 = Yc . Finally if F is of type L− 2n+1 [m] and (a) holds, set J = {1, 2, 3}, G1 = Ff , E1 = Yf , G2 = Ff c , E2 = Yf c , G3 = NF (W ), and E3 = W. We verify that the tuple (Gj , Ej : j ∈ J) satisfies Hypothesis 1.4.7 in [Asc19], and then appeal to 1.4.8 in [Asc19] to conclude that Y  F, except in case (c) where we verify Hypothesis 1.4.3 in [Asc19] and appeal to 1.4.5 in [Asc19]. By 13.2.26, F = Gj : j ∈ J S and Y = Ej : j ∈ J T . Thus in particular Hypotheses 1.4.1 and 1.4.7.3 in [Asc19] are satisfied. By 13.6.3 and 13.6.4.2, Ej  Gj for each j ∈ J, so Hypotheses 1.4.3.1 and 1.4.7.1 in [Asc19] are satisfied. By 13.6.2, Y is saturated, verifying 1.4.7.4 in [Asc19]. Suppose (c) does not hold. In this case to complete the proof by verifying Hypothesis 1.4.7 in [Asc19], it remains to show that 1.4.7.2 in [Asc19] holds. As we observed during the proof of 13.6.4, S, T is Sylow in each Gj , Ej , respectively, so it remains to show AutF (T ) ≤ Aut(Ej ) for each j ∈ J. As O(τ ) ≤ T , NF (T ) ≤ NF (W ), so as W  NF (W ), we have AutF (T ) ≤ Aut(W). From the proof of 13.6.4, AutF (T ) = AutSL0 (T ), so as g ∈ Z(SL0 ) for g ∈ {f, f c}, AutF (T ) centralizes g, and then as Ej  Gj , we have AutF (E) ≤ Aut(Ej ). This completes the proof in (a) and (b), so we may assume (c) holds. In (c) we must verify Hypothesis 1.4.3 in [Asc19], so it remains to check parts (2) and (3) of that hypothesis. The Sylow group of E2 is T , so the lifting condition 1.4.3.3 is trivial when j = 2. Further, as above, AutF (T ) ≤ Aut(E2 ). Next E1 = Yf and |S : CS (f )| = 2 = |T : CT (f )| with f t = f c for t ∈ T − CT (f ). Now, as above, A = AutF (CT (f )) = AutS (CT (f )) so each member of A lifts to cs ∈ AutF (T ) for some s ∈ S, and hence 1.4.3.3 is satisfied when j = 1. Also E1t = Yft = Yf c ≤ Y, so 1.4.3.2 holds by 1.4.6 in [Asc19]. This completes the proof.  Theorem 13.6.6. Assume Hypothesis 13.1.1 and F ◦ is transitive on Ω. Then one of the following holds: (1) ZS (τ ) = {z} is of order 1 and either F ◦ = O(z) or F ◦ ∼ = L2 [2m](1) . ◦ ∼ ˆ (2) m = 8, n = 2, and F = AE5 .

374

13. |Ω(z)| = 1 AND μ ABELIAN − (3) F ◦ ∼ = (P )Sp2n [m], (P )SL− 2n [m], or L2n+1 [m].

Proof. Recall n = |Ω|. If n = 1 then (1) or (3) holds by Theorem 7.1.7. Thus we may assume n > 1. Hence by 13.1.11, Z  = ∅. If F ◦ is the central product of k conjugates of some subnormal subsystem E = E ◦ then as F ◦ is transitive on Ω by hypothesis, it follows that k = 1. Therefore Z m = ∅ by 13.1.12. Hence Hypothesis 13.1.13 is satisfied. Then by Theorem 13.1.15, AutF (O) induces Sym(Ω) on Ω. Therefore Hypothesis 13.2.1 is satisfied. If 1 = c ∈ Z(F) then (2) or (3) holds by 13.2.13.2, so we may assume otherwise. Therefore Hypothesis 13.2.14 holds. Then 13.6.5 completes the proof.  Remark 13.6.7. Observe that Theorem 13.6.6 and Theorem 1 imply Theorem 7. Namely Theorem 1 implies the Extended Inductive Hypothesis , while the Hypotheses of Theorem 7 imply the remaining parts of Hypothesis 13.1.1.

CHAPTER 14

More generation Let τ = (F, Ω) be a quaternion fusion packet such that W ∈ W (τ ) is weakly closed in S with respect to F; for example by 3.3.14, if F is transitive on Ω then W is weakly closed unless possibly m = 8 and μ = μ(τ ) is abelian. In particular if μ is isomorphic to Weyl(Dn ) for some n ≥ 3 or to Weyl(En ) then W is weakly closed. Define B0 (τ ) = NF (W ), Fx : x ∈ F f ∩ W is an involution . In Chapter 14 we prove two theorems which insure that F = B0 (τ ) under suitable hypotheses. We begin in section 14.1 with some general results about B0 (τ ) and the structure and F-automizer of essential subgroups R such that AutF (R) = AutB0 (τ ) (R). Then in Hypothesis 14.1.11 we consider those τ such that |Ω(z)| = 1 for z ∈ ZS (τ ), there exists an involution u ∈ CW (S) with B0 (τ ) = Fu , and either μ ∼ = Weyl(Dn ) ¯ (E7 , m). We for some n ≥ 3 or μ ∼ = Weyl(E7 ) and M ∈ M (τ ) is isomorphic to ω prove in Theorem 14.1.18 that if Hypothesis 14.1.11 is satisfied then u ∈ Z(F), so that in particular F = B0 (τ ). Then in Theorem 14.2.4 we prove F = B0 (τ ) when μ ∼ = Weyl(E8 ).

14.1. A generation lemma In this section we assume the following hypothesis: Hypothesis 14.1.1. τ = (F, Ω) is a quaternion fusion packet such that W ∈ W (τ ) is weakly closed in S with respect to F. Definition 14.1.2. Set B0 = B0 (τ ) = NF (W ), Fx : x ∈ W ∩ F f is an involution . Lemma 14.1.3. (1) For each 1 = X ≤ W , NF (X) ≤ B0 . (2) E(τ ) ≤ B0 . Proof. Let 1 = X ≤ W . By 14.1.1, W is weakly closed in S, so NF (W ) controls fusion in W and for each α ∈ A(X), W α = W , so Xα ≤ W . As NF (W ) ≤ B0 , α is a B0 -map, so, replacing X by Xα, we may assume X ∈ F f . In particular if X is of order 2 then by definition of B0 , we have NF (X) ≤ B0 . Set X = NF (X). By 1.3.2 in [Asc10], X = NS (X)CF (X), NF (CS (X)) . But for x an involution in X ∩ Z(NS (X)), we have NS (X)CF (X) ≤ Fx ≤ B0 from the previous paragraph, and NF (CS (X)) ≤ NF (W ) ≤ B0 , completing the proof of (1). 375

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Recall from 3.4.2 that E(τ ) = SO(τ ), NF (W ), NF (O(θ)) : θ ⊆ ZS and O(θ) ∈ F f . But NF (W ) ≤ B0 and SO(τ ) ≤ NF (ZS ) ≤ B0 and NF (O(θ)) ≤ B0 by (1), so (2) holds.  Until the last result of the section, in the remainder of the section we assume: Hypothesis 14.1.4. F = B0 . Notation 14.1.5. By 14.1.4 and the Alperin-Goldschmidt Fusion Theorem, we have R = ∅, where R consists of those R ∈ F e with AutF (R) = AutB0 (R). Let R∗ consist of those members of R of maximal order. Pick R ∈ R∗ and let G be a model for NF (R). Set WO = Ω2 (K ∩ W ) : K ∈ Ω and W = W or WO . Set G∗ = G/R, WR = W ∩ R, WG = NW (R) and SG = NS (R). Define H = NG (X), NG (Y ) : 1 = X ≤ WG with R ≤ NS (X), R < Y ≤ SG . Lemma 14.1.6. (1) AutH (R) ≤ AutB0 (R), so H = G. (2) H ∗ is strongly embedded in G∗ . Proof. Let R < Y ≤ SG . By maximality of |R| and the Alperin-Goldschmidt Fusion Theorem, NF (Y ) ≤ B0 , so AutNG (Y ) (R) ≤ AutB0 (Y ). If 1 = X ≤ WG with R ≤ NS (X) then by 14.1.3.1, NF (X) ≤ B0 , so AutNG (X) (R) ≤ AutB0 (R). This proves (1). ∗ Let 1 = Y0∗ ≤ SG , and let Y be the preimage of Y0∗ in G. Then NG∗ (Y0∗ ) = ∗ ∗  NG∗ (Y ) = NG (Y ) ≤ H ∗ , and H ∗ = G∗ by (1), so (2) follows. Lemma 14.1.7. (1) WR = 1. (2) WR = WG , so WG∗ = 1. (3) For each X ≤ WG , [X, R] ≤ WR , so if WR ≤ X then R ≤ NS (X). (4) Φ(WG ) ≤ WR , so Φ(WG∗ ) = 1. Proof. As W  S, 1 = Z(S) ∩ W ≤ CW (R) ≤ R ∩ W = WR , so (1) holds. If WR = WG then W = WR ≤ R, so for K ∈ Ω, V = Ω2 (K ∩W ) ≤ R, contrary to 14.1.3.2 and 3.4.5. Thus (2) holds. Let X ≤ WG . Then as W  S, we have [X, R] ≤ W ∩ R = WR , so (3) holds. As W is abelian, WG is quadratic on R by (3), and hence WG is also quadratic ˜ = R/Φ(R). But O2 (G∗ ) = 1, so G∗ is faithful on R ˜ and hence as W ∗ is on R G ∗ ˜  quadratic on R we have Φ(WG ) = 1, proving (4). Lemma 14.1.8. Let g ∈ G with WGg ≤ H and set L = WG , WGg and B = WR WRg . (1) [L, R] ≤ B ≤ L so R ≤ NG (L). (2) WR ∩ WRg = 1, so B = WR × WRg . (3) Either (a) |WG∗ | = 2 and L∗ is dihedral, or (b) L∗ ∼ = L2 (2k ) or Sz(2k ) with 1 < m(WG∗ ) ≤ k. (4) We may choose g ∈ L. (5) For each y ∈ WG − WR , CB (y) = WR = [B, y] = [R, y]. (6) Φ(B) = 1. (7) m(WG∗ ) ≤ m(WR ).

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Proof. By 14.1.7.3, [WG , R] ≤ WR , so also [WGg , R] ≤ WRg and hence (1) holds. Set A = WR ∩ WRg and suppose A = 1. As WG and WGg are abelian, A ≤ Z(L). As WR and WRg are R-invariant, A is normal in R, so L ≤ NG (A) ≤ H, contradicting WGg ≤ H. Therefore (2) holds. By 14.1.6.2, H ∗ is strongly embedded in G∗ , so as L∗ ≤ H ∗ , HL∗ = H ∗ ∩ L∗ is strongly embedded in L∗ . Then (3) follows from the structure of groups with a strongly embedded subgroup. By (3) there is l ∈ L with WG∗l ≤ HL∗ , so, replacing g by l if necessary, (4) holds. Choose y as in (5) and suppose WR < CB (y). Then 1 = B0 = CB (y g ) ∩ WR is centralized by L0 = WG , y g , so B0 ≤ WR ∩WRi , where i ∈ L0 such that WG∗i , y g∗ is a 2-group. But this is contrary to (2) with i in the role of g. Thus CB (y) = WR . By 14.1.7.4, y 2 ∈ WR . By 14.1.7.3, WR = CB (y) = [B, y], so as B is abelian by (2), it follows that Φ(B) = 1, establishing (6). Then as m(CB (y)) = m(WR ) = m(B/WR ), also WR = [B, y], and then WR = [R, y] by 14.1.7.3. For b ∈ B − WR , WG∗ induces a group of transvections of WR b with axis WR , so (7) holds.  Lemma 14.1.9. Take W = W and suppose R acts on a subgroup U of W containing WR . Then CU/WR (R) = (U ∩ WG )/WR . Proof. Set V /WR = CU/WR (R). By 14.1.7.3, U ∩ WG ≤ V . Conversely [V, R] ≤ WR ≤ R, so V ≤ NW (R) = WG , so that V ≤ U ∩ WG .  Lemma 14.1.10. (1) CS (B) ≤ R. (2) CW (WRg ) = CW (B) = WR . Proof. Let S1 = CS (B) and S2 = NS1 (R). Then [S2 , L] ≤ CS (B)∩CS (R/B) ≤ R, so [S2∗ , L∗ ] = 1, Then as L∗ ≤ H ∗ , it follows from 14.1.6.2 that S2∗ = 1, so S2 ≤ R, Hence also S1 ≤ R, completing the proof of (1). As W is abelian and B = WR WRg , we have CW (B) = CW (WRg ). By (1),  CW (B) ≤ W ∩ R = WR , so (2) holds. In the remainder of the section we assume the following hypothesis: Hypothesis 14.1.11. (1) τ = (F, Ω) is a quaternion fusion packet such that either μ(τ ) ∼ ¯ (E7 , m). = Weyl(Dn ) for some n ≥ 3 or μ(τ ) ∼ = Weyl(E7 ) and M (τ ) ∼ =ω (2) |Ω(z)| = 1 for z ∈ ZS . (3) There exists an involution u ∈ CW (S) such that B0 (τ ) = Fu . Notation 14.1.12. Take W = W . Set Σ = AutF (B), WB = NW (B), and ΣW = AutWB (B). Lemma 14.1.13. (1) [ΣW , B] ≤ WR . (2) Φ(WB ) ≤ WR , so Φ(ΣW ) = 1. (3) For each 1 = φ ∈ ΣW , CB (φ) = WR . (4) m(ΣW ) ≤ m(WR ). Proof. First [WB , B] ≤ W ∩ B = WR , so (1) holds. As WB is quadratic on B, Φ(WB ) ≤ CW (B) = WR by 14.1.10.1, so (2) holds.

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Choose φ as in (3) and assume WR < B0 = CB (φ). Then for each conjugate ϕ of φ under Σ, 1 = W0 = CWR (ϕ), so by 14.1.3.1, ϕ ∈ AutB0 (B) = Σ0 . Therefore Ξ = φΣ ≤ Σ0 . Let θ = ΣW ∩ Ξ and U the preimage in WB of θ. Claim θ ∩ AutWG (B) = 1. For if not U ∩ WG = WR . But as Ξ  Σ, R acts on U and as φ ∈ θ, WR < U , so 14.1.9 supplies a contradiction. Let θ ≤ Γ ∈ Syl2 (Ξ); by a Frattini argument, Σ = ΞNΣ (Γ). By the claim and 14.1.8.5, X = CB (Γ) ≤ CB (θ) = WR , so NΣ (Γ) ≤ NΣ (X), and by 14.1.3.1, NΣ (X) ≤ AutB0 (B). Therefore Σ ≤ AutB0 (B). But by 14.1.11.3, B0 = Fu with u ∈ CW (S). Hence as Z(S) ≤ R, we have u ∈ WR , a contradiction as AutL (B) does not centralize any involution in WR . This contradiction completes the proof of (3). The proof of (4) is the same as that of 14.1.8.7.  Lemma 14.1.14. Assume W = WB and let k = m(WR ). Then (1) m(ΣW ) = k. (2) WR = Ω1 (WB ), so WB ∼ = Zk4 . (3) WR = Ω1 (W ). Proof. Set A = WB B and let x ∈ NW (A) − WB . As WB = NW (B), B = B x . As x centralizes WB , WR ≤ B ∩ B x . But by 14.1.13.3, for a ∈ A − B, CB (a) = WR , so for y ∈ B x − B, WR = CB (y). Therefore as B ∩ B x ≤ CB (y) it follows that WR = B ∩ B x . Therefore m(A/B) ≥ m(B x /WR ) = k, while by 14.1.13.4, m(A/B) ≤ k, so (1) holds and A = BB x . Then as CA (b) = B for b ∈ B − WR , it follows that B and B x are the maximal elementary abelian subgroups of A, so in particular (2) holds. Then (2) and 14.1.9, applied to Ω1 (W ) in the role of U , imply (3).  Lemma 14.1.15. WR = Ω1 (W ). Proof. Assume WR = Ω1 (W ) and set A = WRg . Then A centralizes Ω1 (W ), so for each V ∈ η, A centralizes z(V ) and hence A acts on V as |Ω(z(V ))| = 1 by 14.1.11.1. Set V1 = Ω2 (V ); then [A, V1 ] ≤ z(V ) ≤ Ω1 (W ) = WR ≤ B, so V1 ≤ WB . But |A : CA (V1 )| ≤ 2, so in particular CA (V1 ) = 1, contrary to 14.1.13.3.  Lemma 14.1.16. W = WB . Proof. This follows from 14.1.14.3 and 14.1.15.



Lemma 14.1.17. (1) m = 8. (2) W is not homocyclic. (3) μ(τ ) ∼ = Weyl(Dn ) with n ≥ 6 even. (4) WR = Ω1 (W ). Proof. By 14.1.8.6, Φ(WR ) = 1 while by 14.1.16 and 14.1.13.2, Φ(W ) ≤ WR , so W is of exponent 4. Thus (1) holds. If W is homocyclic then Ω1 (W ) = Φ(W ) ≤ WR ≤ Ω1 (W ), so WR = Ω1 (W ), contrary to 14.1.15. Therefore (2) holds. ¯ be the universal group of type Dn or E7 defined in 4.3.1. By 4.3.8 there Let M ¯ → M with Z¯ = ker(π) ≤ Z(M ¯ ). By 4.3.7.2, is a surjective homomorphism π : M ¯ is of type Dn by 14.1.11.3. ¯ ∼ W = Zn4 , so Z¯ = 1 by (2). Hence M As |Ω(z)| = 1, Z¯ = 1 if n ≤ 4, so n ≥ 5. Therefore by the proof of 15.1.6, ¯ But if n is odd then Z(M ¯ ) is ¯0 ∈ / Z. u0 = u(τ ) exists in Z(M ). In particular u

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∼ E4 , ¯) = cyclic, so Z¯ = 1. Therefore n is even, completing the poof of (3). Also Z(M so Z¯ ∼ = Z2 . We’ve shown that Z¯ ∼ × Z2 . Therefore Φ(W ) ∼ = Z2 , so W ∼ = Zn−1 = E2n−1 and 4 W/Φ(W ) ∼ = E2n , so as Φ(W ) ≤ WR , (4) follows from 14.1.13.4, completing the proof.  Theorem 14.1.18. Assume Hypothesis 14.1.11. Then u ∈ Z(F). Proof. Assume otherwise. By 14.1.11.3, B0 = Fu , so Hypothesis 14.1.4 is satisfied. Observe however that 14.1.15 and 14.1.17.4 now supply a contradiction, completing the proof of the theorem. 

14.2. A generation lemma for E8 In this section we assume the following hypothesis: Hypothesis 14.2.1. τ = (F, Ω) is a quaternion fusion packet such that μ(τ ) ∼ = Weyl(E8 ). Notation 14.2.2. Let W ∈ W (τ ) and M ∈ M (τ ). As in 14.1.5, set WO = Ω2 (K ∩ W ) : K ∈ Ω . Set Θ = ZS and for t ∈ Θ let Kt ∈ Ω(t) and Vt = vt = Ω2 (Kt ∩ W ). As μ(τ ) ∼ = Weyl(E8 ), we have Ω(t) = {Kt }. Set nS = |Θ|. Lemma 14.2.3. (1) nS = 8. (2) WS ∼ = E16 . (3) There is a hyperplane U of WS such that Θ = WS − U . (4) WO ∼ = Z44 × E16 , Φ(WO ) = WS , and Ω1 (W ) = Ω1 (WO ) = WΔ . Proof. See 15.8.6 for parts (1)-(3). By definition of WO , we have WS = Φ(WO ) and WO /WS ∼ = E2nS , so as WO is abelian it follows from (1) and (2) that 4 Z × E . In particular m(WO ) = 8 = m(W ), so Ω1 (WO ) = Ω1 (W ) = WΔ , WO ∼ = 4 16 completing the proof of (4).  Define B0 = B0 (τ ) as in Definition 14.1.2. The main result of this section is: Theorem 14.2.4. F = B0 (τ ). Assume that F = B0 . Then Hypothesis 14.1.4 is satisfied, and we adopt Notation 14.1.5 with W = WO . Write Z0 for the set of elementary abelian normal subgroups Z of G such that Z ≤ Z(R), and set Z = {Z ∈ Z0 : WO ∩ Z = 1}. Lemma 14.2.5. Θ ∩ Z(R) = ∅. Proof. See 14.1.3.2 and 3.4.6.



Lemma 14.2.6. (1) If Z ∈ Z0 with [WG , Z] = 1 then Z ∈ Z. (2) For each Z ∈ Z, R = CS (Z). Proof. By 14.1.7.3, [WG , Z] ≤ W0 , so (1) follows. Assume Z ∈ Z. From the  proof of 11.3.5.1, O 2 (CG (Z)) = R. Then CS (Z) = R by 3.4.4.1. 

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Lemma 14.2.7. For each Z ∈ Z, we have |Z| > 4. Proof. Assume otherwise. By 14.1.3.1, Z ≤ WO , so there is s ∈ Z − WR . As |Z| = 4, Z = w, s with w = WR ∩ Z, so as WO is abelian, CWO (s) = CWO (Z). By 14.2.6.2, CWO (s) = CWO (Z) ≤ R. For t ∈ Θ, v = vt vts ∈ CWO (s) ≤ WR , so v 2 = 1 by 14.1.8.6. Therefore as Ω(t) = {Kt } if follows that s centralizes t and inverts vt . Therefore WS ≤ CWO (s) ≤ R and s inverts WO . Hence s centralizes Ω1 (WO ) = WΔ by 14.2.3.4, so WR = WΔ by 14.1.8.6. Let y ∈ WG − WR . Then s inverts y of order 4, so [y, s] = y 2 = w. Recall L is defined in 14.1.8 and observe L∗ = GL(Z) ∼ = S3 . By 14.1.8, O2 (L) = L ∩ R = B = WR × WRg . Let h ∈ L be of order 3. Set Wt = tL . As CB (h) = 1, E4 ∼ = Wt = t, th . As B is abelian, th acts on Va for each a ∈ Θ. As WR = WΔ we can pick y = v1 · · · v4 with vi = vzi for some θ = {z1 , . . . , z4 } ⊆ Θ independent in WS and with z1 = t and z2 = a. Now   t = [th , y] = [th , v1 · · · v4 ] = [th , vi ] = zii , i

i

h

so as θ is independent in WS it follows that t inverts vt and centralizes vi for i > 1. In particular th centralizes va for each a ∈ Θ − {t}, so as WO = va : a ∈ Θ − {t} ,  it follows that th centralizes WO . This is a contradiction as th inverts vt . Lemma 14.2.8. Let Z ∈ Z0 and x ∈ WG . (1) x does not induce a transvection on Z. (2) If |[R, x]| ≤ 2 then x ∈ R. (3) If Y ≤ WO is R-invariant of order 4 then Y ≤ R. (4) |U ∩ R| ≥ 4. Proof. Assume (1) fails. Then by 2.4.1 in [Asc19] there is Z1 ∈ Z with Z1 ≤ Z, m(Z1 ) = 2, and x induces a transvection on Z1 . Now 14.2.7 supply a contradiction. Assume the setup of (2) but x ∈ / R. Let w generate [R, x] and set Zw = wG . As [R, x]  R, w ∈ Z(R), so Zw ∈ Z. But x induces a transvection on Zw , contrary to (1), establishing (2). Part (3) follows from (2), and (4) follows from (3).  Lemma 14.2.9. If R is transitive on Θ then WS ≤ R. Proof. Assume otherwise. Let f ∈ CU (R) and Z = f G , so that Z ∈ Z. By 14.2.6.2, R = CS (Z). As WS ≤ R = CS (Z), for each t ∈ Θ, Z acts on Vt . Then |Z : CZ (Vt )| = 2 as Vt ≤ R by 14.1.8.6. As R is transitive on Θ and Z ≤ Z(R), we have CZ (Vt ) = CZ (Va ) for each a ∈ Θ, so CZ (Vt ) centralizes WO , and for s ∈ Z − CZ (Vt ), s inverts each Va . Therefore s inverts WO , so WΔ ≤ CS (s, CZ (Vt ) ) = CS (Z) = R. Hence WΔ = WR by 14.1.8.6. Pick y ∈ WG −WR ; thus y ∈ W0 . Then y centralizes CZ (Vt ) and [y, s] = y 2 = u, so y induces a transvection on Z, contrary to 14.2.8.1.  Lemma 14.2.10. The orbits of R on Θ are of length 4 or 8. Proof. Assume θ = tR is an orbit of length less than 4. By 14.2.5, |Θ| = 1, so θ = {t1 , t2 } is of order 2. By 14.2.8.3, X = θ ≤ R. Let vi = vti , y = v1 v2 , and u = t1 t2 . As [R, y] ≤ X ≤ R, y ∈ WG and u ∈ Z(R), so Z = uG ∈ Z and hence / R by 14.1.8.6, so [Z, y] = 1. But ti ∈ / Z by R = CS (Z) by 14.2.6.2. As |y| = 4, y ∈ 14.2.5, so [Z, y] = u , contrary to 14.2.8.1. 

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Notation 14.2.11. Let T be the set of subsets θ of Θ of order 4. Observe from 14.2.3 that either θ is independent and T = θ = WS , or T ∼ = E8 and θ = T − (T ∩ U ). In the first case we say θ is of the first kind, and in the second case we say θ is of the second kind. Write Ti for the members of T of the ith kind. Lemma 14.2.12. If θ is an orbits of R on Θ of length 4 then θ ∈ T2 .  Proof. Assume θ ∈ T1 and set u = t∈θ t; then u ∈ CWS (R) ≤ Z(R) is an involution. Set Z = uG , and observe Z ∈ Z, so R = CS (Z) by 14.2.6.2. Suppose first that Z = Zθ is the pointwise stabilizer in Z of θ. Then WS ≤ / R by 14.1.8.6, so |Z : CZ (vt )| = 2. But as R is CS (Z) = R. For t ∈ θ, vt ∈ transitive on θ, CZ (vt ) = CZ (va ) for each a ∈ θ. Thus s ∈ Z − CZ (vt ) inverts each va and hence CZ (vt ) centralizes WO = v a : a ∈ θ while s inverts WO . Therefore, as usual, WR = WΔ . Also s inverts y = t∈θ vt of order 4, and [R, y] ≤ WS ≤ R, so y ∈ WG − WR . But now y induces a transvection on Z with axis CZ (vt ), contrary to 14.2.8.1. Therefore Z = Zθ . Suppose next that Z θ is of order 2 and let s ∈ Z − Zθ . Then s = (t, ts )(a, as ) ∈ Sym(Θ) as R is transitive on θ and Z ≤ Z(R). Therefore s centralizes x = tts , so x centralizes Z = Zθ s and hence x ∈ R. Set y = ta; then U = y, x, u with UR = x, u ≤ R, so [R, y] ≤ UR ≤ R and hence y ∈ WG . But [s, y] = u = 1 and y centralizes the hyperplane Zθ of Z, contrary to 14.2.8.1. Therefore E4 ∼ = Z θ ≤ Z(Rθ ) is regular on θ, so CU (Z) = u . This is a contradiction as Z ≤ Z(R) and |U ∩ R| > 2 by 14.2.8.4.  Lemma 14.2.13. Suppose θ is an orbit of R on Θ of length 4. Then (1) P = θ ≤ R. (2) WS ≤ R. Proof. By 14.2.10, θ  = Θ − θ is also an orbit of R, so if (1) holds then also θ ⊆ R, and hence (2) holds. So we may assume P ≤ R and it remains to derive a contradiction.  By 14.2.12, θ ∈ T2 . Thus σ = t∈θ t = 1. As R is transitive on θ and P ≤ R, t ∈ θ is not in R. By 14.2.8.3, UP = U ∩ P ≤ CU (Z) = U ∩ R. Let 1 = u ∈ CU (R) and Z = uG , so that Z ∈ Z. As R acts on P , [R, t] ≤ UP ≤ R, so t ∈ WG . By 14.2.8.1, |[Z, t]| > 2, so as [R, t] ≤ UP , we conclude that UP = [Z, t] ≤ Z. Therefore Z is transitive on θ = P − UP , so E4 ∼ = Z θ is regular on θ.  Let D be a complement to Zθ in Z and y = d∈D vtd . As Z θ is regular on θ and Z ≤ Z(R), we have R = D × Rθ . Also y ∈ CWO (D) is an involution centralizing Rθ as D ≤ Z(R) is transitive on θ and σ = 1. Therefore y ∈ CS (R) = Z(R). Let x = vt vtd for some d ∈ D# . Then [x, D] = y and [Rθ , x] ≤ ttd ≤ UP ≤ R, so / Z then x induces a transvection on Z, contrary to 14.2.8.1. x ∈ WG . But if y ∈ Therefore [Z, x] = y, ttd . Now L acts on Z ∩ B ≥ Z0 = UP , y ∼ = E8 , so by  14.1.8.5, 2 = m([Z, x] ≥ m(Z0 ) = 3, a contradiction. 

Lemma 14.2.14. R is transitive on Θ. Proof. Assume otherwise. By 14.2.10 and 14.2.12, an orbit θ for R on Θ is in T2 . By 14.2.13.2, WS ≤ R. We argue as in the proof of 14.2.9. Let Z ∈ Z. For t, a ∈ θ, CZ (Vt ) = CZ (Va ) is a hyperplane of Z and s ∈ Z − CZ (Vt ) inverts Y = Va : a ∈ θ . Thus Y acts on R. By 14.1.8.6, Ω1 (Y ) = Y ∩R, so there is y ∈ WG − R of order 4. Then y centralizes CZ (Vt ), contrary to 14.2.8.1. 

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Lemma 14.2.15. Θ ∩ R = ∅. Proof. By 14.2.14, R is transitive on Θ, so if t ∈ Θ ∩ R then Θ ⊆ R, contrary to 14.2.9.  Pick Z ∈ Z. Lemma 14.2.16. Z Θ = 1. Proof. If so Z = ZΘ , so WS ≤ CS (Z) = R by 14.2.6.2, contrary to 14.2.15.  Notation 14.2.17. Let 1 = γ ∈ Z Θ . By 14.2.14, R is transitive on Θ, so as Z ≤ Z(RΘ ), we have Z Θ semiregular on Θ. In particular γ = (t1 , s1 ) · · · (t4 , s4 ) is a fixed-point-free involution in Sym(Θ). Set ui = ti si and Uγ = CU (γ). Then or 2. In the first case we say γ is Uγ ≥ u1 , . . . , u4 . As m(U ) = 3, Uγ is of rank 3  of type 1 and in the second of type 2. Set σ = i ui and observe that σ = 1. If γ is of type 2 then ui = uj for some distinct i, j, so we may choose notation such that u1 = u2 . Then as σ = 1, also u3 = u4 , but as m(Uγ ) = 2, we have u1 = u3 . Θ

Lemma 14.2.18. Z Θ is of order 4 or 8. Proof. From 14.2.17, Z Θ is semiregular on Θ, so |Z Θ | divides |Θ| = 8. Thus we may assume Z Θ = γ is of order 2. Suppose γ is of type 1. Then U = Uγ ≤ CS (Z) = R. Hence for t ∈ Θ, [R, t] ≤ U ≤ R, so t ∈ WG , but t ∈ / R by 14.2.15. However t centralizes the hyperplane ZΘ of Z, contrary to 14.2.8.1. Therefore γ is of type 2. Set y = t1 t3 , so that y ∈ U − Uγ and [y, R] ≤ Uγ ≤ R,  so y ∈ WG . But [Z, y] = [γ, y] = u1 u3 , contrary to 14.2.8.1. Lemma 14.2.19. Z Θ is of order 4. Proof. Assume otherwise; by 14.2.18, Z Θ is regular on Θ, so R = RΘ × D, where D is a complement to ZΘ in Z. Now RΘ centralizes WS , and hence for each t ∈ Θ, RΘ acts on Vt . Hence X = CRΘ (Vt ) is of index at most 2 in RΘ , and as Z is transitive on Θ, we have X = CRΘ (Va ) for each a ∈ Θ, so that X centralizes WO = Va : a ∈ Θ . Similarly if r ∈ RΘ − X then r inverts WO , so RΘ centralizes WΔ .   For F ≤ D set vF = f ∈F vtf and wF = f ∈F tf . Observe F centralizes vF and wF with vF2 = wF . In particular D centralizes vD ∈ WO and wD = 1 so vD ∈ WΔ . Therefore vD ∈ CS (ZRΘ ) = Z(R). Let E be a hyperplane of D. Then E centralizes wE and for d ∈ D − E, [wE , d] = wD = 1, so wE ∈ CWΔ (D) = Z(R). Similarly E centralizes vE and [vE , d] = vD ∈ R. Also X centralizes vE and if r ∈ Rθ − X then [r, vE ] = wE ∈ R. Therefore vE ∈ WG . But [R, vE ] = vD , wE , so by 14.2.8.1 we have wE = 1 and [Z, vE ] = [R, vE ]. As this holds for each hyperplane E of D, it follows that ∗ : |D : E| = 2} is of order 7, so |V| ≥ 8. This contradicts 14.1.8.7 which V = {vE says m(WG∗ ) ≤ m([R, w∗ ]) for 1 = w∗ ∈ WG∗ .  Notation 14.2.20. By 14.2.18 and 14.2.19, Z has two orbits θ1 and θ2 on Θ, each of length 4. As usual let D be a complement to ZΘ in Z, pick t ∈ Θ, and for F ≤ D define vF and wF as in the proof of 14.2.19. Observe F centralizes vF and wF .

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Lemma 14.2.21. θi ∈ T2 for i = 1, 2. Proof. Assume otherwise; then θi ∈ T1 for i = 1, 2. But now CU (D) = wD is of order 2, so as Z ≤ Z(R) we conclude that U ∩R = wD , contrary to 14.2.8.4.  Lemma 14.2.22. (1) wD = 1. (2) vD ∈ R. (3) Set Pi = θi . Then UD = U ∩ P1 = U ∩ P2 ≤ R. Proof. By 14.2.21, θi ∈ T2 . Therefore wD = 1, so (1) holds. Further D centralizes vD and as D is transitive on θi , [vD , ZΘ ] ≤ wD = 1. Hence vD ∈ CS (Z) = R, so (2) holds. Calculating using 14.2.3, UD = U ∩ P1 = U ∩ P2 and D centralizes UD . Thus as ZΘ centralizes WS , UD ≤ CS (DZΘ ) = CS (Z) = R, proving (3).  Lemma 14.2.23. UD = U ∩ R. Proof. Assume otherwise; then U ≤ R by 14.2.22.3. Thus [R, t] ≤ U ≤ R, so t ∈ WG . Also vD ∈ R by 14.2.22.2, so X = U vD ≤ WR with m(X) = 4. This is  contrary to 14.1.8.5 which says m(WR ) = m([R, t]). We are now in a position to obtain a contradiction, and hence establish Theorem 14.2.4. Let u ∈ U −UD ; then [R, u] ≤ UD ≤ R by 14.2.22.3, so u ∈ WG . By 14.2.23, u∈ / R. But X = UD vD ≤ WR by 14.2.22.2, and m(X) = 3. This is contrary to 14.1.8.5 which says m([R, u]) = m(WR ). Theorem 14.2.24. Assume Hypothesis 14.2.1. Then F = Fz , Fu , NF (W ) , where z ∈ ZS and u ∈ WS ∩ F f is not in ZS . Proof. By 15.8.5 and 15.8.6, WΔ is the orthogonal space for μ ˜ = μ/Z(μ) ∼ = O8+ (2). In particular M has two orbits on involutions in W : the singular and nonsingular points, with z nonsingular and u ∈ WS − ZS singular. Hence the theorem follows from Theorem 14.2.4. 

CHAPTER 15

|Ω(z)| = 1 and μ nonabelian Let τ = (F, Ω) be a quaternion fusion packet. In Chapter 15 we assume |Ω(z)| = 1 for z ∈ ZS (τ ) and μ = μ(τ ) is nonabelian. In addition in Hypothesis 15.1.1 we add the assumptions that F is transitive on Ω and τ satisfies the Inductive Hypothesis. Later in Hypothesis 15.2.1 we strengthen this last condition and require that τ satisfy the Extended Inductive Hypothesis. Section 15.1 is devoted to general results about τ under Hypothesis 15.1.1. In section 15.2 we assume Hypothesis 15.2.1 with F = F ◦ , and prove that μ is isomorphic to Weyl(Φ), where Φ is An , Dn , or En . In Remark 15.2.30 we observe that τ is determined when Φ is A2 , the smallest case. Then we go on to determine τ when Hypothesis 15.2.1 holds with F = F ◦ for the various possibilities for Φ. Section 15.3 treats the case where Φ is Dn for n = 3 or n > 4. In section 15.4 we determine τ when Φ is D4 . In section 15.5 we assume Φ is An with n ≥ 4 and either Hypothesis 15.2.1 holds or O 2 (F) = F ◦ is the 2-fusion system of a central factor group of SLd (q). Various preliminary results are established, leading to the generational result in Theorem 15.6.3. Then the case Φ = An is completed in Theorem 15.7.7. Finally the case Φ = En is handled in Theorem 15.8.16.

15.1. |Ω(z)| = 1 In this section we assume the following hypothesis:

Hypothesis 15.1.1. (1) τ = (F, Ω) is a quaternion fusion packet such that for all z ∈ ZS , |Ω(z)| = 1. (2) F is transitive on Ω. (3) For z ∈ ZS ∩ F f , D∗ (z) = D(z). (4) The Inductive Hypothesis is satisfied by τ . Lemma 15.1.2. There exists a unique η ∈ η(τ ), so W = η is the unique member of W (τ ) and W is weakly closed in S with respect to F. Proof. By 15.1.1.2, F is transitive on Ω, while by 15.1.1.3, Δ(τ ) = Ω, so the lemma follows from 3.3.14.  Notation 15.1.3. Let G be a model for NF (CS (W )) and M = ΩG . Set μ = μ(τ ), so that μ = M/CM (W ). Let K ∈ Ω such that z = z(K) ∈ F f . Set D = {dV : V ∈ η} ⊆ μ. 385

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Lemma 15.1.4. (1) Let {ηi : 1 ≤ i ≤ r} be the orbits of M on η, Ωi = {J ∈ Ω : J ∩ W ∈ ηi }, Mi = JiM for Ji ∈ Ωi , Wi = ηi , and Di = {dV : V ∈ ηi }. Then D is a set of 3-transpositions of μ, Di is a conjugacy class of 3-transpositions of μi = AutMi (W ), and M = M1 · · · Mr is a central product. (2) W = CM (W ) and Wi = Mi ∩ W = CMi (Wi ), so μ = M/W , μi = Mi /Wi , and μ = μ1 × · · · × μr . (3) NG (O(τ )) is transitive on Ω and {Ω1 , . . . , Ωr }, and G is transitive on {M1 , . . . , Mr }. (4) For 1 ≤ i ≤ r, τi = (FS∩Mi (Mi ), Ωi ) is a quaternion fusion packet with μi = μ(τi ). (5) For each 1 ≤ i ≤ r, μi ∼ = Weyl(Φ), where Φ is An , Dn , or En . Proof. By 15.1.1.1, Ω(z) = {K} is of order 1 for each z ∈ ZS , so A(τ ) = ∅. Hence D is a set of 3-transpositions of μ by 4.2.10.3. Next M = M1 ∗ · · · ∗ Mr is a central product by 4.2.10.2, so as M is transitive on ηi , so is Mi . Then Di is a conjugacy class of 3-transpositions of μi by 4.2.8.1. Thus (1) is established. Set O = O(τ ). By 15.1.1.2, F is transitive on Ω, so NG (O) is transitive on Ω by 6.1.6.2. Then (3) follows as M  G, so G permutes the orbits of M on η. We’ve seen that M = M1 ∗ · · · ∗ Mr , so Mi  M and hence Si = S ∩ Mi ∈ Syl2 (Mi ) and Mi has 2-fusion system Fi = FSi (Mi ). Also Ωi is Fi -invariant, so by 2.5.2, τi = (Fi , Ωi ) is a quaternion fusion packet. As Mi is transitive on ηi , Fi is transitive on Ωi . Then by 4.3.8.1, νi = μ(τi ) ∼ = Weyl(Φ) for some Φ ∈ {An , Dn , En }. By 4.3.8.4, Wi = CMi (Wi ). Therefore νi = Mi /Wi . Then as in addition, M = M1 ∗ · · · ∗ Mr , also W = CM (W ), so μ = M/W and Wi = W ∩ Mi , so μi = AutMi (W ) = Mi W/W ∼ = Mi /Wi = νi , and μ = μ1 × · · · × μr . That is (2), (4), and (5) hold.  Lemma 15.1.5. Let ρ = (Y, Γ) be a subpacket of τ with Γ ⊆ Ω. (1) ρ◦ = ρ1 ∗· · ·∗ρs is a central product of quaternion fusion packets ρi = (Yi , Γi ) such that Yi = Yi◦ is transitive on Γi . Set ni = |Γi |. (2) For each J ∈ Γi , Γi (z(J)) = {J}. (3) If μ(ρi ) is abelian then one of the following holds: (i) ni = 1 and either Yi = O(zi ) for {zi } = Z(ρi ), or Yi ∼ = L2 [2m](1) . ˆ 5. (ii) m = 8, ni = 2, and Yi ∼ = AE − (iii) Yi is (P )Sp2ni [m], (P )SL− 2ni [m], or L2ni +1 [m]. (4) Assume ρi is proper in τ and μ(ρi ) is nonabelian. Then one of the following holds: (i) W (ρi )  Yi . (ii) μ(ρi ) ∼ = Weyl(Dn ). for some n ≥ 3. (iii) μ(ρi ) ∼ = E6+ [m]. = Weyl(E6 ) and Yi ∼ ˜7 [m]. (iv) μ(ρi ) ∼ = Weyl(E7 ) and Yi ∼ = E7 [m] or E ∼ ∼ (v) μ(ρi ) = Weyl(E8 ) and Yi = E8 [m]. + (1) (vi) μ(ρi ) ∼ , G2 [m], or M12 , or m = 8 and = S3 and Yi ∼ = L+ 3 [m], L3 [m] ∼ Yi = L3 (2)/E8 . (vii) μ(ρi ) ∼ = Sn for some n ≥ 5 and Yi is a central factor system of SL+ n [m]. Proof. Part (1) follows from 15.1.1.4 and 6.6.6. Part (2) follows as the condition |Ω(z)| = 1 in 15.1.1.1 inherits to subpackets (E, Θ) of τ with Θ ⊆ Ω. If μ is abelian then ρi is proper and hence appears as a conclusion in Theorem 1 with an abelian Thompson group by 15.1.1.4. Then (3) follows. (cf. Theorem 7) Thus

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it remains to prove (4), so we may assume ρ = ρi is proper in τ and ν = μ(ρ) is nonabelian. Now ρ satisfies Hypothesis 15.1.1 with Y = Y ◦ , so ρ satisfies the hypothesis of Theorem 1, and hence also satisfies one of its conclusions by 15.1.1.4. Then (4) follows (cf. 15.2.29, Remark 15.2.30, Theorem 15.8.16, and Theorem 15.7.8 for details).  Lemma 15.1.6. Assume μj ∼ = W eyl(Dn ). (1) If n = 4 there exists a unique involution uj = u(τj ) ∈ Zj = Z(Mj ) such that for each t ∈ ZS (τj ), t = tuj ∈ ZS (τj ). (2) If n = 4 then Zj ∼ = E4 and for each t ∈ ZS (τj ), tZj = ZS (τj ). ¯ (Dn , m). (3) If n is odd then Zj ∼ = Z4 and Mj ∼ =ω ¯ be the universal group ω Proof. Let G ¯ (Dn , m) defined in 4.3.8.1. By 4.3.8.2 ¯ π = Wj , K ¯ i π = Ki ∈ Ω j , ¯ → Mj with W there is a surjective homomorphism π : G ¯ by 4.3.8.3. Set z¯i = z(K ¯ i ) and zi = z¯i π. Set u and ker(π) ≤ Z(G) ¯j = z¯n−1 z¯n and ¯ As Ω(zi ) = {Ki }, u uj = u ¯j π. By 5.8.2, u ¯j ∈ Z(G). ¯j ∈ / ker(π) by 5.8.5, so uj is an involution in Zj . Also zn uj = zn−1 ∈ ZS (τj ), so as uj ∈ Zj and FS∩Mj (Mj ) is transitive on Ωj , it follows that for each t ∈ ZS (τj ) we have t = tuj ∈ ZS (τj ). ¯ ∼ By 5.8.4, Z(G) = Z4 or E4 for n odd or even, respectively. In particular if n ¯ so as u ¯j ∈ / ker(π), (3) holds. Thus is odd then u ¯j is the unique involution in Z(G), we may assume n is even. Suppose u ∈ Zj −uj with tu ∈ ZS (τj ) for each t ∈ ZS (τj ). Set Mj+ = Mj /Zj . Then |Ω(t+ )| > 2, so by 10.2.13, n = 4. This completes the proof of (1). Part (2) follows from 5.8.7.1.  Lemma 15.1.7. Assume ρ = (Y, Γ) is a proper subpacket of τ with Γ ⊆ Ω such that Y = Y ◦ is transitive on Γ and ν = μ(ρ) is nonabelian. Let J ∈ Γ. Then (1) ν ∼ = Weyl(Σ) for some Σ ∈ {Ak , Dk , Ek }. (2) J ∈ Ωj for some 1 ≤ j ≤ r, Γ ⊆ Ωj , η(ρ) ⊆ ηj , Z(ρ) ⊆ Z(τj ), W (ρ) ≤ Wj , M (ρ) ≤ Mj , and ν is a Dj -subgroup of μj . Proof. As Y = Y ◦ is transitive on Γ, it follows from 15.1.5.1 that ρ is coconnected. Hence by 15.1.5.4 either (1) holds or W (ρ)  Y. In the latter case, (1) follows from 4.3.8.1 and the hypothesis that Y = Y ◦ is transitive on Γ. Next Γ ⊆ Ω by hypothesis, and then η(ρ) ⊆ η, so W (ρ) ≤ W , M (ρ) ≤ M , and ν is a D-subgroup of μ. Let E be the set of 3-transpositions generating ν. By (1), D(E)c is connected,  so as J ∈ Ωj , (2) follows from 15.1.4.1. Lemma 15.1.8. Assume ρ = (Y, Γ) is a proper subpacket of τ with Γ ⊆ Ω such that Y = Y ◦ is transitive on Γ and ν = μ(ρ) is nonabelian. Then (1) For some 1 ≤ j ≤ r, Γ ⊆ Ωj and ν is a Dj -subgroup of μj . (2) If μj ∼ = Sn+1 then ν ∼ = Sk+1 for some k ≤ n. = Weyl(An ) ∼ (3) If μj ∼ = Weyl(Ek ) for some 6 ≤ k ≤ n; or Weyl(Al ) or = Weyl(En ) then ν ∼ Weyl(Dl ) for some l < n; or Weyl(D8 ) if n = 8. (4) If μj ∼ = Weyl(Dk ) or Sk for some k ≤ n. = Weyl(Dn ) then ν ∼ (5) Suppose ν ∼ = Weyl(Dn ). Set ζ = = Weyl(Dk ) for some k ≥ 4 and μj ∼ (NY (W (ρ))◦ , Γ). Then one of the following holds: (i) 5 ≤ k ≤ n and uj = u(ζ). (ii) k = 4, n ≥ 5, and uj ∈ Z(M (ρ)). (iii) k = n = 4 and M (ρ) = Mj .

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Proof. Part (1) follows from 15.1.7.2. Let E be the set of 3-transpositions generating ν. By 15.1.7.1, ν ∼ = Weyl(Σ), where Σ ∈ {Ak , Dk , Ek }. In particular D(E)c is connected. Suppose μj ∼ = Sn+1 . As ν is a Dj -subgroup of μj , ν is generated by a class E of transpositions, so as D(E)c is connected, ν ∼ = Sk+1 with k ≤ n, proving (2). Suppose μj ∼ = Weyl(En ). Then as ν is a subreflection group of μj , (3) follows. Similarly (4) follows. Assume the setup of (5). By (4), 4 ≤ k ≤ n. If n = 4 then as ν ≤ μj , (5iii) holds, so we may assume n ≥ 5. Set Q = O2 (μj ), Z = Z(μj ), W0 = ZS (τj ) , and + ∼ + + μ+ j = μj /Q, so that μj = Sn and Dj is the set of transpositions in μj . Thus, as in the proof of (2), ν + ∼ = Sl for some l. In particular if k > 4 then O2 (ν) = ν ∩ Q. But by 5.8.3.2 and 15.1.6, Q/Z is a group of transvections on W0 with center uj , so u(ζ) = [O2 (ν), W0 (ρ)] ≤ [Q, W0 ] = uj , so u(ζ) = uj and (5i) holds. Therefore we may take k = 4. Here as μ+ ∼ = Sl for some l, we have l = 3 or 4, so P = O2 (ν) ∩ Q is of order 32 or 8, and uj = [P, W0 (ρ)] ≤ Z(M (ρ)), so (5ii) holds. This completes the proof of (5).  Lemma 15.1.9. Assume ρ = (Y, Γ) is a proper subpacket of τ with Γ ⊆ Ω such that Y = Y ◦ is transitive on Γ and ν = μ(ρ) ∼ = Weyl(Dk ) for some k ≥ 3. Then (1) Either (i) there exists an involution u ∈ Z(Y) such that for each t ∈ ZS (ρ), t = tu ∈ ZS (ρ), or (ii) k = 4 and Y ∼ = F4 [m] or E6− [m]. (2) If k = 4 or 1 = Z(Y) is cyclic, then u = u(ρ) is the unique such involution and one of the following holds: (i) W (ρ)  Y. + − (ii) Y is Spin+ 2k [m], HSpin2k [m], Spin2k+1 [m], or Spin2k+2 [m], with k = 4 in the first case. ˆ 2k , H AE ˆ 2k , or AE ˆ 2k+1 , with k = 4 in the first case. (iii) m = 8 and Y ∼ = H AE ∼ ˆ 6 (2). ˆ (iv) (n, m) = (3, 8) and Y = L3 (2)/E64 or Sp (3) If k = 4 and Z(Y) is noncyclic, then Z(Y) = Z(M (ρ)) = Z ∼ = E4 and each u ∈ U # satisfies (1i). Moreover one of the following holds: (i) W (ρ)  Y. (ii) Y ∼ = Spin+ 8 [m]. ˆ 8, L ˆ 3 (2)/23+6 , or O ˆ + (2). (iii) m = 8 and Y ∼ = AE 8 (4) Γ ⊆ Ωj for some 1 ≤ j ≤ r. Suppose k ≥ 4 and μj ∼ = Weyl(Dn ). Then one of the following holds: (i) 5 ≤ k ≤ n and uj = u(ρ). (ii) k = 4, n ≥ 5, and uj ∈ Z(Y). ˆ / Z(Y), and Y ∼ (iii) k = 4, n ≥ 5, uj ∈ = Spin9 [m], Spin− 10 [m] or AE9 . (iv) k = n = 4, M (ρ) = Mj , and Z(Mj ) ∩ Z(Y) = 1. (v) k = 4 and Y ∼ = F4 [m] or E6− [m]. Proof. Set Y + = Y/Z(Y) and ρ+ = (Y + , Γ+ ). Let t ∈ ZS (ρ) and set ζ = (NY (W (ρ))◦ , Γ). As ν ∼ = Weyl(Dk ) for some k ≥ 3 and ρ is proper in τ , it follows from 15.1.1.4 (cf. Theorems 15.3.10, 15.3.14, and 15.4.30) that one of the following holds: (a) Z(Y) = 1, |Γ+ (t+ )| = 2, and Y + ∼ = X /Z(X ) for some X listed in (2).

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(b) k = 4, Z(Y) ∼ = E4 , |Γ+ (t+ )| = 4, and Y + ∼ = X /Z(X ) for some X listed in (3).

(c) k = 4 and Y ∼ = F4 [m] or E6− [m]. If (c) holds then so does (1ii) and (4v), while (2) and (3) are vacuously true. Thus we may assume (a) or (b) holds. Suppose (a) holds. Then Ω(t+ ) = {K1+ , K2+ }, where Ki = Ω(ti ), t1 = t, and t2 = tu for a uniquely determined involution u ∈ Z(Y). Thus (1i) and (2) hold in this case. Similarly in case (b), Γ+ (t+ ) = {Ki+ : 1 ≤ i ≤ 4}, and setting ti = z(Ki ), Z(Y) = Z with {ti : 1 ≤ i ≤ 4} = tZ. Thus (1i) and (3) hold in this case. In particular (1)-(3) are established. Finally the first remark in (4) follows 15.1.8.1. If k ≥ 5 then 15.1.8.5.i holds, so uj = u(ζ), while as k ≥ 5, u(ζ) = u(ρ) by (2), so that (4i) holds. Thus we may assume k = 4. If n ≥ 5 then 15.1.8.5.ii holds, so uj ∈ Z(M (ρ)). If uj ∈ Z(Y) then (4ii) holds, so we may assume otherwise; in particular Z(M (ρ)) = Z(Y), so Z(Y) is cyclic by (3), and then (4iii) holds by (2). It remains to treat the case k = n = 4, where (4iv) holds by (1). 

15.2. The case r > 1 In this section we assume the following hypothesis: Hypothesis 15.2.1. Hypothesis 15.1.1 holds and τ satisfies the Extended Inductive Hypothesis. Adopt Notation 15.1.3 and the notation in Lemma 15.1.4. The main result in this section is the following theorem: Theorem 15.2.2. Assume F = F ◦ . Then r = 1. Until the proof of Theorem 15.2.2 is complete, we assume that F = F ◦ but r > 1. Lemma 15.2.3. (1) We can choose z ∈ Z(τ1 ), W1 , and O(τ1 ) in F f . (2) As W1 ∈ F f , for each w ∈ W1 there is a member of wF ∩ W1 in F f . Proof. See the proof of 12.3.17.



Lemma 15.2.4. Fz◦ = O(z) ∗ Cz ∗ C2 ∗ · · · ∗ Cr where (1) For 2 ≤ i ≤ r, ρi = (Ci , Ωi ) is a quaternion fusion packet, ηi ∈ η(ρi ), and Mi ∈ M (ρi ), so μi = μ(ρi ). (2) If Φ = A2 then Cz = 1, while otherwise ρz = (Cz , Ωz ) is a quaternion fusion packet, where Ωz = Ω1 − Ω(z), ηz ∈ η(ρz ), and Mz ∈ M (ρz ), where ηz = {V ∈ η1 : C (z) ∼ CD (dK ) − {dK } . V ≤ CS (O(z))} and Mz = Ωz M1 . Further μz = μ(ρz ) = 1 ∼ Weyl(An−2 ) or (3) If Φ = An with n ≥ 4 or Dn with n ≥ 5 then μz = Weyl(Dn−2 ), respectively. If Φ = Dn for n = 3 or 4 then μz ∼ = Z2 or E8 , respectively. If Φ is En then μz ∼ = S6 , Weyl(D6 ), Weyl(D7 ) for n = 6, 7, 8, respectively. (4) If Φ is not A2 or D4 then ρz is a coconnected component of τz◦ .

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Proof. By 15.1.5.1, τz◦ = ξ1 ∗ · · · ∗ ξa where ξj = (Fj , Γj ) with (Γj : j) a partition of Ω and (ξj : j) the coconnected components of τz◦ . For 2 ≤ i ≤ r, FS∩Mi (Mi ) is coconnected and centralizes z, so by 6.6.11, FS∩Mi (Mi ) ≤ Fj(i) for some j(i). Therefore Ωi ⊆ Γj(i) , while Γj(i) ⊆ Ωi by 15.1.8.1. We conclude Γj(i) = Ωi and μ(ξj(i) ) = μi . Thus we may choose notation such that for 2 ≤ j ≤ r, Fj = Cj and Γj = Ωj ; that is ξj = ρj . Now for k ∈ / {2, . . . , r}, Γk ⊆ Ω1 . We may choose notation so that ξ1 = (O(z), {K}), and Cz is the central product of the Fk with r < k ≤ a. In particular (1) and (2) hold. From 15.1.4.5, μ1 ∼ = Weyl(Φ) for some Φ ∈ {An , Dn , En }. If Φ = A2 then μ1 ∼ = S3 , so Cz = 1. If Φ = An for some n ≥ 4 then μ1 ∼ = Sn+1 and DdK ∼ = Sn−1 , so μz ∼ = Weyl(An−2 ). A similar argument establishes (3) when Φ is Dn with n ≥ 3. Finally if Φ = En then (3) follows from 5.5.4, completing the proof of (3). Part (4) follows from (3).  Notation 15.2.5. Let I = {1, . . . , r} and I1 = I−{1}. For i ∈ I1 , let zi ∈ Z(ρi ) with zi ∈ Cif , and, when Φ is not A2 , set Di = CCi (O(zi ))◦ , Γi = Ωi − Ω(zi ), and δi = (Di , Γi ). Lemma 15.2.6. Assume Φ is not A2 . Then for all i ∈ I1 and αi ∈ A(zi ) with zi αi = z, we have Di αi∗ = Cz , so Di ∼ = Cz .
1. It follows that for j ∈ J = I1 − {i}, ρj α∗ ≤ ρjσ for some injection σ : J → I1 by 15.2.4 and as ρj is coconnected. Similarly δi α∗ ≤ ρz and ρz α∗ ≤ δk , where < < I1 = Jσ ∪ {k}. Thus for each i ∈ I1 , δi ∼ ρz ∼ δk . Let I0 = {i0 ∈ I1 : δi0 ∼ = ρz } and observe that (I0 − {i})σ ⊆ I0 . Suppose that i ∈ I1 − I0 . Then I0 σ ⊆ I0 , so as σ is an injection, we have I0 σ = I0 . But this is < < a contradiction as also ρz ∼ δk ∼ ρz , so that k ∈ I0 . This contradiction completes the proof.  Lemma 15.2.7. Assume Φ = Dn for some n ≥ 3. If n = 4 set ui = u(τi ) for each i ∈ I and u = u1 . If n = 4 pick u ∈ Z(M1 )# ∩ F f . (1) τu◦ = ξ1 ∗ · · · ∗ ξr , where ξi = (Ui , Ωi ) is a quaternion fusion packet with Mi ∈ M (ξi ) and μi = μ(ξi ). Let Pi be Sylow in Ui and P = P1 · · · Pr . (2) For i ∈ I1 , Ui = Ci , so ρi = ξi . (3) If n = 4 then for i ∈ I1 , ui ∈ uF , ui = u(ξi ), and for αi ∈ A(ui ) with ui αi = u, we have Ui αi∗ = U1 and αi permutes U = {Uj : j ∈ I}. In particular AutF (P ) is transitive on U. (4) If n = 4 then for i ∈ I1 and βi ∈ A(Zi ) with Zi βi = Z1 , we have CUi (Zi )◦ αi = CU1 (Z1 )◦ and αi permutes V = {CUj (Zj ) : J ∈ I}. In particular AutF (P ) is transitive on V. (5) Suppose n = 4. For i ∈ I1 , vi ∈ uF ∩ Zi , and αi ∈ A(vi ) with vi αi = u, we have CCi (vi )◦ αi∗ ≤ U1 , U1 αi∗ ≤ Ck(1) for some k(1) ∈ I1 , and for each i = j ∈ I1 , Cj αi∗ ≤ Ck(j) for some k(j) ∈ I1 − {k(1)}. Proof. Part (1) follows from 15.1.5, 15.1.7, and the fact that u ∈ Z(M ); the argument essentially appears in the proof of 15.2.4.

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Let i ∈ I1 . As z ∈ W1 ≤ CS (Ui ) and ξi is coconnected on Ωi , it follows from 6.6.11 that Ui ≤ Ci . Similarly u ∈ K · O(ρz ) ≤ CS (Ci ), so Ci ≤ Ui , establishing (2). Suppose vi ∈ uF ∩Zi , so that there is αi ∈ A(vi ) with vi αi = u. Then Ωi αi = Ω1 and as vi centralizes Mi , CCi (vi )◦ is coconnected, so we have CCi (vi )◦ αi∗ ≤ U1 . Similarly as vi centralizes U1 , we have U1 αi∗ ≤ Ck(1) for some k(1) ∈ I1 . Also for each i = j ∈ I1 , vi centralizes Cj , so Cj αi∗ ≤ Ck(j) for some k(j) ∈ I1 − {k(1)}. In particular (5) holds. Suppose n = 4. Then from parts (2) and (4) of 15.1.9 and the proof of 15.1.9.4 in the case n = 3, we have u(ξi ) = ui , and from 15.1.4.3, NG (W ) is transitive ∗ on {uj : j ∈ I}, so u ∈ uF i . Hence from the previous paragraph, Ci αi ≤ U1 , ∗ ∗ U1 αi ≤ Ck(1) , and for i = j ∈ I1 , Cj α ≤ Ck(j) . Now, arguing as in the proof of 15.2.6, (3) holds. A similar argument establishes (4).  Lemma 15.2.8. Assume Φ = D4 , for i ∈ I1 let Ti be Sylow in Ci , and set Q = Ti : i ∈ I1 , τQ = (CF (Q), Ω1 ), and U = CF (Q)◦ . Let T1 be Sylow in U and T = T1 Q. Then (1) τQ is a quaternion fusion packet with M1 ∈ M (τQ ) and μ1 = μ(τQ ). Further U is transitive on Ω1 . (2) If Z(U) is noncyclic then Z(U) = Z1 and for each i ∈ I1 , Z(Ci ) = Zi . More◦ over for each u ∈ Z1# ∩ F f , {τQ , ρi : i ∈ I1 } is the set of coconnected components ◦ of τu . (3) If 1 = Z(U) = u then for each i ∈ I1 , Z(Ci ) = vi with vi ∈ uF , ◦ {τQ , ρi : i ∈ I1 } is the set of coconnected components of τu◦ , and αi ∈ A(vi ) with vi αi = u permutes U = {U, Cj : j ∈ I1 }. In particular AutF (T ) is transitive on U. (4) If Z(U) = 1 then for each i ∈ I1 , U ∼ = F4 [m] or E6− [m]. Further for = Ci ∼ ∗ βi ∈ A(Wi ), Ci βi = U, so βi permutes U. In particular AutF (T ) is transitive on U. (5) CU (z)◦ = O(z)Cz , so CU (z)◦ ∼ = O(zi )Di for each i ∈ I1 . Proof. By 2.5.2, τQ is a quaternion fusion packet. By parts (1) and (2) of 15.2.7, Q centralizes M1 , so (1) holds as usual. Similarly by 15.2.4, Q centralizes O(z)Cz , so O(z)Cz ≤ CU (z)◦ , and the opposite inclusion follows from 15.2.4. Then 15.2.6 completes the proof of (5). Let i ∈ I1 . Suppose Z(U) is noncyclic; then by 15.1.9.3, Z1 = Z(U) and U is described in 15.1.9.3. Choose u ∈ Z1# ∩ F f and define U1 as in 15.2.7. Then U1 ≤ U, while as u ∈ Z1 = Z(U) we also have the opposite inclusion. Thus U1 = U and Z1 = Z(U1 ). As this holds for each such choice of u, it follows from 15.2.7.5 that if Z(Ci ) = 1 then for vi an involution in Z(Ci ) and αi ∈ A(vi ) with vi αi = u, we have Ci αi∗ ≤ U1 = U, so Zi = Z(Ci ). On the other hand if Z(Ci ) = 1 then by 15.1.9.1, Ci ∼ = F4 [m] or E6− [m]. Then from 5.5.3.1, Di ∼ = Sp6 [m] or SL− 6 [m], ∼ respectively. Then by (5), U = Ci , a contradiction. Therefore Zi = Z(Ci ) for each i ∈ I1 , so that (2) holds. Suppose next that 1 = Z(U) = u is of order 2. Then from 15.1.9.2, U ∼ = − ˆ 9 . Hence by (5), Cz ∼ ˆ [m], or AE [m], Spin [m], or AE Spin Spin9 [m], Spin− = 5 5, 10 6 as is Di for each i ∈ I1 . This implies that Ci ∼ = U. Let vi generate Z(Ci ) and αi ∈ A(vi ) with vi αi ∈ U1 ; by 15.2.7.5, Ci αi∗ ≤ U, so it follows that vi αi = u. Now (3) holds. This leaves the case Z(U) = 1, where by 15.1.9.1, U ∼ = F4 [m] or E6− [m]. Then by (5), for each i ∈ I1 , Di ∼ = Cz is isomorphic to Sp6 [m] or SL− 6 [m], respectively,  so Ci ∼ = Ci . But = U. Let O1 = Ω − Ω1 and Y = CF (O1 )◦ . From 15.2.9.2, Y ∼

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∼ Ci = ∼ Y, we have Y = U. Let βi ∈ A(Wi ). Then O1 ≤ Q, so U ≤ Y and then as U =  βi is also in A(Oi ), so 15.2.9.2 completes the proof of (4).  Lemma 15.2.9. For k ∈ I let Ok = O(τk ) and Ok = k =j∈I Oj . Set Y = CF (O1 )◦ and θ = (Y, Ω1 ). For i ∈ I1 set Oi = O 2 (CCi (Z(Oi ))) and O1 = O 2 (CY (Z(O1 ))). Then (1) Oi ∈ F f and θ is a quaternion fusion packet with M1 ∈ M (θ) and μ1 = μ(θ). Further Y is transitive on Ω1 . (2) For i ∈ I1 and αi ∈ A(Oi ), Ci αi∗ = Y and αi permutes U = {Y,  Ci : i ∈ I1 }. (3) For i ∈ I1 let Ti be Sylow in Ci and T1 Sylow in Y. Set T = j∈I Tj . Then AutF (T ) is transitive on U.  (4) Let Sk = S ∩ Mk and Sk = k =j∈I Sj . Assume S1 centralizes Ci for some i ∈ I1 . Then Y = CF (S1 )◦ . (5) Suppose T2 = S2 . Then Tj = Sj for each j ∈ I and if S1 centralizes Ci for some i ∈ I1 then F contains a central product Y ∗ C2 ∗ · · · ∗ Cr . (6) Suppose T2 = S2 and C2 = S2 O2 , FS2 (M2 ) . Then F contains a central product Y ∗ C2 ∗ · · · ∗ Cr . Proof. As W1 ∈ F f , also O1 ∈ F f . By 2.5.2, θ is a quaternion fusion packet. As O1 centralizes M1 , M1 ∈ M (θ) and μ1 = μ(θ), so (1) holds. < As in the proof of 15.2.6, for β, γ ∈ {θ, ρi : i ∈ I1 }, write β ∼ γ if β is isomorphic to a subpacket of γ. < Let i ∈ I1 and γi ∈ A(Oi ). Then Ci γi∗ ≤ Y, so ρi ∼ θ. Let αi ∈ A(Oi ). Then Oi centralizes Y and Cj for j ∈ I1 − {i}, while Oi αi = O1 , so Yαi∗ ≤ Ck(1) and
1 and C0 γ ∗ ≤ X , for each j ∈ I, ρj γ ∗ = θj , so C0 γ ∗  X , and hence NS (Y ) ≤ NS (U γ). Now for δ ∈ A(U γ) with U γδ = U , γδ ∈ A(X), so that (2) holds.  Lemma 15.2.13. Let X ≤ T1 with X ∈ F f and assume C  = C2 · · · Cr  NF (X).  (1) O 2 (NC1 (X))  NF (X). (2) Each coconnected component of NF (X) not in {Ci : i ∈ I1 } is a subcomponent of (NC1 (X), CΩ1 (X)). 

Proof. Let X = NF (X), E = O 2 (NC1 (X)), and E Sylow in E. Then E ≤ CS (C  ), so E = [E]E ≤ D = CX (C  ). But D ≤ CX (O1 ) and E  CX (O1 ), so by 1.1.10 in [Asc19], we have E  D. Let D be Sylow in D; as X is saturated, AutX (D) = AutG (D), where G = NX (DO1 . Then since E is strongly closed in D with respect to G, it follows from 7.4 in [Asc11] that (1) holds. Then (1) implies (2).  Lemma 15.2.14. Let X be the set of involutions x of T1 in F f but not in Z(C0 ). Then (1) for each i ∈ I1 , ρi is a coconnected component of Fx . (2) C  = C2 · · · Cr  Fx . Proof. Assume (1) fails. Then for some i ∈ I1 , ρi < θi = (Ti , Ωi ) a component of τx◦ . Observe (a) x ∈ / z F and r = 2. If r > 2 or x ∈ z F then some t ∈ z F ∩ CS (x) centralizes Ti . Let α ∈ A(t) with tα = z. Then θi α∗ ≤ ρ for some cocomponent ρ = (C, Γ) of τz . This is a contradiction as C is isomorphic to a subsystem of Ci . (b) C2 is not FS2 (M2 ).  Assume otherwise. By 15.2.13.1, O 2 (CC1 (z))  Fz . But W1  CC1 (z), so as W1 is weakly closed in CS (z) with respect to Fz , we conclude that W1  Fz . As FS2 (M2 ) = T2 , it follows from 15.1.5 that W2 is not normal in CT2 (z2 ), a contradiction. (c) T2 has more than one class of involutions. If not, as xF ∩ T2 = ∅, we have x ∈ z F , contrary to (a). (d) Φ is not A2 . Assume otherwise. We have the proper containment C2 < T2 . By (b), C2 is not FS2 (M2 ), so it follows from 15.1.5.4.vi that T2 is G2 [m], contrary to (c). (e) Φ is Dn for some n ≥ 3. If not then by (d) and 15.1.4.5, Φ is An for some n ≥ 4 or En for some n ∈ {6, 7, 8}. But then by 15.1.5.4 and (b), we have no proper containment C2 < T2 . (f) If Φ is D4 then C2 is not F4 [m] or E6− [m]. Assume otherwise. As C2 < T2 it follows from 15.1.9.1 that C2 is F4 [m] and T2 is E6− [m]. Then by 5.5.3.1, CT2 (z2 ) has a coconnected component E ∼ = SL− 6 [m]. But Cz ∼ = Sp6 [m] by 5.5.3.1, so E is isomorphic to no subsystem of Cz , a contradiction. (g) Φ is D4 .

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If not then by (e), Φ is Dn for some n = 4. Then by parts (1) and (2) of 15.1.9 and 15.1.6, u2 = u(ρ2 ) = u(θ2 ), whereas by 15.2.7, C2 is a component of Fu2 , contradicting C2 < T2 . (h) T2 is F4 [m] or E6− [m]. Suppose otherwise; then by 15.1.9.1, there is an involution u ∈ Z(T2 ) ∩ Z2 . Hence from 15.2.8, T2 is isomorphic to a subsystem of U ∼ = C2 , contradicting C2 < T2 . (i) C2 = CT2 (u)◦ for some involution u in Z2 . By (f) and 15.1.9.1, there is an involution u ∈ Z(C2 ), so C2 ≤ CT2 (u)◦ . On the other hand by aC.8, CT2 (u)◦ is isomorphic to a subsystem of U, so the remark follows as U ∼ = C2 . We now obtain a contradiction, establishing (1). As x ∈ T1 ≤ C1 , x is conjugate in F to z2 or u, the representatives for the two classes of involutions in T by (h) and 5.5.6.1. By (a), x is not conjugate to z2 , so x ∈ uF . Therefore by (i) there is a component D of Fx◦ isomorphic to C2 . But now there is a conjugate of z in D centralizing T2 , a contradiction. We’ve established (1). Thus if (2) fails and D is the coconnected component of Fx containing FS1 (M1 ), then D is conjugate to some Ci in Fx . Therefore D ∼ = Ci and D ≤ Y ∼ = Ci , so D = Y. Therefore x ∈ Z(C0 ), contrary to x ∈ X. Therefore (2) holds.  Lemma 15.2.15. Z(C1 ) = 1. Proof. Assume Z(C1 ) = 1; we claim that Hypothesis 8.1.8 holds. Then by 8.1.10, F ◦ = C0 , for our usual contradiction. First, Hypothesis 8.1.7 is satisfied by 15.2.11.3, so it remains to check the additional four conditions in 8.1.8. Condition (1) holds by 15.2.11.2. By 15.2.12, we may choose the involution x in 8.1.14 to be in X, and we must show that ρi is a component of τx for each i ∈ I1 , and all other components are subcomponents of (NC1 (x)◦ , CΩ1 (x)). The first condition follows from 15.2.14.1. Then the second condition holds by 15.2.14.2 and 15.2.13.2.  Notation 15.2.16. Set U = Z(C0 ). Observe that by 15.2.11.3, U = Z(Cj ) : j ∈ I , so U = 1 by 15.2.15. Set U = NF (U ). Lemma 15.2.17. (1) U  S, so U ∈ F f . (2) S permutes {Tj : j ∈ I} and {Cj : j ∈ I}. (3) NF (O1 ) ≤ U. Proof. As W O(τ )  S, S permutes {Oj : j ∈ I}. In particular for s ∈ S, O1 cs = Oi for some i ∈ I, so φ = cs αi is a NF (O1 )-map by 15.2.9.2. As Y  NF (O1 ), we have T1 φ = T1 and Yφ∗ = Y, so by 15.2.9.2, T1s = T1 cs = T1 αi−1 = Ti . As z ∈ O1 , {ρk : k ∈ I1 } are the coconnected components of CF (O1 ), so C  = C2 · · · Cr  N = NF (O1 ) and φ permutes {Tj : j ∈ I} and {Cj : j ∈ I}, as does αi by 15.2.9.2, so cs = φαi−1 does too. This proves (2). By (2), S acts on Z(Cj ) : j ∈ I = U , so (1) holds. As C   N , we have N = NS (O1 )C  , NN (O1 ) by 1.3.2 in [Asc19]. Thus Z(Y)  N by 15.2.11.1, so U  N , completing the proof of (3).  Lemma 15.2.18. (1) Z(F) = 1. (2) O2 (F) = 1. Proof. Assume (1) fails and set F + = F/Z(F). Then τ + = (F + , Ω+ ) is a quaternion fusion packet by 3.3.2. By 15.1.1.2, F + is transitive on Ω+ , and

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as F = F ◦ , also F + = (F + )◦ . By 6.6.9.2, τ + satisfies the Extended Inductive Hypothesis. Thus if k = |Ω(z + )| = 1 then the hypotheses of Theorem 15.2.2 are satisfied, so by induction on the order of τ we have a contradiction to r > 1. Therefore k > 1. If k > 2 then 10.2.13 supplies a contradiction, so k = 2. Then 9.3.11 and 12.3.8 supply a contradiction to r > 1, completing the proof of (1). Let Q = O2 (F). If z ∈ Q then Theorem 2 supplies a contradiction, so we may assume otherwise. Then by 3.3.2, F ◦ centralizes Q, so as F = F ◦ , Q ≤ Z(F) = 1 by (1).  Lemma 15.2.19. (1) For each 1 = X ≤ U with X ∈ F f , {ρj : j ∈ I} is the set of components of NF (X), so NF (X) ≤ U. (2) C0  U. Proof. By 15.1.5, for j ∈ I, ρj ≤ θj = (Tj , Ωj ) a component of NF (X). As r > 1 and Tj centralizes zk for k =  j, it follows that Tj = Cj , so (1) holds. Then (1) and 15.2.17.1 imply (2).  Lemma 15.2.20. (1) NF (W ) ≤ U. (2) NF (O(τ )) ≤ U. Proof. Set O = O(τ ), W = NF (W ), S0 = S1 · · · Sr , and M = FS0 (M ). Then M  W, so by 1.3.2 in [Asc19], W = SM, NW (S0 ) . By construction, M ≤ U so SM ≤ U by 15.2.17.1. Thus to prove (1), it suffices to show that N = NF (S0 ) ≤ U. Let P = CS (S0 ). Next CF (S0 ) ≤ NF (O1 ) ≤ U by 15.2.17.3 and by 1.3.2 in [Asc19], N = SCF (S0 ), NN (P ) , so it suffices to show NN (P ) ≤ U. Let H be a model for NN (P ); it suffices to show that U  H. But as Oj αj = O1 , NH (U ) is transitive on {Oj : j ∈ I}, so H = NH (O1 )NH (U ) = NH (U ) by 15.2.17.3, completing the proof of (1). Set D = NF (O) and let E be the kernel of the action of S on Ω. By 2.8.11 in [Asc19] there is a normal subsystem E of D on E such that each member of Ω is normal in E. By 1.3.2 in [Asc19], D = SE, ND (E) . By 15.2.17, SE ≤ U. As W ≤ E, ND (E) ≤ W ≤ U, proving (2).  Lemma 15.2.21. (1) U controls fusion in U . (2) U is tightly embedded in F. Proof. To prove (2), we must verify the three conditions (T1)-(T3) of Chapter 3 of [Asc19]. Condition (T3) is trivial in this case. As W is weakly closed in S and abelian, NF (W ) controls fusion in W , so as U ≤ W and NF (W ) ≤ U by 15.2.20.1, it follows that U controls fusion in U , so that (1) and (T2) hold. It remains to verify (T1). Let 1 = X ≤ U . Using 15.2.20.2 we may assume  X ∈ F f . We must show that E = O 2 (NF (X)) ≤ U. But this follows from 15.2.19.1, completing the proof.  Lemma 15.2.22. (1) Φ is not An with n even, E6 , or E8 . ˜7 [m], and in either case (2) If Φ is E7 then either Ci = FSi (Mi ) or Ci ∼ = E Z(Mi ) = Z(Ci ) ∼ = Z2 . (3) If Φ is An with n odd then either Ci = FSi (Mi ) or Ci is a central factor system of SL+ n+1 [m] and in either case Z(Mi ) = Z(Ci ) is cyclic. (4) If Φ is Dn with n = 4 then u(τi ) = u(ρi ) ∈ Z(Ci ). (5) If Φ is D4 then 1 = Z(Ci ) ≤ Z(Mi ) ∼ = E4 . (6) For 1 = X ≤ Z(Ci ), NF (X) ≤ U.

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Proof. Part (1) follows from 15.2.15 and 15.1.5.4. Similarly if Φ is E7 then ˜7 [m] or E7 [m], and the last case is out by by 15.1.5.4, either Ci = FSi (Mi ) or Ci ∼ =E Z and as S is Sylow in Ci by 5.3.10 and 5.5.8, we have 15.2.15. Therefore Z(Ci ) ∼ = 2 i Z(Ci ) ≤ Z(Mi ), so we conclude that (2) holds. A similar argument using 5.7.2.6 establishes (3). Part (4) follows from 15.2.7.3, while (5) follows from 15.2.15 and 15.2.8. Let 1 = X ≤ Z(Ci ). Then X ≤ U . By 15.2.21.1 we may assume X ∈ F f . Then 15.2.19 establishes (6).  Remark 15.2.23. From 15.2.16, 12.3.24 is satisfied in our setup. Most of the lemmas in section 12.3 following 12.3.24 hold in our situation, and indeed usually their proofs go through verbatim, or sometimes we have already proved the lemma. In particular 12.3.25 follows from 15.2.18.1 and 12.3.26 follows from 15.2.20.1. Then the proofs of 12.3.27-12.3.29, go through verbatim. We now establish 12.3.30 in the following lemma: Lemma 15.2.24. Assume Θ = (T , Γ) is a proper subpacket of τ with Ωi ⊆ Γ ⊆ Ω and Ci ≤ T . Then (1) ρi is a coconnected component of Θ◦ . (2) T ≤ U. Proof. To prove (1) we may assume Γ = Ωi and Ci < T . Let X = Z(Ci ); observe X = 1 by 15.2.15. By 15.2.22.6, Ci = CT (X)◦ . By 15.1.4.5 and 15.2.22.1, Φ is E7 , Dn for some n ≥ 3, or An for some odd n ≥ 3. In the first case Z(Mi ) = Z(Ci ) ∼ = Z2 by 15.2.22.2, so as Mi ∈ M (Θ) it follows that T is not E7 [m] and hence X = Z(T ) by 15.1.5.4, contradicting T = Ci = CT (X)◦ . Similarly if Φ is An then X = Z(T ) and we have the same contradiction. Therefore Φ is Dn . By 15.1.9.1, either there is an involution u = u(Θ) ∈ Z(T ) or Φ is D4 and T is F4 [m] or E6− [m]. However in the latter case, NT (Wi ) is transitive on Zi# , so as Z(Ci ) = 1 is cyclic by 5.5.6, and as NF (Wi ) ≤ U by 12.3.28.2, we have a contradiction. Therefore u = u(Θ) ∈ Z(T ). If n = 4 then by 15.2.7.3, u = ui ∈ Z(Ci ) ≤ U , contrary to Ci = T and 15.2.22.6. Therefore n = 4 and at least u ∈ Zi , so u ∈ Z(Ci ) ≤ U , yielding our usual contradiction. This completes the proof of (1). Then (2) follows from (1) as in the proof of 12.3.30.2.  Remark 15.2.25. By 15.2.18.2, O2 (F) = 1, so as U ≤ O2 (U) we have U = F. This is 12.3.31.1, while the remaining two parts of 12.3.31 follow from 15.2.19.2. Observe 12.3.32 is 15.2.21.2. Given that 12.3.25-12.3.32 hold, the proofs of 12.3.33 and 12.3.34 go through verbatim. Lemma 12.3.35 is unnecessary, while the proofs of 12.3.36-12.3.42 go through verbatim. Indeed 12.3.38 is 15.2.18.2. The proofs of 12.3.43 and 12.3.44 go through verbatim. Lemma 12.3.45 is also unnecessary. The proof of 12.3.46 goes through verbatim. We only need parts of 12.3.47, which we establish in the next lemma. Lemma 15.2.26. Assume A ∈ ΞP and let J ∈ Ω. Then (1) |NJ (P )| ≤ 2. (2) AP acts on J. ˆ Proof. We follow part of the proof of 12.3.47. By 12.3.46.2, CU (P )α ≤ Q ∈ P. Assume (1) fails. Then there is x of order 4 in NJ (P ). By 12.3.46.5, xα ∈ ΣQ . But xα ∈ D ∈ Δ, so xα ∈ Sj for some j ∈ I, contrary to 12.3.42.3. This proves (1).

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To prove (2) we proceed as in the proof of 12.3.47.1. Let t = z(J). If A centralizes t then so does AP by 12.3.44.3, so (2) holds. Thus we may assume for some a ∈ A, s = ta = t. Let L ∈ Ω(s), so that L = J a . Then Y = JL = J × L, and arguing as in the proof of 12.3.47.1, B = NA (J) is of index 2 in A and centralizes  Y . But then J = NJ (P ), contrary to (1). This completes the proof of (2). Theorem 15.2.27. ΞP = ∅. Proof. Assume A ∈ ΞP . By 15.2.26.2, AP acts on K and hence on V = K ∩W . By 12.3.44.3, P centralizes the subgroup V0 of V of order 4, so V0 ≤ NK (P ), contrary to 15.2.26.1.  Lemma 15.2.28. (1) P = a ∼ = Z2 . (2) r = 2. (3) W1a = W2 . Proof. By 12.3.46.4, each nontrivial subgroup of P is in ΣP , so P is cyclic by 15.2.27. Let a be the involution in P . ˆ Suppose ∅ = J ⊂ I such that Y =  By 12.3.46.2, CU (P )α ≤ Q ∈ P. j∈J Wj is a-invariant. Then Y ≤ CS (Ci ) for i ∈ I − J, so by 12.3.46.5, each nontrivial subgroup of X = CY (a)α is in ΣQ . Hence X is cyclic by 15.2.27, so m(Y ) ≤ 2. But if a acts on W1 we may take J = {i} and Y = W1 , whereas m(W1 ) ≥ 3. Therefore we may assume W1a = W2 , so (3) holds. Similarly (2) holds or else we could choose Y = W1 W2 . Finally as r = 2, Φ(S) acts on W1 , so a ∈ / Φ(P ) and therefore P = a , establishing (1). 

We are now in a position to obtain a contradiction, and hence finally establish Theorem 15.2.2. Namely Lemma 15.2.28 is just 12.3.49, and the last paragraph of the proof of Theorem 12.3.8 (which appears just after the proof of 12.3.49) supplies our contradiction. Thus the proof of Theorem 15.2.2 is finally complete. Theorem 15.2.29. Assume Hypothesis 15.2.1 with F = F ◦ . Then μ(τ ) ∼ = Weyl(Φ), where Φ is An , Dn , or En . Proof. By Theorem 15.2.2, r = 1, so W = W1 and μ(τ ) = μ1 . Thus the theorem follows from 15.1.4.5.  Remark 15.2.30. Assume Hypothesis 15.2.1 with F = F ◦ , and suppose Φ = A2 . We claim that either (a) z ∈ O2 (F) and F = FS (M ) or F ∼ = L3 (2)/E8 , or + (1) (b) F ∼ [m], L [m] , G [m], or M = L+ 2 12 . 3 3 For Ω = {K} is of order 1 and μ = μ(τ ) ∼ = S3 . Hence Hypothesis 7.1.1 holds, so we can appeal to the results in section 7.1. In particular if z ∈ O2 (F) then (a) holds by Theorem 7.1.6, while if a ∈ / O2 (F) then (b) holds by Theorem 7.1.29.

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15.3. Φ = Dn In this section we assume the following hypothesis: Hypothesis 15.3.1. Hypothesis 15.2.1 holds with F = F ◦ and μ = μ(τ ) ∼ = Weyl(Φ), where Φ is of type Dn for some n ≥ 3. In addition we adopt Notation 15.1.3, set SM = S ∩ M , M = FSM (M ), and we adopt the following notation: Notation 15.3.2. We adopt a modified version of the notation in section 5.3. ¯ = Spin+ (q). There is a Let q be a prime power with (q − 1)2 = m/2 and let L 2n ¯ ¯ representation π ¯ : L → L = Ω(U, q) of L on an orthogonal space (U, q) over F = Fq ¯ F¯ = FS¯ (L), ¯ and τ¯ = (F, ¯ Ω) ¯ a Lie of sign +1 and dimension 2n. Let S¯ ∈ Syl2 (L), ¯ packet for L. Then π ¯ induces a morphisms π ¯L : τ¯ → τL = (FL , ΩL ), where τL is ¯ ∈ W (¯ ¯ ∈ M (¯ ¯L . Let W τ ) and M τ ). a Lie packet for L with Sylow group SL = S¯π ¯ ¯ ). By 4.3.8 there is a surjective homomorphism π : M → M with ker(π) ≤ Z(M ¯ → ML , where WL = W ¯π ¯L Similarly π ¯L induces a surjective homomorphism πL : M and ML ∈ M (τL ). We use these maps to study M and τ , via the representation of ML on U . Let I = {1, . . . , n}. In particular, as in 5.3.4 and 5.3.5, there is an ML -invariant decomposition U = U1 ⊥ · · · ⊥Un of U such that for i ∈ I, Ui is a line of sign +1 and WL is the kernel of the action of ML on Γ = {Ui : i ∈ I}. Notation 15.3.3. If n = 4 then by 15.1.6 there exist unique involutions u ¯= ¯ ) and u = u(τ ) ∈ Z(M ) such that for each t in ZS (¯ τ ) or ZS (τ ), t¯ = t¯ u u(¯ τ ) ∈ Z(M τ ) or ZS (τ ), respectively. If n = 4 then ker(πL ) is of order or t = tu is also in ZS (¯ 2 and we write u ¯ for a generator of the kernel. For 1 ≤ k ≤ n, write i(2k, +) for the set of involutions j ∈ L such that [U, j] is nondegenerate of dimension 2k and sign +1, and write ¯i(2k, +) for the set of ¯ of such involutions. A projective involution in L is an element x preimages in L of order 4 such that the eigenspaces Vλ and V−λ for x on V are totally singular of dimension n, where λ ∈ F is of order 4. ¯ of w under π or πL . ¯ for a preimage in W For w in W or WL , write w Lemma 15.3.4. (1) If n = 4 then u ¯π = u, so u ¯∈ / ker(π). τ ), t¯ πL = t¯. (2) ¯ u = ker(πL ), so for t¯ ∈ ZS (¯ (3) For tL ∈ ZΔ (τL ), [U, tl ] = Ui + Uj for some distinct i, j ∈ I. (4) If n is odd or n = 4 then π is an isomorphism. (5) If n > 4 is even then either π is an isomorphism or ker(π) ∼ = Z2 . Proof. The uniqueness of u and u ¯ imply (1). ¯ ) is cyclic by 5.8.4, so as u If n is odd then Z(M ¯∈ / ker(π) by (1), it follows that ker(π) = 1, so that π is an isomorphism. If n = 4 then π is an isomorphism as ¯) ∼ |Ω(z)| = 1. Thus (4) holds. Similarly if n > 4 is even then Z(M = E4 by 5.8.4, so (1) implies (5). If n = 4 then (2) holds by construction. For tL ∈ ZS (τL ), we have Ω(tL ) = {KL , KL } of order 2 with [U, KL ] = [U, KL ] = [U, tL ] = Ui + Uj for some distinct i, j ∈ I. Then (2) follows from the uniqueness of u ¯. Also as ML is transitive on  ZS (τL ), (3) follows.

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Lemma 15.3.5. (1) Members ¯j of ¯i(2k, +) are of order 2,4, for k even, odd, ¯ if k is odd. respectively, with ¯j 2 = u (2) If j is an involution in WL then [U, j] is the sum of k members of Γ, so j ∈ i(2k, +). (3) If t is an involution in W then one of the following holds: (a) t¯πL ∈ i(2k, +) for some even k. (b) t¯πL is a projective involution. (c) t = u. f (4) If x is a projective involution in L then CL (x) ∼ = GLn (q), and if x ∈ FL then τ (CFL (x), CΩL (x)) is a Lie packet of CL (x) with Thompson group Weyl(An−1 ). Proof. For i ∈ I let ji be the involution in L with [U, ji ] = Ui . Then j1 is a representative for i(2, +). Let B = U1 + U2 and regard j1 as a member of Ω(B, q) ∼ = SL2 (q) × SL2 (q), ¯j1 is of order 4 with = SL2 (q) ∗ SL2 (q). As Spin(B, q) ∼ ¯j12 = u ¯. Now a representative x for i(2k, +) is x = j1 · · · jk , so as [¯jr , ¯js ] = 1 for r, s ∈ I, (1) follows. the kernel of the action of ML on Γ, so each involution Next from 15.3.2, WL is  y in WL is the product y = ji ∈I(y) ji for some I(y) ⊆ I. Hence (2) holds. Let t be an involution in W and tL = t¯πL . Then tL ∈ WL , so if tL is an involution then tL ∈ i(2k, +) for some k by (2). But if k is odd then t¯2 = u by (1), so as u ¯∈ / ker(π) by 15.3.4.1, we have a contradiction to t an involution. Thus (3a) ¯ holds if tL is an involution, so we may assume otherwise. If tL = 1 then t¯L = u by 15.3.4.2, so (c) holds by 15.3.4.1; thus we may assume tL = 1, so |tL | ≥ 4 and hence |t¯| = |t¯L | ≥ 4. Then as t is an involution, π is not an isomorphism, and therefore n > 4 is even and ker(π) ∼ = Z2 by 15.3.4. Then t¯2 = y¯ generates ker(π), but y¯πL = y = 1 is in Z(ML ), so y = −1U . Therefore tL is a projective involution, completing the proof of (3). Part (4) is an exercise; see for example 15.12 in [Asc77].  Lemma 15.3.6. Let t be an involution in W ∩ F f with tL = t¯πL ∈ i(2k, +). Set U (1) = [U, tL ], U (2) = CU (tL ), and 2ki = dim(Ui ). (1) θt = (CM (t)◦ , CΩ (t)) is a quaternion fusion packet. ¯ : K ∈ ΩL and [U, K] ≤ U (i)}, (2) θt = θ1 ∗ θ2 , where θi = (Mi , Ωi ), Ωi = {Kπ and Mi = [Ωi ]CM (t)◦ . (3) μ(θi ) ∼ = Weyl(Φi ) where Φ = Dki , unless k2 ≤ 1 where τ2 = ∅. (4) If n ≥ 5 and ki ≥ 2, then u ∈ Z(Mi ). If in addition ki = 4 then u = u(θi ). Proof. Part (1) follows from 2.5.2. As t ∈ F f also tL ∈ FLf . As tL ∈ i(2k, +), k is even by 15.3.5.3, so in particular k1 ≥ 2. Now CL (t)◦ = L1 ∗ L2 , where Li = Ω(U (i), q) ∼ = Ω+ 2ki (q), except if k2 ≤ 1 then L2 = 1. Let Si = SL ∩ Li and ξi = (FSi , Λi ), where Λi = {K ∈ ΩL : [U, K] ≤ U (i)}. Then as tL ∈ WL ∩ FLf , ξi is a Lie packet of Li with μ(ξi ) = Dki . Let N = FSL (ML ), Ni ∈ M (ξi ), and Ni = FSi (Ni ). Then ¯ using πL and projecting onto M ¯ using CN (tL )◦ = N1 ∗ N2 , so pulling back to M π, we conclude that (2) and (3) hold. Assume n ≥ 5 and ki ≥ 2, so that u is defined. Then there exist K, K  ∈ Λi ¯  ) = z(K)¯ ¯ u is in Z(N ¯i ), and hence u = u with z(K) = z(K  ), so z(K ¯π ∈ Z(Mi ). Then if ki = 4, u(θi ) is defined and u = u(θi ) by uniqueness of u(θi ). This completes the proof of (4) and the lemma. 

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Lemma 15.3.7. Assume n ≥ 5 and t is an involution in W ∩ F f with tL = t¯πL ∈ i(2k, +). Then u ∈ Z(Ft ). Proof. Adopt the notation from 15.3.6. If u ∈ Z(F) the lemma is trivial, so we may assume otherwise. Suppose t ∈ Z(F) and set F + = F/t . Then, using 6.6.9, τ + = (F + , Ω+ ) satisfies Hypothesis 15.3.1, so u+ ∈ Z(F + ) by induction on the order of τ (cf. 15.3.10). But then pulling back to F, the lemma holds in τ . Therefore t ∈ / Z(F), so τt = (Ft◦ , Ω) is proper in τ . Then by 15.1.5 and an argument in the proof of 15.2.4, τt = τ1 ∗ τ2 where τi = (Fi , Ωi ) and for Mi ∈ M (τi ), Mi = FSi (Mi ), so that μ(τi ) = Weyl(Φi ) with Φi = Dki by 15.3.6. (a) If 4 = ki ≥ 3 then u = u(τi ). Namely by 15.3.6.4, u = u(θi ). But by 15.1.9.1, u(θi ) ∈ Z(Fi ), so (a) holds. (b) For i = 1, 2, either ki ≤ 2 or ki = 4. If not then we can choose i with 4 = ki ≥ 3. Then by (a), u = u(τi ) ∈ Z(Fi ). But observe that either Fi  Ft , or k1 = k2 = n/2 and Fi is normal in a subsystem of Ft of index 2. Then as u ∈ Z(S), in any event u ∈ Z(Ft ), contrary to assumption. This proves (b). Then as k1 + k2 = n ≥ 5 and k1 is even, we conclude from (b) that: (c) (k1 , k2 ) is (2, 4), (4, 1), (4, 2), or (4, 4). Observe that by 15.3.6.4: (d) u ∈ Z(Mi ) for each i with ki = 1. (e) If (k1 , k2 ) = (4, 1) then u ∈ Z(Fi ) for i = 1, 2. By (d), u ∈ Sj for j = 1, 2, so as S3−i centralizes Fi , part (e) follows. (f) u ∈ / Z(Fi ) for i ∈ {1, 2}. Assume otherwise and set Zi = Z(Fi ). If Zi = u then the proof of (b) shows that u ∈ Z(Ft ), contrary to assumption. Then from (c), Zi ∼ = E4 , and the proof of (b) shows that there is a 3-element φ ∈ AutFt (S1 S2 ) irreducible on Zi . Then φ acts on W1 W2 and is induced in NF (CS (W1 W2 ), so as W is weakly closed in CS (W1 W2 ), φ acts on W . This is a contradiction as u ∈ Z(NF (W )). Observe that by (e) and (f): (g) n = 5 and (k1 , k2 ) = (4, 1). ˆ (h) F1 is isomorphic to Spin9 [m], Spin− 10 [m], or AE9 . This follows from (g) and 15.1.9.2. We now obtain a contradiction, establishing the lemma. By (g), n = 5, so Z(M ) = y ∼ = Z4 . Now by (f), y acts faithfully on F1 centralizing M1 . But then (h) and 11.5.17.3 supply a contradiction.  Lemma 15.3.8. Assume n ≥ 5 and t is an involution in W ∩ F f with tL = t¯πL a projective involution. Then (1) τt = (Ft◦ , CΩ (t)) is a quaternion fusion packet with μ(τt ) ∼ = Weyl(An−1 ). (2) Either (a) W (τt )  Ft , or (b) Ft◦ is a central factor system of SL+ n [m]. (3) u ∈ Z(Ft ). ∼ Proof. By 2.5.2, τt is a quaternion fusion packet, and by 15.3.5.4, μ(τt ) = W eyl(An−1 ), so (1) holds. Then (2) follows from (1) and 15.1.5. Assume u ∈ / Z(Ft ). Claim u centralizes Y = Ft◦ . In case (a) of (2) this is clear, so assume case (b) holds. Then by (b) and 15.7.1 applied in an inductive setting,

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Y = FT (N ), T Oτt , where N ∈ M (τt ), so as u centralizes N and Oτt , the claim also holds in case (b). Let X = CFt (Y) and X Sylow in X . We may take K ∈ CΩ (t), so X ≤ CFt (z) ≤ CFt (u) by 15.3.7. Let R be Sylow in X Y and St = CS (t). By 1.3.2 in [Asc19], Ft = St X Y, NFt (R) , so it suffices to show that N = NFt (R) centralizes u. Let P ≤ St and φ ∈ homN (P, St ); we may take R ≤ P and it suffices to show uφ = u. Now φ acts on O = O(τt ) and X, so it suffices to show that α = φ|XO ∈ AutN (XO) fixes u. But NY (O) is transitive on CΩ (t), so there is ψ ∈ AutY (XO) with zαψ = z. As CFt (z) centralizes u, we have uαψ = u, and then as Y centralizes u, uα = uψ −1 = u, completing the proof of (3) and the lemma.  Lemma 15.3.9. If n ≥ 5 then u ∈ Z(F). Proof. Let T be the set of fully centralized involutions in W . Let t ∈ T and set tL = t¯πL . By 15.3.5.3, either tL ∈ i(2k, +) for some even k or tL is a projective involution, or t = u. Hence by 15.3.7 and 15.3.8.3, u ∈ Z(Ft ). Also u ∈ Z(NF (W )). Therefore Fu = NF (W ), Ft : t ∈ T , so the lemma follows from Theorem 14.1.18.  Theorem 15.3.10. Assume Hypothesis 15.3.1 with n ≥ 5. Then u ∈ Z(F) and one of the following holds: (1) W  F. − (2) F ∼ = Spin+ 2n [m], Spin2n+1 [m], or Spin2n+2 [m]. + (3) n is even and F ∼ = HSpin2n [m]. ∼ ˆ ˆ (4) F = AE2n or AE2n+1 . ˆ 2n . (5) n is even and F ∼ = H AE Proof. By 15.3.9, u ∈ Z(F). Set F + = F/u ; then τ + = (F + , Ω+ ) is a quaternion fusion packet with F + = (F + )◦ transitive on Ω+ . By 6.6.9.2, τ + satisfies the Extended Inductive Hypothesis. Further z + = z2+ where z2 = zu, so Ω(z + ) = {K + , K2+ } is of order 2. Therefore F + appears in one of cases (1)-(3) of Theorem 12.3.50, using Theorem 9.3.24 to verify that Z ∩ O(z) = {z}. Then lifting back to F we conclude the theorem holds using 5.1.21 and 3.3.16.  Notation 15.3.11 (aE.11). In the remainder of the section we assume that ¯ → M is an isomorphism, so we identify M with n = 3. Hence by 15.3.4.4, π : M ¯ the universal group M of 4.3.1 via this isomorphism and adopt the notation found there, except we suppress the “bars”. Thus M = K1 , K2 , K3 with Ki = ki , vi and Vi = vi = W ∩ Ki . Set zi = z(Ki ); we choose notation so that z = z3 , so that K = K3 and u = z2 z3 . Lemma 15.3.12. Assume n = 3. (1) τz = (Fz , Ω) is a quaternion fusion packet with τz = τ1 ∗ τ2 , where τ1 = (O(z), {K}) and τ2 = (F2 , {K2 }). (2) Either (a) F2 = O(z2 ) and u ∈ Z(F), or (1) (b) F2 ∼ . = L− 3 [m] or L2 [2m] Proof. By 3.3.1, τz is a quaternion fusion packet. As n = 3, Ω = {K, K2 }, so (1) follows.

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From 14.1.2, B0 = NF (W ), Ft : t ∈ T , where T is the set of fully centralized ¯ , up to conjugation in M , T = {u, z}. Also involutions in W . As n = 3 and M = M NF (W ) ≤ Fu , so B0 = Fu , Fz . If F2 = O(z2 ) then z2 is in the center of Fz , so u = zz2 is too, and hence B0 = Fu . But then (2a) holds by Theorem 14.1.18. Thus we may assume z2 is not in the center of F2 . But as μ ∼ = S4 , μ(τz ) ∼ = E4 and hence μ(τ2 ) ∼ = Z2 , so (3b) holds by Theorems 7.1.6 and 7.1.7.  Lemma 15.3.13. Assume n = 3. Then u ∈ Z(F). Proof. Assume otherwise; by 15.3.12.2, F2 ∼ = L2 [2m](1) or L− 3 [m]. In either case a Sylow subgroup S2 of F2 is semidihedral of order 2m, with S2 = x, y where |y| = m with y 2 = v2 , x is an involution, and y x = y −1 z2 . Now E = Ω1 (W ) = z1 , z2 , z with V3 centralizing S2 and z1 centralizing V3 and acting on S2 . Thus [x, z1 ] ∈ W ∩ S2 = V2 , so [x, z1 ] = z2a for some a ∈ {0, 1}. If a = 0 set t = z1 , and observe t centralizes z1 . If a = 1 set t = jx where j ∈ K2 − V2 with j = yx; then as [k2 , z1 ] = z2 , again t centralizes z1 and this time t = y. Define G as in 15.1.3 and form the subgroups U = W t and X = M t of G. Then U = CX (E), so U  X. Then as t centralizes V3 and M is transitive on η, it follows that U , and hence also t, centralizes W . Hence as x does not centralize V2 , we conclude that t = y. But then U m/2 = y m/2 = z2 is normal in M , a contradiction.  Theorem 15.3.14. Assume Hypothesis 15.3.1 with n = 3. Then u ∈ Z(F) and one of the following holds: (1) W  F. − (2) F ∼ = Spin+ 6 [m], Spin7 [m], or Spin8 [m]. ˆ 6 or AE ˆ 7. (3) F ∼ = AE ˆ 3 (2)/E64 or Sp ˆ 6 (2). (4) m = 8 and F ∼ =L Proof. The proof is much like that of Theorem 15.3.10. By 15.3.13, u ∈ Z(F). Set F + = F/u ; then τ + = (F + , Ω+ ) is a quaternion fusion packet with F + = (F + )◦ transitive on Ω+ . By 6.6.9.2, τ + satisfies the Extended Inductive Hypothesis. Further z + = z2+ , so Ω+ = Ω(z + ) = {K + , K2+ } is of order 2. Therefore F + appears in one of cases (1)-(3) or (5) of Theorem 12.3.50, using Theorem 9.3.24 to verify that Z ∩ O(z) = {z}. Then lifting back to F we conclude the theorem holds using 3.3.16, 5.10.7.2, and 5.1.21. 

15.4. Φ = D4 In this section we assume the following hypothesis: Hypothesis 15.4.1. Hypothesis 15.2.1 holds with F = F ◦ and μ = μ(τ ) ∼ = Weyl(Φ), where Φ is of type D4 . Notation 15.4.2. As usual let SM be Sylow in M and adopt the notation from ¯ → section 15.3, particularly Notation 15.1.3, 15.3.2, and 15.3.3. By 15.3.4.4, π : M ¯ , except we suppress M is an isomorphism, so, just as in 15.3.11, identify M with M the “bars”. Thus M = Ki : i ∈ I with Ki = ki , vi , and Vi = vi = Ki ∩ W .

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Set zi = z(Ki ). We choose notation so that z = z4 and hence K = K4 . Let K0 be the fourth member of Ω other than K1 , K3 , K4 . Set U = Z(M ). Lemma 15.4.3. (1) U = zz3 , zz1 ∼ = E4 . (2) Let T be the set of fully centralized involutions in W . Then up to conjugation in M , T = U # ∪ {z}. (3) F = NF (U ), Ft : t ∈ T . ˜ Ω) ˜ is the Lie packet with F˜ isomorphic to F4 [m] or E − [m]. (4) Assume τ˜ = (F, 6 ˜ )), F˜z˜, F˜u˜ , where M ˜ = M (˜ Then F˜ = NF˜ (Z(M τ ), z˜ ∈ ZS˜ , and u ˜ is an involution ˜ ). in Z(M Proof. Part (1) follows from 5.8.7.2, while (2) follows from 5.8.3. Observe Hypothesis 11.3.1 is satisfied with nS = 4 and WS = U z of rank 3. Then from 11.3.2, B = B(τ ) = E(τ ), Fx : x ∈ T . From 14.1.3.1, E(τ ) ≤ B0 = NF (W ), Fx : x ∈ T , so as NF (W ) ≤ NF (U ), (3) follows from 11.3.10. Assume the hypothesis of (4). From 5.5.2.1 and 5.5.4.1, Hypothesis 15.4.1 is satisfied by τ˜, with the exception of 15.1.1.4. Similarly by 5.5.4.1, τ˜ satisfies Hypothesis 11.3.8.3. By 5.5.6.1, F˜ has two classes of involutions with representatives z˜ and u ˜. Now the proof of (3) establishes (4).  Lemma 15.4.4. Let u ∈ U # ∩ F f . Then τu◦ = (Fu◦ , Ω) is a quaternion fusion packet and one of the following holds: (1) W  Fu . (2) Fu◦ ∼ = Spin+ 8 [m]. ˆ + (2), L ˆ 3 (2)/23+6 , or AE ˆ 8. (3) m = 8 and Fu◦ ∼ =Ω 8 − ◦ ∼ (4) Fu = Spin9 [m] or Spin10 [m]. ˆ 9. (5) m = 8 and Fu◦ ∼ = AE Moreover in cases (1)-(3), U = Z(Fu◦ ), while in cases (4) and (5), u = Z(Fu◦ ). Proof. First, τu◦ is a quaternion fusion packet by 2.5.2. Then if τu is proper in τ , the lemma follows from parts (2) and (3) of 15.1.9, so assume otherwise; then u ∈ Z(F). Set F + = F/u ; then τ + = (F + , Ω+ ) is a quaternion fusion packet by 3.3.2, and as F is transitive on Ω, we have F + transitive on Ω+ . By 6.6.9.2, τ + satisfies the Extended Inductive Hypothesis. Further z = zi+ , where i ∈ {0, 1, 3} with u = zzi . Hence |Ω(z + )| = 2, so F + appears as one of conclusions (1)-(4) of Theorem 12.3.50, using Theorem 12.3.24 to verify that Z + ∩ O(z + ) = {z + }. Now, lifting back to F and appealing to 3.3.16, 5.10.12, and 5.1.16, the lemma follows.  Lemma 15.4.5. (1) τz◦ = (Fz◦ , Ω) is a quaternion fusion packet and τz = τ1 ∗ τ2 , where τi = (Ti , Ωi ), T1 = O(z), Ω1 = {K}, and Ω2 = {K0 , K1 , K3 }. Further μ(τ2 ) ∼ = E8 and z ∈ Z(T2 ). (2) Up to choice of notation, one of the following holds: (i) zi ∈ Z(T2 ) for each i ∈ {0, 1, 3, 4} and U  Fz . (ii) τ2 = ξ0 ∗ ξ1 , where ξ0 = (O(z0 ), {K0 }) and ξ1 = (G, {K1 , K3 } with μ(ξ1 ) ∼ = ∼ ˆ 5 . Moreover zz0 ∈ [m], or m = 8 and G AE E4 and either G ∼ = Sp4 [m] or SL− = 4 Z(Fz ). (iii) T2 ∼ = Sp6 [m] or SL− 6 [m] and z = Z(T2 ).

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Proof. By 3.3.1, τz◦ is a quaternion fusion packet; then (1) follows. Observe that 1 = z0 z1 z3 z, so z = z0 z1 z3 ∈ T2 and hence z ∈ Z(T2 ). If zi ∈ Z(Ti ) for each i ∈ {0, 1, 3} then (2i) holds, so we may assume z1 ∈ / Z(T2 ). / Z(T2 ). Then as z1 = zz0 z3 and z ∈ Z(Fz ), we may also assume z3 ∈ Suppose that z0 ∈ Z(T2 ). Then τ2 = ξ0 ∗ ξ1 , where ξ0 = (O(z0 ), {K0 }) and ξ1 = (G, {K1 , K3 }) with μ(ξ1 ) ∼ = E4 . Then (2ii) holds by 15.1.5.3, observing that zz0 = z1 z3 ∈ Z(G). / Z(T2 ). Here (2iii) holds This leaves the case where for each i ∈ {0, 1, 3}, zi ∈  by 15.1.5.3, again keeping in mind that z ∈ Z(T2 ). Lemma 15.4.6. Assume U  Fu for each u ∈ U # . Then U ≤ Z(F). Proof. If U  F then as U centralizes Ω and F = F ◦ , also U ≤ Z(F). So it suffices to show that U  F. By 15.4.3.3, F = NF (W ), Ft : t ∈ T , so as U  NF (W ) it remains to show U  Ft for each t ∈ T. This holds by hypothesis if t ∈ U , so by 15.4.3.2 it suffices to show U  Fz . Thus we may assume case (i) of 15.4.5.2 does not hold. If case (ii) of that lemma holds, then u = zz0 ∈ Z(Fz ), so as U  Fu , our lemma holds in that case. Thus we may assume 15.4.5.2.iii is satisfied. Let u = zz0 ; we may assume u ∈ F f . By hypothesis, U  CFu (z). But CFu (z) = CFz (u) and U is not normal in CFz (u) = CFz (z0 ) in 15.4.5.2.iii, completing the proof of the lemma.  In light of 15.4.6, in the remainder of the section we assume: Hypothesis 15.4.7. Hypothesis 15.4.1 holds and U is not normal in Fu0 for some 1 = u0 ∈ U ∩ Z(S). Lemma 15.4.8. We have u0 = Z(Fu0 and either (1) Fu◦0 ∼ = Spin9 [m] or Spin− 10 [m], or ˆ 9. (2) m = 8 and Fu◦0 ∼ = AE Proof. This is a consequence of 15.4.7 and 15.4.4.



Lemma 15.4.9. (1) For T0 Sylow in Fu◦0 , u0 = CU (T0 ). (2) Either (a) AutNF (W ) (U ) = GL(U ), or (b) u0 is weakly closed in U with respect to F and for each u ∈ U −u0 , u ∈ F f and u0 ∈ Z(Fu ). Proof. Part (1) follows from 15.4.8 and 11.5.17.2. By (1), either AutF (U ) = AutT0 (U ) ∼ = Z2 , or AutF (U ) = GL(U ). As N = NF (W ) controls fusion in W , (2a) holds in the latter case, so we may assume the f / uF former, where u0 is weakly closed in U . Let u ∈ U − u0 ; then u ∈ 0 and u ∈ F . ◦ Let Tu be Sylow in Fu . As u0 is weakly closed in U , Tu centralizes u0 , so by 11.5.17.2, Fu◦ does not satisfy one of cases (4) or (5) of 15.4.4. Hence U  Fu by  15.4.4, so u0 ∈ Z(Fu ) as u0 is weakly closed in U , and hence (2b) holds. Lemma 15.4.10. Assume u0 is weakly closed in U with respect to F. Then u0 ∈ Z(F) and either (1) F ∼ = Spin9 [m] or Spin− 10 [m], or ˆ 9. (2) m = 8 and F ∼ = AE

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Proof. If u0 ∈ Z(F) then (1) or (2) holds by 15.4.4 and 15.4.7. Thus it remains to show u0 ∈ Z(F). As u0 is weakly closed in U , NF (W ) ≤ Fu0 , so by 15.4.3 and 15.4.9.2, it suffices to show u0 ∈ Z(Fz ). Assume otherwise; then by 15.4.9.2, U ∩ Z(Fz ) = 1, and as u0 is weakly closed in U , U is not normal in Fz . Therefore by 15.4.5.2, conclusion (iii) of that lemma holds. But now u0 is not  weakly closed in U with respect to T2 , a contradiction. Given lemmas 15.4.9.2 and 15.4.10, in the remainder of the section we assume:

Hypothesis 15.4.11. Hypothesis 15.4.1 holds, AutNF (W ) (U ) = GL(U ), and for 1 = u0 ∈ Z(S) ∩ U , U is not normal in Fu0 . Lemma 15.4.12. Fu0 is Spin9 [m] or Spin− 10 [m]. ˆ 9 . Recall Proof. Assume otherwise; then by 15.4.8, m = 8 and Y = Fu◦0 is AE Notation 15.4.2, and choose u0 = zz0 . Let u = zz3 . As AutF (U ) = GL(U ) there is α ∈ A(u) with uα = u0 . Then α acts on W and hence also on U and CS (U ). Then as O = O(τ ) ≤ CS (U ), we have Oα = O. Indeed as AutNF (W ) (U ) = GL(U ), we can choose α to centralize z, z1 and interchange u and u0 , so α = (z0 , z3 ) as a permutation of ZS . There is φ ∈ AutY (O) of order 3 centralizing K1 K3 with CKK0 (φ) = z, z0 . Let ϕ = φα∗ ; then as α = (z0 , z3 ), ϕ centralizes K1 K0 with CKK3 (ϕ) = z, z3 . Now Y  Fz0 , so and argument in the proof of 5.10.13 supplies a contradiction.  Definition 15.4.13. Define F to be of type F4 [m] if Fu◦0 is Spin9 [m] and Fz◦ ◦ is O(z) ∗ Sp6 [m]. Define F to be of type E6− [m] if Fu◦0 is Spin− 10 [m] and Fz is − O(z) ∗ SL6 [m]. Lemma 15.4.14. F is of type Ψ[m], where Ψ is F4 or E6− . Proof. Set Y = Fu◦0 ; by 15.4.12, Y is Spin9 [m] or Spin− 10 [m]. As usual take u0 = zz0 . Let X = CFu0 (z)◦ ; then X = (O(z) × O(z0 )) ∗ C, where C is Sp4 [m] or ◦ SL− 4 [m], respectively. Now X = CFz (u0 ) , so we conclude from 15.4.5.2 that case (ii) or (iii) of that lemma holds. By 15.4.11, and arguing as in the proof of 15.4.12, there is γ ∈ AutF (CS (U )) centralizing z with |γ|U | = 3. Then γ is transitive on {K0 , K1 , K3 }, so 15.4.5.2.iii holds. But this implies the lemma.  Lemma 15.4.15. Let Y = CF (U )◦ and Y¯ = Y/U . + + ¯ ∼ ¯ ∼ (1) Y ∼ = Spin+ 8 [m] and Y = P Ω8 [m] is tamely realized by LU = P Ω8 (q) for suitable q. ¯ (2) Y = FSM (LU ) where LU ∼ = Spin+ 8 (q) is the universal covering group of LU . (3) CS (SM ) = CS (Y). (4) O 2 (CAutF (SM ) (U )) = 1. ∼ Spin+ [m] by 15.4.14. Therefore Y¯ ∼ Proof. As Y = CF (U )◦ , Y = = P Ω+ 8 8 [m], and then (1) holds by 3.5 in [AO16]. Now (1) and 3.3.16 imply (2). ¯ so by (1) and 2.22.3 in [AO16], S¯Y¯ = SY/CS (Y) = Next SY reduces to Y, ¯ for some L ¯U ≤ X ¯ ≤ Aut(L ¯ U ). Hence if s ∈ S − CS (Y) centralizes SM , then FS¯ (X) / Z(SM ), so s¯ = z¯, so s = zc for some c ∈ CS (SM ). This is a contradiction as z ∈ (3) is established.

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Let Σ = O 2 (CAutF (SM ) (U )) and σ of odd order in Σ. Set E = ZS . Then E∼ = E8 and ZS = zU , so as σ centralizes U and is of odd order, we conclude that σ centralizes E. Therefore σ acts on each Ki ∈ Ω and then on the cyclic subgroup Ki ∩ W of index 2 in Ki , so σ centralizes Ki . Thus σ centralizes O = O(τ ). Now W/(W ∩ O(τ )) is cyclic, so σ centralizes W . Then [SM , σ] ≤ CSM (W ) = W ≤  CSM (σ), so σ = 1, proving (4). Notation 15.4.16. For x ∈ {z, u0 } let Tx be Sylow in Fx◦ . Set T0 = Tu0 and T = T0 Tz . Lemma 15.4.17. Assume Ψ = F4 . Then (1) |T0 | = (m/2)4 · 27 . (2) |Tz | = (m/2)4 · 24 . (3) Tz ≤ T0 so T = T0 . Proof. Part (1) follows from 5.2.7.3. Next Fz◦ = O(z) ∗ T2 with T2 ∼ = Sp6 [m], so Tz = K ∗ T2 with T2 Sylow in T2 . By 5.2.6, |T2 | = (m/2)3 · 24 , so (2) follows. Indeed CFz (u0 )◦ = (O(z) × O(z0 )) ∗ C where C ∼ = Sp4 [m], so a Sylow group of  CFz (u0 )◦ is of order (m/2)4 · 24 , and hence is Tz by (2). Therefore (3) holds. Lemma 15.4.18. Assume Ψ = E6− . Then (1) |T0 | = (m/2)4 · 28 . (2) |Tz | = (m/2)4 · 26 . (3) |T : T0 | = 2 and CT (SM ) = U . Proof. Part (1) follows from 5.2.7.5. By 5.2.5.2 a Sylow group of T2 is of order (m/2)3 · 26 , so (2) follows. This time CFz (u0 )◦ = (O(z) × O(z0 )) ∗ C with C ∼ = SL− 4 [m], so its Sylow group P is (K × K0 ) ∗ P1 where P1 is Sylow in C of order (m/2)2 · 24 . Hence |P | = (m/2)4 · 25 , so |Tz : P | = 2. Indeed Tz = P s where s ∈ Z(τ2 ) induces / T0 , as an outer automorphism on K0 and K1 and centralizes KK3 . Then s ∈ involutions in T0 centralizing K also centralize K0 , Therefore s induces an outer automorphism on Fu◦0 , so |Tz : Tz ∩ T0 | = 2, proving (3).  Lemma 15.4.19. (1) Fz = CS (z)Fz◦ , CFz (u0 ) . (2) CT (z)Fz◦  Fz . Proof. Set Sz = CS (z) and Y = Fz◦ . By 1.3.2 in [Asc19], Fz = Sz Y, NFz (Tz ) , so to prove (1) it suffices to show NFz (Tz ) ≤ Fu0 . But this holds as K0 is the unique member of Ω − {K} normal in Tz , so NFz (Tz ) ≤ CFz (z0 ) = CFz (u0 ). Hence (1) holds. Let X = T Fu◦0 , Xz = CX (z), and T1 = CT (z). Then NXz (Tz ) = T1 if m > 8, while if m = 8 at least NXz (Tz ) = T1 NY (Tz ). Hence as NFz (Tz ) ≤ CFz (u0 ), we conclude that T1 NY (Tz )  NFz (Tz ). Then (2) follows from 1.5.2 in [Asc19].  Lemma 15.4.20. T  S. Proof. As u0 ∈ Z(S), we have T0  S. Similarly Tz  CS (z), so CS (z) acts on T0 Tz = T . As SM ≤ T0 ≤ T , SM acts on T . Finally as SM is transitive on ZS , S = SM CS (z), completing the proof.  Lemma 15.4.21. T Fu◦0  Fu0 .

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Proof. Set Y = Fu◦0 and X = Fu0 . If Ψ = F4 then T = T0 so T Y = Y  X , so we may assume that Ψ = E6− . By 1.5.2 in [Asc19], it suffices to show that T NY (T0 )  NX (T0 ) = N . Observe that NY (T0 ) = T0 , so we must show that T  N . Now CT (z)  CN (z) by 15.4.19.2. Also SM is transitive on ZS , so  N = T0 CN (z). Therefore T = T0 CT (z)  T0 CN (z) = N . Lemma 15.4.22. T is strongly closed in S with respect to NF (U ). Proof. Set N = NF (U ) and U = CF (U )◦ . By 1.3.2 in [Asc19], N = SU, NN (SM ) . As SU ≤ Fu0 , T is strongly closed in S with respect to SU by 15.4.21. Therefore it suffices to show that T is strongly closed in S with respect to M = NN (SM ). Suppose t ∈ T − SM and φ ∈ homM (t, S); we must show tφ ∈ T . Set P = SM t ; we may assume φ ∈ homM (P, S). As SM is transitive on ZS , we may take t ∈ CT (z). Now zφ ∈ ZS so there is s ∈ SM with zφcs = z. By 15.4.19.2, CT (z) is strongly closed in CS (z) with respect to Fz , so tφcs ∈ T . Then by 15.4.20,  tφ ∈ T c−1 s = T , completing the proof. Notation 15.4.23. Set F1 = Fu0 , E1 = T Fu◦0 , F2 = NF (U ), and E2 = [T ]F2 . Set E = E1 , E2 . ¯ 2 ), where P Ω+ (q) ∼ ¯U ≤ L ¯2 ≤ Lemma 15.4.24. (1) E¯2 = E2 /U = FT¯ (L = L 8 ¯ ¯ ¯ Aut(LU ) and L2 is generated by conjugates of reflections under Aut(LU ). ¯ 2 /L ¯U ∼ (2) If Ψ = F4 then L = S3 . − ¯ 2 /L ¯ U is the subgroup of Out(L ¯ U ) generated by all (3) If Ψ = E6 then L ¯ Out(LU )-conjugates of images of reflections. (4) T is Sylow in E2 . ¯ 2 via the universal property of (5) E2 = FT (L2 ) where L2 is the pull back of L ¯ the universal cover LU of LU . Proof. We appeal to 2.1.14 applied to N = NF (U ), Y = CF (U )◦ , SM , T in the roles of F, F0 , S0 , T . Hypotheses (a), (c), and (d) of 2.1.14 are satisfied by 15.4.15. Hypothesis (b) holds by 15.4.11. Finally we check that (e) holds: First, T is strongly closed by 15.4.22. By 15.4.9.1, [T, U ] = 1. Recall SM is Sylow in Y, so to verify (e) it remains to show that CT (SM ) = U . If Ψ = E6− this follows from 15.4.18.3, while if Ψ = F4 it follows from 15.4.17.3 and the structure of Spin9 [m]. This completes the verification of the hypotheses of 2.1.14. By 2.1.14.1, (4) holds. By 2.1.14.2, F ∗ (E2 ) = Y. By 2.1.14.5, red(E2 ) = Y¯ = Y/U , so by 2.22.3 in [AO16], E2 = FT (L2 ) for some group L2 with F ∗ (L2 ) = LU . ∼ ¯ ¯ ¯ Then, using 15.4.15, P Ω+ 8 (q) = LU ≤ L2 ≤ Aut(LU ). Now (5) follows from the ¯U . universal property of the universal covering group LU of L From 15.4.14 and 5.9.1, there is r ∈ T − SM with r 2 = u0 and r¯ induces ¯ U . Set H = NL (S2 ). From 2.1.14.4, CH (U ) = CT (U ) and a reflection on L 2 H/CT (U ) ∼ = S3 . If Ψ = F4 then SM = CT (U ), so as r ∈ T − SM induces a reflection, (1) and (2) hold, completing the proof in this case. So we may take ¯ U ) = Σ × Γ where Σ ∼ Ψ = E6− . Then Out(L = S4 is the subgroup described in (3) and Γ is the group of field automorphisms. Now T /SM = Ω1 (T /SM ) is of order 8, ¯ 2 /L ¯U ∼ ¯ U ∈ Σ, so we conclude T /SM ∼ = D8 and L = S4 . As r¯ is a reflection, we have r¯L ¯ U = Σ. So in this case (1) and (3) hold, completing the proof. ¯ 2 /L  so L Lemma 15.4.25. (1) F = F1 , F2 .

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(2) Assume the setup of 15.4.3.4. Then F˜ = F˜1 , F˜2 , where F˜1 = F˜u˜ and ˜ )). F˜2 = NF˜ (Z(M Proof. We first prove (1). By 15.4.3 and 15.4.11, F = F1 , F2 , Fz , so it remains to show that Fz ≤ F1 , F2 . Set Sz = CS (z). From Theorem 13.6.3, Sz T2 = Sz CT2 (z0 ), NSz T2 (z0 , z1 , z3 ) . By 1.3.2 in [Asc19], Fz = Sz T2 , NFz (T2 ) , where T2 is Sylow in T2 . Now F1 contains Sz CT2 (z0 ) and NT2 (z0 , z1 , z3 ), so Sz T2 ≤ F1 . Further NFz (T2 ) ≤ F2 , so (1) holds. The proof of (2) is essentially the same, given 15.4.3.4.  Notation 15.4.26. Let F˜ = Ψ[m]. Then F˜ satisfies our hypotheses (with the exception of aB.1.4), with F˜ of type Ψ[m]. We will show that F ∼ = F˜ . By 15.4.14, Fu◦0 = FT0 (L0 ) for L0 ∼ (q) for suitable q. The same is true for = Spin9 (q) or Spin− 10 ◦ ◦ ˜ ˜ ˜ F by 5.5.6.2, so we may take Fu0 = Fu˜0 , SM = SM , etc. Lemma 15.4.27. If Ψ = F4 then E = F˜ . ˜ so E1 = FT (L0 ) = F˜ ˜ (L0 ) = F˜1 . Proof. From 15.4.17.3, T = T0 = S, S ∞ Also LU = CL0 (U ) and T = r SM , where r 2 = u0 and r¯ acts as a reflection ¯ 2 is generated by reflections. Therefore ¯ U . From 15.4.24.2, L2 /LU ∼ on L = S3 and L ¯ L2 is determined up to conjugacy under CAut(L¯ U ) (¯ r). Hence L2 is determined up to conjugation via α ∈ NAut(L¯ U ) (LU r ), and α is induced by β ∈ Aut(L0 ), so, adjusting our identification of E1 with F˜1 via β if necessary, we may assume E2 = FT (L2 ) = F˜2 . By 15.4.25.2 we have F˜ = F˜1 , F˜2 , so the lemma holds.  Lemma 15.4.28. E = F˜ . Proof. By 15.4.27, we may assume Ψ = E6− . By 15.4.24, E2 = FT (L2 ) with ¯ ¯ ¯ U ) of the image of a L2 /LU = Σ ∼ = S4 , where Σ is the normal closure in Aut(L reflection. As the same holds for F˜ , it follows that T = S˜ and E2 = F˜2 . ¯ U , so T0 /SM Next T0 /SM ∼ = E4 contains rSM where r¯ is a reflection on L is the 4-subgroup of T /SM distinct from O2 (NL2 (SM )/SM ). Let hSM generate ¯ can be chosen to be an involution of type i(2, −) on L ¯ U . Hence Φ(T /SM ). Then h CE2 (h)◦ ∼ [m], and of course a conjugate g of h under N (S ) in T −T0 also = Spin− L2 M 6 [m]. Hence g induces an outer diagonal automorphism satisfies CE2 (g)◦ ∼ = Spin− 6 on L0 , so T L0 /u0 is determined as a subgroup of Aut(L0 /u0 ). Then as L0 is the universal covering group of L0 /u0 , T L0 is also determined. Thus E1 = ˜ 0 ) = F˜1 . Then, as in the proof of the previous lemma, E = FT (T L0 ) = FS˜ (SL ˜ ˜ E1 , E2 = F1 , F2 = F˜ .  Theorem 15.4.29. Assume Hypothesis 15.4.11. Then F ∼ = F4 [m] or E6− [m]. Proof. We show that the tuple F1 , F2 , E1 , E2 satisfies Hypothesis 1.4.7 in [Asc19]. Then we appeal to 1.4.8 in [Asc19] to conclude that E  F. Hence F = F ◦ = E, so the theorem follows from 15.4.28. Hypothesis 1.4.1 in [Asc19] is satisfied by 15.4.25. By 15.4.21, E1  F1 , while by construction in 15.4.23, E2  F2 . Hence condition (1) of 1.4.7 in [Asc19] holds. Next NF (T ) ≤ NF (W ) ≤ NF (U ) = F2 , so as E2  F2 , we have AutF (T ) ≤ Aut(E2 ). Similarly u0 = CU (T ), so as E1  F1 , also AutF (T ) ≤ Aut(E1 ). That is condition (2) of 1.4.7 in [Asc19] holds.

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Condition (3) follows from the definition of E as E1 , E2 , and finally E is saturated by 15.4.28, so condition (4) holds. This completes the verification of Hypothesis 1.4.7 in [Asc19] and hence also the proof of the theorem.  Theorem 15.4.30. Assume τ = (F, Ω) is a quaternion fusion packet such that F = F ◦ and μ(τ ) ∼ = Weyl(D4 ). Assume for z ∈ ZS that |Ω(z)| = 1, and let M ∈ M (τ ) and U = Z(M ). Assume the Extended Inductive Hypothesis. Then Z(F) ≤ U and one of the following holds: (1) U = Z(F) and one of the following holds: (i) W  F. (ii) F ∼ = Spin+ 8 [m]. ˆ + (2), L ˆ 3 (2)/23+6 , or AE ˆ 8. (iii) m = 8 and F ∼ =O 8 (2) Z(F) ∼ = Z2 and either (a) F ∼ = Spin9 [m] or Spin− 10 [m], or ˆ (b) m = 8 and F ∼ AE . = 9 (3) Z(F) = 1 and F ∼ = F4 [m] or E6− [m]. Proof. By assumption, conditions (1) and (4) of Hypothesis 15.1.1 hold, and as μ(τ ) ∼ = W eyl(D4 ), conditions (2) and (3) of that hypothesis are also satisfied. That is Hypothesis 15.1.1 holds, and then also Hypothesis 15.4.1 is satisfied. If U  Fu for each u ∈ U # then by 15.4.6, U ≤ Z(F). Then (1) holds by 15.4.4 applied to any u ∈ U # . Therefore we may assume Hypothesis 15.4.7 is satisfied. If u0 is weakly closed in U with respect to F then by 15.4.10, u0 ∈ Z(F) and then (2) holds by that lemma. Therefore we may assume u0 is not weakly closed, so by 15.4.9, Hypothesis 15.4.11 holds. Now Theorem 15.4.29 says that (3) holds, completing the proof. 

15.5. Φ = An In this section we assume the following hypothesis: Hypothesis 15.5.1. Either (1) τ = (F, Ω) is a quaternion fusion packet with F ◦ = O 2 (F) and τ ◦ is the ¯ Z, ¯ where L ¯ = SLd (q) with (q − 1)2 = m/2, d = n + 1 ≥ 5, and Lie packet of L/ ¯ or Z¯ ≤ Z(L), (2) Hypothesis 15.2.1 holds with F = F ◦ and μ = μ(τ ) ∼ = W eyl(Φ), where Φ is of type An for some n ≥ 4. In addition we adopt Notation 15.1.3, set SM = S ∩ M , M = FSM (M ), and we adopt the following notation: Notation 15.5.2. We adopt a modified version of the Notation 5.4.12 and 5.7.1. Let q be a prime power with (q − 1)2 = m/2, d = n + 1, and U a d¯ = SL(U ) = SLd (q), S¯ ∈ Syl2 (L), ¯ dimensional vector space over F = Fq . Let L ¯ and τ¯ = (F, ¯ Ω) ¯ a Lie packet for L. ¯ Let W ¯ ∈ W (¯ ¯ ∈ M (¯ F¯ = FS¯ (L), τ ) and M τ ). By ¯ → M with ker(π) ≤ Z(M ¯ ). We 4.3.8 there is a surjective homomorphism π : M ¯ on U . use π to study M and τ , via the representation of M

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Let I = {1, . . . , d}, B = {ui : i ∈ I} a basis for U , and Γ = {F ui : i ∈ I}. Let M = NL¯ (Γ) and W the kernel of the action of M on Γ. From 5.4.13 and 5.7.2, ¯ O(M), W ¯ =M ¯ ∩ W, and M ¯ and W ¯ are described in 5.7.2. In particular M=M ¯ is represented as Sym(Γ) ∼ ¯. M = Sym(I) on Γ with kernel W ¯ 0 = Ω1 ( W ¯ ). Lemma 15.5.3. Set W ¯ /W ¯ ∼ ¯ (1) W0 is the core of the permutation module for M = Sym(I). ¯ ¯ ¯ (2) For t ∈ W0 of weight k, dim([U, t ]) = k. (3) ZΔ is the set of vectors eα , α ∈ I2 , of weight 2. Proof. This is essentially 5.7.3.



Notation 15.5.4. Recall from 3.4.11 that E(τ ) = SO(τ ), NF (O(τ )), NF (O(ξ)), NF (W ) : ξ ∈ Ξ , where Ξconsists of those ∅ = ξ ⊆ ZS such that NS (ξ) is transitive on ξ and O(ξ) = s∈ξ O(s) ∈ F f . Further from Definition 11.3.2, W(τ ) consists of those  1 = fθ = t∈θ t for θ ⊆ ZS such that |θ| is a power of 2, unless nS = 4 and m(WS ) = 3, where W(τ ) also includes the involutions in W∗ (τ ). In the latter case by iJ.3.2 we have n = 7 and the members of W∗ (τ ) are the M -conjugates of c0 x0 ¯ ) and x0 = fθ for some 2where c0 ∈ Z(M ) with c¯0 = c0 π −1 of order 4 in Z(M / WS . subset θ of ZS . In particular either c0 = 1 and x = fθ ∈ WS or c0 = 1 and x ∈ Moreover B(τ ) = E(τ ), Fx : x ∈ W(τ ) ∩ F f . Lemma 15.5.5. (1) nS = |ZS | = d/2 if d is even and (d − 1)/2 of d is odd. (2) Let Θ = {{2i − 1, 2i} : 1 ≤ i ≤ nS }. We can choose notation so that ZS = {eθ : θ ∈ Θ}. (3) Let WS = ZS . Then either m(WS ) = nS , or d is even, eI ∈ ker(π), so that fZS = eI π = 1, and m(WS ) = nS − 1. Proof. This follows from 15.5.3.



Lemma 15.5.6. F = B(τ ). Proof. The lemma follows from 11.3.10, once we verify that Hypothesis 11.3.8 holds; note that conclusion (2) of 11.3.10 does not hold as μ ∼ = Weyl(An ). We first observe that Hypothesis 11.2.1 holds; this is immediate, except for 11.2.1.2, which follows from 15.5.5.3. Similarly 11.3.8.2 holds in 15.5.1.1, while 11.3.8.1 holds in 15.5.1.2 by 15.1.1.4.  Lemma 15.5.7. F = NF (W ), Fx : x ∈ W(τ ) ∩ F f . Proof. Set B1 = NF (W ) : Fx : x ∈ W(τ ) ∩ F f . By 15.5.6, it suffices to show that E(τ ) ≤ B1 . For x ∈ W(τ ), x is conjugate under M ≤ NF (W ) ≤ B1 to a member of W(τ ) ∩ F f , so Fx ≤ B1 . Let ξ ∈ Ξ. If fξ = 1 then fξ ∈ W(τ ) and NF (O(ξ)) ≤ CF (fξ ) ≤ B1 . On ¯ 0 is f¯ξ = the other hand if fξ = 1 then d is even, ξ = ZS , and it preimage in W −1U ∈ ker(π). As ξ = ZS , O(ξ) = O(τ ) = O, so it remains to show that SO(τ ) and NF (O) are contained in B1 . Observe that WS ≤ Z(O(τ )) and there exists x ∈ W(τ ) ∩ Z(S) ⊆ CWS (S), so SO(τ ) ≤ Fx ≤ B1 .

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The same argument shows that SCF (WS ) ≤ Fx ≤ B1 . By 1.3.2 in [Asc19], N = NF (O) = SCF (WS ), NN (T ) , where T = CS (WS ). But W ≤ T is weakly closed, so NN (T ) ≤ NF (W ) ≤ B1 , completing the proof.  Lemma 15.5.8. For each involution t ∈ W0 and each t0 ∈ tF ∩ F f , we have t0 ∈ t M ∩ W S . ¯ /W ¯ = ¯ 0 is the core of the permutation module for M Proof. By 15.5.3, W Sym(I). Therefore t = eθ for some θ ⊆ I of even order k. Set βi = {2i − 1, 2i} ∈ Θ; from 15.5.5.3, ti = eβi ∈ ZS . Let β = β1 ∪ · · · ∪ βk/2 ; then s = eβ = t1 · · · tk/2 ∈ ¯ 0 is the core of the permutation module, s ∈ tM , so we may take WS , and as W t = s ∈ WS . In particular, O = O(τ ) ≤ CS (t). Let α ∈ A(t); then W = W α and  O = Oα, so tα ∈ tM and tα ∈ WS , completing the proof. ¯ ) is cyclic of order (d, m/2). In particular Remark 15.5.9. From 5.7.2.6, Z(M ¯ ¯ if d is odd then Z(M ) = 1 so π : M → M is an isomorphism, and we identify ¯ ) = eI = f¯Z , ¯ 0 ∩ Z(M the two groups via this isomorphism. If d is even then W S ¯ which we denote by c¯. Set c = c¯π; if c = 1 then as Z(M ) is cyclic, again π is an isomorphism. Finally if c = 1 then each x ∈ W0 has two preimages; we pick one and call it x ¯; the second preimage is x ¯c¯. Recall the exceptional case from 15.5.4, where n = 7 and m(WS ) = 3. In this ¯ ) be of order 4 and set c0 = c¯0 π. From case c = 1, and, as in 15.5.4 let c¯0 ∈ Z(M 15.5.4, W∗ (τ ) ⊆ W(τ ) consists of the W -conjugates of c0 fθ , where θ ⊆ ZS is of order 2. Recall that if c0 = 1 then c0 fθ = fθ ∈ WS . Lemma 15.5.10. Let t be an involution in W0 ∩ F f , set U (1) = [U, t¯], U (2) = CU (t¯), and ki = dim(Ui ). ¯ 0 , so k1 is even. Further k2 = d − k1 . (1) k1 is the weight of t¯ in W (2) θt = (CM (t)◦ , Ω) is a quaternion fusion packet. ¯ ≤ U (i)}, and (3) θt = θ1 ∗ θ2 , where θi = (Mi , Ωi ), Ωi = {K ∈ Ω : [U, K)] Mi = [Ωi ]CM (t)◦ . (4) μ(θi ) ∼ = Weyl(Aki −1 ), unless k2 ≤ 1 where τ2 = ∅. = Sk i ∼ ¯ 0. Proof. By 15.5.8, t ∈ WS , so t centralizes Ω and t¯ is of even weight k in W By 15.5.3.2, k = k1 , so (1) holds. As t centralizes Ω, (2) follows from 2.5.2. As k is even, k ≥ 2. Now CL¯ (t¯)◦ = L1 ∗L2 , where Li = CL¯ (U (3−i))◦ ∼ = SLki (q), except if k2 ≤ 1 where L2 = 1. Let Si = SL ∩Li and ξi = (FSi , Λi ). As t ∈ W0 ∩F f , ξi is a Lie packet of Li with μ(ξi ) ∼ = Ski . Let Mi ∈ M (ξi ) and Mi = FSi (Mi ). Then CM (t)◦ = M1 ∗ M2 , so we conclude that (3) and (4) hold.  Lemma 15.5.11. Let t be an involution in W0 ∩ F f of weight k1 and set U (1) = ¯ [U, t], U (2) = CU (t¯), and k2 = d − k1 . (1) τt◦ = τ1 ∗ τ2 is a central product of quaternion fusion packets τi = (Yi , Ωi ) ¯ ≤ U (i)}. such that Yi = Yi◦ is transitive on Ωi = {K ∈ Ω : [U, K)] ∼ (2) Mi ∈ M (θi ) is in M (τi ) and μi = μ(τi ) = Ski , unless k2 ≤ 1 where τ2 = ∅. (3) One of the following holds: (i) W (τi )  Yi . (ii) Yi ∼ = SLki [m], or ki = 2 and Yi ∼ = Qm . (1) (iii) ki = 3 and Yi ∼ or G2 [m], or m = 8 and Yi ∼ = L+ = M12 or L3 (2)/E8 . 3 [m] ˆ 7, ˆ 6 , AE (iv) ki = 4 and Yi ∼ [m], or m = 8 and Yi ∼ = Spin7 [m] or Spin− = AE 8 ˆ 3 (2)/E64 , or Sp ˆ 6 (2). L

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Proof. Adopt the notation from 15.5.10. By 15.5.10.3, θt = θ1 ∗ θ2 with μi = μ(θi ) ∼ = Ski , unless k2 ≤ 1, where θ2 = ∅. If 15.5.1.1 holds then the lemma follows from the discussion in section 5.4, so we may assume 15.5.1.2 holds. Set ρ = τt◦ ; by 15.1.5.1, ρ = ρ1 ∗· · ·∗ρs is a central product of quaternion fusion packets ρi = (Yi , Γi ) such that Yi = Yi◦ is transitive on Γi . As Mi is transitive on Ωi , we can choose notation so that Mi ≤ Yi and Ωi ⊆ Γi , for i = 1, 2. If t ∈ Z(M ) then k2 ≤ 1, so θ2 = ∅, M1 = M, and Ω1 = Ω. Thus s = 1 and / Z(M ) ρ = ρ1 = τ1 , so that (1) and (2) hold in this case. On the other hand if t ∈ then t ∈ / Z(F), so by induction on the order of τ , μ(Yi ) is coconnected, and hence μ(ρi ) = μ(θi ) and Mi ∈ M (θi ) is in M (ρi ). As Ω = Ω1 ∪ Ω2 , we conclude that s = 2 (unless Ω = Ω1 where s = 1). This completes the proof of (1) and (2). It remains to prove (3). Suppose first that t ∈ Z(F) and set F + = F/t . By induction on the order of τ , either W +  F + or F + is the image of SLd [m] modulo its central subgroup of order 2. But then, pulling back to F and using 3.3.16, either W  F or F ∼ = SLd [m]. That is (3i) or (3ii) holds in this case. Therefore we may assume that ρ is proper in τ . Then (3) follows from 15.1.5.4 and 15.1.9.2.  Theorem 15.5.12. Assume d is even. Then (1) c ∈ Z(F), and (2) if c = 1 then either F ∼ = SLn+1 [m] or W  F. Proof. If c = 1 then the lemma is trivial, so we may assume c = 1. In particular from 15.5.9 the exceptional case n = 7 and m(WS ) = 3 does not hold. ¯ ), so From 15.5.9, c = eI = fζ , where ζ = ZS , and c¯ is the involution in Z(M c ∈ Z(NF (W ). Let t ∈ W(τ ) ∩ F f with t = c. As the exceptional case does not hold, it follows from 15.5.8 and 15.5.9 that t ∈ WS , and we adopt the notation of 15.5.11. Then c = t · s, where s is the involution in Z(Y2 ). Now either Y2  Ft , so s ∈ Z(Ft ), or t is of weight d/2 and t = s. But in the latter case c = 1, so that (1) holds; thus we may assume the former holds, so that c = ts ∈ Z(Ft ). We’ve shown that c ∈ Z(Ft ) for each t ∈ W(τ ) ∩ F f . Therefore by 15.5.7, c ∈ Z(F), completing the proof of (1). Then (2) follows from 15.5.11 (and its proof) applied to c in the role of t.  Notation 15.5.13. By 15.5.11, τz◦ = τK ∗ τ  where τK = (O(z), {K}) and τ = (T , Ω − {K}) with μ(τ  ) ∼ = Sd−2 . 

Lemma 15.5.14. Assume n = 7, m(WS ) = 3, and c0 = 1. (1) τc◦0 = (Y, Ω) is a quaternion fusion packet with μ(τc◦0 ) ∼ = S8 . (2) c0 ∈ Z(S). ¯ (3) Either W  Y or Y is the 2-fusion system of L/−1 U . +  ∼ (4) Either W (τ )  T or T = SL6 [m]. Further W  Y iff W (τ  )  T . (5) There is a 2-subset θ of ZS with f = fθ ∈ Z(S). Set x = c0 f . (6) c0 is in the center of Ff and Fx . (7) If c0 ∈ Z(Fz ) then c0 ∈ Z(F). Proof. From 15.5.9, c0 is the involution in Z(M ), so (1) and (2) hold. In ¯ while in 15.5.1.2, (3) follows from (1) 15.5.1.1, (3), (6), and (7) follow as c¯0 ∈ Z(L), and 15.1.5, unless c0 ∈ Z(F). But in the latter case, passing to F/c0 , inducting on the order of τ , and appealing to 3.3.16, again (3) follows.

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From 15.5.9, c = 1, so (5) follows. The first sentence in (4) follows from 15.5.11.3. Suppose Y is the 2-fusion ◦ ∼ ¯ system of L/−1 U , and let X = CY (K) . Then X = SL6 [m] and X ≤ T , so X = T . Conversely assume T ∼ = SL6 + [m] and write f = zt for some t ∈ ZS . Then CT (f )◦ = O(z) ∗ O(t) ∗ E with E ∼ = SL+ 4 [m]. In a moment we will use this fact to show W is not normal in Y, completing the proof of (4). Next, using 15.5.11.2, we have M (τf ) = M (τx ) = M1 × M2 , with μ(Mi ) ∼ = S4 and Z(Mi ) = ci ∼ = Z4 , where f = c21 and c0 = c1 c2 . From 15.5.11.1, τf◦ = τ1 ∗ τ2 with τi = (Yi , Ωi ) and Mi ∈ M (τi ). Some member of S interchanges τ1 and τ2 , so Y1 ∼ = Y2 . By 15.5.11.3, one of the following holds: (a) W (τi )  Yi . (b) Yi ∼ = SL+ 4 [m]. ˆ 6, L ˆ 3 (2)/E64 , or Sp ˆ 6 (2). (c) m = 8 and Yi is AE ∼ Here we eliminate the possibilities Yi = Spin7 [m] or Spin− 8 [m] using f ∈ ˆ 6 as Z(Yi ). In case (c), z, t ∈ ZS (τ1 ), so Y2 ≤ CF (O(z)O(t))◦ = E, so Y2 ∼ = AE + ◦ ∼ E = SL4 [m]. But similarly E ≤ CF (f ) , so E ≤ Y2 , a contradiction. Therefore (a) or (b) holds. Further if T ∼ = SL+ 6 [m] then E ≤ CF (O(z)O(t)) = Y2 , so (b) holds. Then as c0 = c1 c2 , we have c0 ∈ Z(Ff ), so c0 centralizes E, and hence W is not normal in Y, completing the proof of (4). Notice we’ve also shown that c0 ∈ Z(Ff ). Recall (6) and (7) hold in 15.5.1.1, so we may assume 15.5.1.2 holds. As c0 ∈ Z(Ff ) we have Ff ≤ Fx . By 15.1.5 and 15.1.9.2, τx◦ = ρ1 ∗ ρ2 with ρi = (Xi , Ωi ), Mi ∈ M (ρi ), Yi ≤ Xi , and either Yi = Xi or Xi appears in 15.5.11.3.iv. As x = c0 f ∈ Z(M1 M2 ), we have x ∈ Z(X1 X2 ), and then arguing as above, Xi = Yi , completing the proof of (6).  Finally if c0 ∈ Z(Fz ) then c0 ∈ Z(F) by 15.5.7 and (6), establishing (7). Lemma 15.5.15. Let t be an involution in W0 ∩ F f and for i = 1, 2, define τi = (Yi , Ωi ) and ki as in aG.11. Assume W (τ  )  T . Then for i = 1, 2, W (τi )  Yi . Proof. Assume W  = W (τ  )  T but Wi = W (τi ) is not normal in Yi for some i. Suppose first that J ∈ Ω3−i and let j = z(J) and α ∈ A(j) with jα = z. Then X = Yi α∗ ≤ Fz , and as Yi is coconnected, it follows that either X ≤ O(z) or X ≤ T , while as jα = z and J ∈ / Ωi , it must be the latter. Therefore X ≤ T , so W (τi )α ≤ W   T and hence W (τi )  Yi , contrary to assumption. Therefore we may assume Ω3−i = ∅, so Ω = Ωi and hence i = 1 and either k1 = d is even or d is odd and k1 = d − 1. In the first case t = c, and then 15.5.12.2 says that either F ∼ = SLd [m], contradicting W   T , or W  F, where the lemma holds. Therefore we may assume that the second case holds. Now t ∈ Z(S), so z ∈ Ftf and CY1 (z)◦ = O(z) ∗ C, where ρ = (C, Ω − {K}) is a quaternion fusion packet with μ(ρ) ∼ = Sd−3 . Observe C ≤ T , so W (ρ) ≤ W  , and  then as W  T , also W (ρ)  C. Then as W (τ1 ) is not normal in Y1 and d ≥ 5, we conclude from 15.5.11.3 that d = 5, m = 8, and Y1 appears in 15.5.11.3.iv. Set O = O(τ ), R = OCS (O), and let GR be a model for NF (R). As d = 5, Ω = {K, K0 } is of order 2, and from 15.5.11.3.iv, there exists X of order 3 in the normal submodel of GR for NY1 (R) with CO (X) = Z(O). But as W   T , X acts on W  ∩ K0 of index 2 in K0 , contradicting CK0 (X) = Z(K0 ). The proof is complete.  Theorem 15.5.16. If W (τ  )  T then W  F.

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Proof. Assume W  = W (τ  )  T . Let r ∈ z F ∩ W  with r ∈ Fzf , and let α ∈ A(r) with rα = z. By 15.5.15, Wz = W (τz )  Fz◦ , so Wz  Fz . As W ≤ CS (r), we have W α = W . Then as Wz  Fz , we have Wr = Wz α−1  CFz (r). Therefore W = Wz Wr  CFz (r). However r ∈ W  , so CFz (Wz ) ≤ CFz (r), and hence W  CFz (Wz ). Thus W ≤ O2 (Fz ), so as W is weakly closed, we have W  Fz . Now choose t as in 15.5.11 and adopt the notation of that lemma. By 15.5.15, Wt = W (Ft◦ )  Ft . Let s ∈ z F ∩ Ftf with s ∈ ZS (τt ), and let β ∈ A(s) with sβ = z. As above, Ws = Wz β −1  CFt (s). Then W = Wt Ws  CFt (Wt ), so as above, W  Ft . Now, if the exceptional case in 15.5.14 does not hold, then W(τ ) ⊆ WS , so by 15.5.7, F ≤ NF (W ), completing the proof. So assume the exceptional case holds and adopt the notation of 15.5.14; in this case we must also show that Fx ≤ NF (W ). But W (τ  )  T , so by 15.5.14.4, W  Fc0 , and by 15.5.14.6. Fx ≤ Fc0 , completing the proof.  Theorem 15.5.17. Assume n = 7, m(WS ) = 3, and c0 = 1. Then c0 ∈ Z(F). ¯ Proof. By 15.5.14.3 either W  Y or Y is the 2-fusion system of L/−1 U . In the first case by 15.5.14.4 and Theorem 15.5.16, we have W  F, so the lemma holds as c0 is the involution in M . Thus we may assume Y is the 2-fusion system of + ∼ ¯ L/−1 U . Hence by 15.5.14.4, T = SL6 [m]. By 15.5.14.7, it suffices to show that c0 ∈ Z(Fz ). Set W  = W (τ  ). Applying 15.5.7 to T is the role of F, we have T = CT (f ), NT (W  ) . By 15.5.14.6, Ff ≤ Fc0 . As W  is selfcentralizing in T , we have W = CW T (W  )  NW T (W  ), so NT (W  ) ≤ Fc0 . Therefore T centralizes c0 . By 1.3.2 in [Asc19], Fz = CS (z)T , NFz (T ) for T Sylow in T . Thus it remains to show NFz (T ) ≤ Fc0 . But NFz (T ) ≤ CFz (f ) ≤ Fc0 by 15.5.14.6, completing the proof.  In light of theorems 15.5.12, 15.5.16, and 15.5.17, during the remainder of the section we assume: Hypothesis 15.5.18. Hypothesis 15.5.1 holds. Further (1) W (τ  ) is not normal in T . (2) If n is odd then eI = 1. (3) If n = 7 then c0 = 1. Lemma 15.5.19. Suppose m > 8, t is in the setup of 15.5.11, and for some i with ki ≥ 3, we have Yi ∼ = SL2 [m]. = SLki [m]. Then O(z) ∼ Proof. Let J ∈ Ωi and j = z(J). As Yi ∼ = SLki [m] with ki ≥ 3, it follows [m], so the lemma follows.  that O(j) ∼ SL = 2 Lemma 15.5.20. Assume d ≥ 7. Then (1) T ∼ = SLd−2 [m]. (2) Assume the setup of 15.5.11 with ki ≥ 3. Then Yi ∼ = SLki [m]. Proof. Part (1) follows from 15.5.18.1 and 15.5.11.3. Assume the hypothesis of (2). Suppose first that J ∈ Ω3−i , and choose J so that j = z(J) ∈ Ftf . Let α ∈ A(j) with jα = z. Then Yi α∗ is a coconnected

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∼ Yi α ∗ ∼ component of CT (tα)◦ , so by (1), Yi = = SLki [m]. Thus we may assume Ω3−i = ∅, so Ω = Ωi and ki = d or d − 1. Indeed t ∈ Z(Yi ), so ki is even, and hence by aG.16, d is odd and ki = d − 1. Hence as d ≥ 7 we have ki ≥ 6, so (2) follows from 15.5.11.  Lemma 15.5.21. Assume d = 6. Then (1) z M = W0# , and (2) T ∼ = SL+ 4 [m]. Proof. By 15.5.18.2, eI = 1 in W0 , so W0 is a natural module for M/W ∼ = S6 . This implies (1). Assume (2) fails. Then by 15.5.18.1 and 15.5.11.3, T appears in case (iv) of the latter lemma. As d = 6, Ω = {K1 , K2 , K3 } is of order 3. Set zi = z(Ki ) and choose z = z1 . Let α ∈ A(z2 ) with z2 α = z. Set O = O(τ ), D = O 2 (CT (O)), and B = Dα∗ . Then O = Oα, so B ≤ CF (O). ∼ Suppose first that T ∼ = Spin7 [m] or Spin− 8 [m]. Then D = SL2 [m] or SL2 [2m], respectively, with z = z(D). Therefore z(B) = zα = z, so B = D. But B is a component or solvable component of CFz (zα), so by 2.2.6.4, either B centralizes T or the Sylow group B of B is contained in the Sylow group Tz of T . As zα = z(B) does not centralize T , we conclude that B ≤ Tz , so B ≤ CTz (O) = Dz2 , where D is Sylow in D. But then z(B) = z(D), contradicting z = zα. In the remaining cases, m = 8. Let R = OCS (O) and GR a model for NF (R). ˆ 7 Then GR is transitive on Ω. Let H  CGR (z) be the model for NT (R). If T ∼ = AE ∼ then D = SL2 (3) with z(D) = z, and then the argument in the previous paragraph ˆ 6 or L ˆ 3 (2)/E64 . Then H = Tz X where supplies a contradiction. Suppose T ∼ = AE ∼ X = Z3 with CO (X) = Kz2 . As GR is transitive on Ω, there is a GR -conjugate X2 of X with CO (X2 ) = K2 z . But by 5.10.13.3, R admits no such group of ˆ 6 or L ˆ 3 (2)/E64 . automorphisms when T is AE ˆ 6 (2). Set Q1 = [CT (O), O 2 (H)]; then This leaves the case where T ∼ = Sp z Q1 ∼ = Q8 with z(Q1 ) = z1 = z. Further Q1  CGR (z) with CH (z) irreducible on Q1 /z , so [R, Q1 ] ≤ z . Let Qi = Qg1i for gi ∈ GR with z g = zi for i = 2, 3.  R Then [Q1 , Qi ] ≤ z ∩ zi = 1, so Q = QG 1 = Q1 ∗ Q2 ∗ Q3 . Let t ∈ Z(τ ) t with Q2 = Q3 ; then [Q2 , t] ≤ Q2 Q3 ∩ Tz ≤ CTz (O) = Q1 z2 , so [Q2 , t] = Q1 z2 ,  whereas Q2 Q3 ∩ Q1 = z . Lemma 15.5.22. Assume d = 5. Then Ω = {K, K0 } is of order 2, and: (1) M has two orbits on W0# with representatives z and t = zz(K0 ). (2) T ∼ = L+ 3 [m]. ◦ ∼ (3) Ft = SL+ 4 [m]. Proof. As d = 5, Ω = {K, K0 } is of order 2; set z0 = z(K0 ) and t = zz0 . Then t ∈ Z(S), so t ∈ F f . By 15.5.3.1 and 15.5.9, W0 is the core of the permutation module for M/W ∼ = Sym(I) ∼ = S5 , so (1) holds. By 15.5.11, τt◦ = (Y, Ω) and either (3) holds or Y appears in 15.5.11.3.iv. Similarly either (2) holds or T appears in 15.5.11.3.iii. Set O = O(τ ) = KK0 and D = O 2 (CY (O)). (1) Suppose T ∼ or G2 [m]. Then C = O 2 (CT (O)) ∼ = L+ = SL2 [m] is a com3 [m] ponent or solvable component of CF (z, t ) = CFt (z0 ) with z0 = z(D). By 2.2.6.4, either C centralizes Y or the Sylow group T of C is contained in the Sylow group Tt of Y. The former is impossible as z0 does not centralize Y. Therefore T ≤ Tt , so

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T ≤ CTt (O), so by inspection of the list in 15.5.11.3.iv, Y is Spin7 [m], Spin− 8 [m], ˆ ˆ ˆ AE7 , L3 (2)/E64 , or Sp6 (2), and z0 = z(T ) = t, a contradiction. Hence by 15.5.11.3, either T is L+ 3 [m] or m = 8 and T is M12 or L3 (2)/E8 . In particular if m > 8 then (2) holds. ∼ ˆ Suppose Y is Spin7 [m], Spin− 8 [m], or AE7 . Then D = SL2 [m] or SL[2m] with z(D) = t and D is a component or solvable component of CF (z, t ) = CFz (t). Therefore by 2.2.6.4, either D centralizes T or the Sylow group D of D is contained in the Sylow group Tz of T . As t does not centralize T , we conclude that D ≤ Tz , contradicting t ∈ / Tz . In particular if m > 8 then Y ∼ = SL+ 4 [m], and hence the ˆ lemma holds. Thus we may assume m = 8. Also by 15.5.11.3, Y is SL+ 4 [8], AE6 , ˆ 3 (2)/E64 , or Sp ˆ 6 (2). L Set R = OCS (O) and let GR be a model for NF (R) and H  GR a model for NY (R). Observe there is X of order 3 in GR induced in T such that X centralizes K and XK0 ∼ = SL2 (3). By 5.10.13.3, R admits no such group of automorphisms ˆ 3 (2)/E64 , so Y is SL+ [8] or Sp ˆ 6 (2). Assume the latter and set ˆ when Y is AE6 or L 4 2 Q = [CTt (O), O (H)]. As in the proof of 15.5.21, Q ∼ = Q8 with t = z(Q). As t does not centralize T , Q is faithful on T . It follows that T is M12 or L3 (2)/E8 and Q ≤ K1 × CS (T ) with K1 = CTz (O) ∼ = Q8 and z(K1 ) = z0 . But we can choose Y of order 3 in H centralizing K0 with Q = [Q, Y ], whereas by 5.10.13.3, R admits no such group of automorphisms when T is M12 or L3 (2)/E8 . Therefore Y ∼ = SL+ 4 [8], completing the proof of (3). Moreover we may assume that (2) fails, so T ∼ M = 12 or L3 (2)/E8 . Now H contains A of order 3 with [R, A] = K0 . Again by 5.10.13.3, R admits no such group of automorphisms when Y is M12 or L3 (2)/E8 . This finally completes the proof of the lemma.  Lemma 15.5.23. Assume the setup of 15.5.11 with ki ≥ 3. Then Yi ∼ = SLki [m]. Proof. If d ≥ 7 this is a consequence of 15.5.20.2, while if d = 6 it follows from 15.5.21 and if d = 5 it follows from 15.5.22.  Lemma 15.5.24. Let ZS = {zi : 1 ≤ i ≤ nS }. Then for 1 ≤ i ≤ nS there exists SL2 [m] ∼ = Oi = Ozi ≤ F with Sylow group O(zi ) such that (1) Oi = O(zi ) if m > 8. (2) Oτ = O1 ∗ · · · ∗ OnS  NF (WS ) and NF (WS ) permutes Λ = {Oi : 1 ≤ i ≤ nS } as Sym(Λ). (3) Oz  Fz . Proof. If m > 8 then O(zi ) ∼ = SL2 [m] by 15.5.19, and the remainder of the lemma follows from 2.6.9 and 3.3.7. Therefore (1) holds and we may assume that m = 8. Let O = O(τ ) and G0 be a model for N = NF (OCS (O)). Choose notation so that z = znS . By 15.5.23, T ∼ = SLd−2 [8], so there is a model H for NFz◦ (E) normal in CG0 (z) such that H = K ∗ H0 and H0 = H1 × · · · × Hk where k = nS − 1, O(zi ) = O2 (Hi ), and Hi ∼ = SL2 (3). As G0 acts transitively on ZS as Sym(ZS ) and K  CG0 (z), it follows that there is SL2 (3) ∼ = HnS ≤ G0 with K = O2 (HnS ) centralizing H0 , and G0 permuting Γ = {Hi : 1 ≤ i ≤ nS } as Sym(Γ). This establishes (1) and (2) with Oi = FO(zi ) (Hi ). Let X ∈ Syl3 (HnS ), T Sylow in T , W1 = W (τ  ), and M1 = M (τ  ). Observe that W is in the kernel of the action of G0 on Γ and [HnS , W1 ] ≤ CHnS (K) = z , so W1 centralizes HnS = O 2 (HnS ). Then X acts on M1 with [M1 , X] ≤ CM1 (W1 ) =

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W1 ≤ CS (X), so X centralizes the Sylow group T of M1 and T . Therefore by 9.5 in [Asc11], X centralizes T . We’ve shown that Oz ≤ C = CFz (T ). Then C centralizes WS , so by (2), we have Oz  C. Now as K  Fz , we conclude from 7.4 in [Asc11] that (3) holds, completing the proof.  Lemma 15.5.25. Assume the setup of 15.5.11 with k1 = d/2. Then (1) W  NFt (W (τt )). (2) Ft = Xi , NFt (W ) : i = 1, 2 , where Xi = CS (t)Yi , unless t = z and m = 8 where X1 = CS (z)Oz . (3) Suppose for i = 1, 2 that Xi = CS (t)Oτi , NXi (W (τi )) . Then Ft ≤ CS (t)Oτ , NFt (W ) . Proof. As k1 = d/2, we have Yi  Ft for i = 1, 2. Let St = CS (t), Ti Sylow in Yi , Wi = W (τi ), and Wt = W (τt ). Then T = T1 T2 is Sylow in Ft◦ and Wt = W1 W2 . By 1.3.2 in [Asc19], Ft = X1 , NFt (T1 ) and NFt (T1 ) = X2 , NFt (T ) . But unless t = z and m = 8, Wt is weakly closed in T , and in the exceptional case NFt (T ) = X1 , NFt (Wt ) , so: (a) Ft = Xi , NFt (Wt ) : i = 1, 2 . Hence (1) implies (2). Further for i = 1, 2, St Oτi ≤ St Oτ and as W3−i  NXi (Wi ), we have NXi (Wi ) ≤ NFt (Wt ), so (1) and (2) imply (3). Therefore it remains to prove (1). Pick r ∈ Z(τt ) ∩ Ftf and let α ∈ A(r) with rα = z. If t = z pick r ∈ Z(τ  ). Set Wz = W (τz ). As in the proof of 15.5.16, Wr = Wz α−1 ≤ W with W = Wr Wt . Let C = CFt (Ft◦ ) and P = CS (Ft◦ ). Then Cα∗ centralizes Oτ , so by 15.5.23 and x.y, Cα∗ ≤ CFz (Fz◦ )P α, so C ≤ CFr (Wr )P . Therefore W = Wr Wt ≤ O2 (W C) and hence: (b) W  St C. Let Mt be the 2-fusion system of M (τt ). Then Mt  NFt (Wt ), so by 1.3.2 in [Asc19], NFt (Wt ) = St Mt , NFt (T ) as T is Sylow in Mt by 5.3.5. Now W ≤ CSt Mt (Wt )  St Mt , so as W is weakly closed in St we have W  St Mt . Finally NFt (T ) = St C as NAut(L1 L2 ) (T ) = RO(NAut(L1 L2 ) (T )), where R ∈ Syl2 (Aut(L1 L2 )) and Li ∼ = SLki (q) tamely realizes Yi . Therefore W  NFt (Wt ) by (b), so (1) follows, completing the proof.  Lemma 15.5.26. Let X consist of those x ∈ W(τ ) which are not of weight d/2. Assume (a) For each t ∈ X ∩ F t , and for each i = 1, 2, Xi = CS (t)Yi = CS (t)Oτi , NXi (W (τi )) . (b) If d is a power of 2 and t ∈ W(τ ) ∩ Z(S) is of weight d/2 then X = SFt◦ ≤ CX (x), NX (W (τt )), SOτ : x ∈ X . Then F = SOτ , NF (W ) . Proof. Set A = SOτ , NF (W ) . By (a) and 15.5.25.3, for each x ∈ X ∩ F f , we have Fx ≤ A. Then as NF (W ) ≤ A and M controls fusion in W , for each x ∈ X we have Fx ≤ A. Next suppose d is a power of 2 and let t ∈ W(x) ∩ F f be of weight d/2. Then t ∈ Z(S). Let Wt ∈ W (τt ), T Sylow in Ft◦ , and Mt the 2-fusion system of Mt ∈ M (τt ). By 1.3.2 in [Asc19], Ft = X , NFt (Wt ) . Then the proof of 15.5.25.3 shows that W  NFt (Wt ), so NFt (T ) ≤ NF (W ) ≤ A. Next NX (Wt ) = SMt

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and W ≤ CS (Wt ) ≤ O2 (SMt ), so W  NX (Wt ). Finally we’ve shown that CX (x) ≤ A for each x ∈ X, so it follows from (b) that X ≤ A. But now by 15.5.7, F = NF (W ), Fx : x ∈ W(τ ) ∩ F f ≤ A, completing the proof. 

15.6. Generating linear systems In this section we assume the following hypothesis: Hypothesis 15.6.1. τ = (F, Ω) is a quaternion fusion packet such that F ◦ = O (F) and either ¯ Z, ¯ where L ¯ ∼ (I) F ◦ is isomorphic to the 2-fusion system of L/ = SLd (q) with ¯ ¯ (q − 1)2 = m/2, d ≥ 5, and Z ≤ Z(L), or ¯ i ) for L ¯i ∼ (II) d ≥ 8 is a power of 2 and F ◦ = F1 ∗ F2 where Fi = FT¯i (L = ¯ ¯ SLd/2 (q) and Ti ∈ Syl2 (Li ), and Z(F1 ) ∩ Z(F2 ) = 1. Further the Sylow group S of F is transitive on {F1 , F2 }. 2

Notation 15.6.2. Suppose for the moment that (I) holds; then we adopt No¯ In particular L ¯ = SL(U ) for U a d-dimensional vector tation 15.5.2 in discussing L. space over Fq . ¯ = SL(U ) Suppose on the other hand that (II) holds. We continue to take L and choose a direct sum decomposition U = U1 ⊕ U2 with dim(Ui ) = d/2. We ¯1 × L ¯ 2 ) = F¯1 × F¯2 , with F¯i = FT¯ (L ¯ i ), ¯ i = CL¯ (U3−i ). Let F¯ ◦ = FT¯ (L take L i ◦ ◦ ¯ ¯ ¯ ¯ ¯ ¯ Ti ∈ Syl2 (Li ), and T = T1 T2 . Then F is a central factor system of F determined ¯1 × L ¯ 2. by a central factor group of L As in 15.5.11, τ = τ1 ∗ τ2 , where τi = (Fi , Ωi ) and Ωi = {K : [U, K] ≤ Ui }. In either case let W ∈ W (τ ) and M ∈ M (τ ). In case (II), W = W1 ∗ W2 and M = M1 ∗ M2 , where Wi ∈ W (τi ) and Mi ∈ M (τi ). Define Oτ as in 15.5.24; more precisely in (II), Oτ = Oτ1 ∗ Oτ2 . ¯ ), and W0 is the image of W ¯ 0 in the Sylow group T ¯ 0 = Ω1 ( W As in 15.5.3, W ◦ ¯ 0 has a subgroup of of F . Thus in (II), W0 = W0,1 ∗ W0,2 , and we can view W ¯ W ∈ W (¯ τ ) from (I), and calculate weights in the latter module. In (II) define X0 to consist of those involutions x ∈ W(τ ) such that x is of weight wt(x) a power of 2 and either wt(x) = d/2 or CS (x) ≤ NS (F1 ). In this section we prove: Theorem 15.6.3. Assume Hypothesis 15.6.1 and adopt notion 15.6.2. Set A0 = SOτ , NF (W ) S . In case (I) set A = A0 , and in case (II) set A = A0 , CF (x) : x ∈ X0 S . Then F = A. Assume the theorem is false and pick a counter example with τ of minimal order. ¯ so F ◦ ∼ Lemma 15.6.4. Assume (I) holds. Then Z¯ = Z(L), = L+ d [m].

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Proof. Assume otherwise and set F + = F/c where c is the involution in Z(F ◦ ). Then τ + = (F + , Ω+ ) is a quaternion fusion packet satisfying Hypothesis 15.6.1, so by minimality of τ , F + = S + Oτ + , NF + (W + ) , where W + ∈ W (τ + ). But now F = A using 2.1.15, contrary to the choice of τ as a counterexample.  Lemma 15.6.5. Let t ∈ W(τ ). Then either (1) Ft ≤ A, or (2) d is a power of 2, wt(t) = d/2, and in case (II), CS (t) does not act on F1 . Proof. Assume that neither (1) nor (2) holds. Assume first that (I) holds. By 15.5.8 and as NF (W ) ≤ A, we may assume t ∈ F f . Adopt the notation of 15.5.11. As (2) fails, Yi  Ft for i = 1, 2. If ki = 3 then the hypothesis of 15.5.25.3 is satisfied by 15.5.7, while if ki = 4 it holds by 9.3.20. If ki ≥ 5 it holds by minimality of τ . Hence Ft ≤ A by 15.5.25.3, contrary to assumption. Therefore (II) holds. By definition of A, wt(t) = d/2 and CS (t) does not act on F1 , so t satisfies (2), again contrary to assumption.  Lemma 15.6.6. Assume (I) holds. Then (1) d is a power of 2. (2) Let f ∈ Z(S) ∩ W0 be of weight d/2. Then Ft ≤ A. Proof. By 15.5.7, F = NF (W ), Fx : x ∈ W(τ ) ∩ F f . By definition of A, NF (W ) ≤ A, and by 15.6.5, and 15.5.14.6 when d = 8, for each x ∈ W(τ ), either Fx ≤ A or d is a power of 2 and we may choose x ∈ W0 with wt(x) = d/2. In particular if d is not a power of 2 then F = A, contrary to the choice of τ as a counter example. Therefore (1) holds. Similarly (using 15.5.14.6 when d = 8) there is x ∈ W(τ ) ∩ F f of weight d/2 such that Fx ≤ A. But as d is a power of 2 and x ∈ Fx , we have x ∈ Z(S), so (2) holds.  Lemma 15.6.7. Case (II) holds. Proof. Assume instead that (I) holds and choose f as in 15.6.6.2. Then by 15.5.11 and 15.6.4, Ff = SFf◦ satisfies Hypothesis 15.6.1.II. Therefore (II) holds by minimality of τ .  Notation 15.6.8. We will abuse notation and work in SL(U ) rather than in the projective group P SL(U ). Let θ1 = {z ∈ ZS : [U, z] ≤ U1 }. Now Π1 = {θ1 , ZS −θ1 } is an S-invariant partition of ZS . Proceeding recursively, assume Πk is an S-invariant partition of ZS into 2k blocks of size n/2k . Pick θk ∈ Πk and let θk+1 ⊆ θk be of order n/2k+1 such that S , so that Πk+1 is an S-invariant |NS (θk ) : NS (θk+1 )| = 2, and set Πk+1 = θk+1 k+1 k+1 partition of ZS into 2 blocks of size n/2 . Continuing this process, we arrive at θr = {z} for some z ∈ ZS ; that is n = 2r . Let Πk = {θk,1 , . . . , θk,2k } and Uk,i = [U, fθk,i ]. Thus  U = Uk,1 ⊕· · ·⊕Uk,2k and we set Uk = {Uk,1 , . . . , Uk,2k }, Lk,i = CUk,i and Lk = i Lk,i the direct product of ¯ of U k . 2k copies of SL2r−k (q). Let U k = {U1 , . . . , Uk } and Gk the stabilizer in S L Set Dk = FS (Gk ). Lemma 15.6.9. Gk = SLk O(Gk ). Proof. By construction, SLk ≤ Gk and Lk is contained in the kernel X of the k action of Gk on Uk . By 2.1.16, GU k is contained in a Sylow 2-group of Sym(Uk ).

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For x ∈ X of odd order, x|Uk,i ∈ Lk,i O(Gk ) as Lk,i acts faithfully as SL(Uk,i ) on Uk,i . Therefore X/Lk O(Gk ) is a 2-group, completing the proof.  ∼ SL2 (q) has Sylow 2-group O(si ) for that si ∈ ZS Lemma 15.6.10. (1) Lr,i = such that si ∈ Z(Lr,i ) and Ut,i = [U, si ]. (2) Oτ = FO(τ ) (Lr ). (3) Dr = FS (Gr ) = SOτ ≤ A. Proof. By construction, Ur,i = [U, si ] for some si ∈ ZS , and then (1) follows. In particular by (1), O(τ ) is Sylow in Lr and (2) holds. By 15.6.9, Gr = SLr O(Gr ),  so FS (Gr ) = FS (SLr ) = SOτ , establishing (3). Lemma 15.6.11. E(τ ) ≤ Y. Proof. From 15.5.4, E = E(τ ) = NF (O(ξ)), SOτ , NF (O), NF (W ) : ξ ∈ Ξ , where Ξ consists of those ξ ⊆ ZS with O(ξ) ∈ F f and NS (O(ξ)) transitive on ξ. Now NF (O) ≤ SOτ , NF (W ) ≤ A. Thus it remains to show that N = NF (O(ξ)) ≤ A for ξ ∈ Ξ. But N ≤ Fx , where x = fξ , and by definition of A, Fx ≤ Y unless ξ = θ1 and U1 = [U, x], so we may the latter holds. Now N = FS (X), where X = NL1,1 (O(ξ))L1,2 S. Therefore N = FS (SL1,2 ), NN (S2 ) , where S2 is Sylow in L1,2 . By minimality of τ , FS (SL1,2 ) ≤ A. Further NN (S2 ) ≤ NN (O), which is contained in A by the previous case.  Notation 15.6.12. Observe that D1 = F, so D1 ≤ A. On the other hand by 15.6.10.3, Dr ≤ Y. Thus there exists a least k such that Dk ≤ A. Set D = Dk , . . . , 2k } and ρ = (D, Ω), let S0 = S ∩ Lk , and write D0 for FS0 (Lk ). Let I = {1, for i ∈I set Si = S ∩ Lk,i and Li = FSi (Lk,i ). From 15.6.8, Lk = i∈I Lk,i , so D0 = i∈I Li is a direct product. Set Ωi = Ω ∩ Si and ρi = (Li , Ωi ). Observe: Lemma 15.6.13. The pair D, D0 satisfies Hypotheses 11.4.1 and 11.4.6. Notation 15.6.14. As in 11.4.2 and 11.4.6, for X ≤ S and 0 ≤ i ≤ 2k , write Xi for X ∩ Si . As D ≤ Y, it follows from the Alperin-Goldschmidt Fusion Theorem that the set R of subgroups R ∈ D e such that AutD (R) ≤ AutY (R) is nonempty. Pick R ∈ R and let G be a model for ND (R) and set G∗ = G/R. By 15.6.13 and 11.4.4.1, R0 ∈ F f rc , so there is a model G0 for ND0 (R0 ). By 11.4.4, we can embed G and G0 in a certain group Gc with G0  Gc and G = NGc (R). Thus we can form H0 = NG0 (R) and by 11.4.4, H0  G with R0 = O2 (H0 ) and H0∗ ∼ = H0 /R0 . Set Q = NR (Lk,1 ) and X = FQS1 (QLk,1 ). Let 1 ∈ Σ be a set of coset representatives for Q in R. Write R∗ for the set of those R ∈ R of maximal order. Lemma 15.6.15. For each K ∈ Ω, |K ∩ R| ≤ 2. Proof. Assume otherwise; then from 3.4.5, AutD (R) = AutE(ρ) (R). But E(ρ) ≤ E(τ ) and by 15.6.11, E(τ ) ≤ A, contrary to the choice of k.  Lemma 15.6.16. (1) For each i ∈ I, Ri = Si . (2) R is transitive on {Lk,i : i ∈ I}. (3) Q ∈ X e . Let GQ be a model for NX (Q). (4) GQ /Q ∼ = H0∗ .

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 (5) σLet E1 = (ZS (ρ1 )) and define α : W1 ∩ Z(Q) → WS ∩ R by α : x → k σ∈Σ x . Then xα ∈ Z(R) and if x is of weight p in E1 then xα is of weight 2 p in WS . (6) If p = |ZS (ρ1 )| then CD (xα) ≤ Y. Proof. If Ri = Si then Ωi ⊆ R, contrary to 15.6.15. Thus (1) holds. By 15.6.13, we can appeal to results in section 11.4. In particular (2) follows from (1) and 11.4.8.1, (3) follows from 11.4.8.4, and (4) follows from 11.4.8.5. From 15.6.14, 1 ∈ Σ is a set ofcoset representatives for Q in R. If σi ∈ Σ with R1 σi = Ri then R0 = i∈I Ri = i∈I R1σi , so for x ∈ E1 ∩ Z(Q) of weight p, xα centralizes Q, Σ = R and xα is of weight |Σ|p = |I|p = 2k p in WS . Thus (5) holds. Assume the hypothesis of (6) and set y = xα. Observe that |ZS (ρ1 )| = n/2k = r−k 2 . As p = |ZS (ρ1 )|, y = 1 so y is an involution. Then by definition of A, either Fy ≤ A or 2k p = d/2 = 2r−1 , and we may assume the latter, so that p = 2r−k−1 . R ∪ (θk − θk+1 )R . Then conjugating in A, we may take x = fθk+1 , so Πk+1 = θk+1 But then CSLk (y) = CS (y)Lk+1 , so as Dk+1 ≤ Y, again CD (y) ≤ Y.  Lemma 15.6.17. ρ1 = (L1 , Ω1 ) satisfies Hypothesis 11.2.1 and if R ∈ R∗ then ρ1 , Q satisfies Hypothesis 11.2.7. Proof. Visibly ρ1 satisfies Hypothesis 11.2.1 with |Ω(z)| = 1. So assume R ∈ R∗ . By 15.6.16.4, Q ∈ Le1 . From 11.2.6, W(Q) consists of those involutions x ∈ Z(Q) ∩ E1 such that x = fα for some α ⊆ ZS (ρ1 ) of weight a power of 2. Passing to L1 /Z(L1 ) if necessary we may assume fZS (ρ1 ) = 1. Hence by 15.6.16.6, for each x ∈ W(Q), CD (xα) ≤ A. Let x ∈ W(Q), y = xα and Z = y G . As R ∈ R∗ and CD (y) ≤ Y, the proof of  11.3.5.1 shows that O 2 (CG (Z)) = R. Set Zx = xGQ . Then Z is a full diagonal   σ subgroup of σ∈Σ Zx and the isomorphism of 15.6.16.4 maps O 2 (CGQ (Zx ))/Q to   O 2 (CG (Z))/R = 1, so O 2 (CGQ (Zx )) = Q. Hence ρ1 , Q satisfies 11.2.7.1. Similarly if ρ, R satisfies the setup of 11.2.7.2 and U = ∅ then the proof of 11.3.5.2 shows that m2 (G∗ ) = 1 and U is not normal in G∗ . Let B ∗ be the subgroup of G∗ generated by its involutions. By 11.4.5.2, B ∗ is the subgroup of H ∗ generated by its involutions. Thus the isomorphism of 15.6.16.4 maps the subgroup ∗ ∗ BQ of G∗Q = GQ /Q generated by involutions to B ∗ , so m2 (G∗Q ) = 1 and UQ = BQ .  Therefore ρ1 , Q satisfies 11.2.7.2. This completes the proof of the lemma. We are now in a position to obtain a contradiction, completing the proof of Theorem 15.6.3. Namely choose R ∈ R∗ . By 15.6.17, the pair ρ1 , Q satisfies Hypotheses 11.2.1 and 11.2.7. Further condition (a) of Theorem 11.2.33 is satisfied, as is condition (b) by 15.6.15. Condition (c) of 11.2.33 holds as X = FQS1 (SLk,1 ). But then by Theorem 11.2.33, |ZS (ρ1 )| = 1, a contradiction. This completes the proof of Theorem 15.6.3. 15.7. Wrapping up Φ = An In this section (until the final theorem) we assume Hypothesis 15.5.18 and adopt the notation from section 15.5. We complete our treatment of the case Φ = An . Define Oτ as in 15.5.24.

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Lemma 15.7.1. F = SOτ , NF (W ) . Proof. This follows from 15.5.26 and 15.6.3. We verify condition (a) of 15.5.26 using aH.3 in case (I) applied to Xi for i = 1, 2. By 15.5.23, Yi ∼ = SLki [m]. Then 15.6.3 implies (a) as long as ki ≥ 5. When ki = 2, condition (a) is trivial, while if ki = 3 or 4 then (a) follows from 15.5.7 or 9.3.20, respectively. Similarly condition (b) of 15.5.26 follows from an application of 15.6.3 in case (II).  Lemma 15.7.2. If Z(M ) = 1 then F is an image of SL+ d [m] modulo some central subgroup. Proof. Assume Z(M ) = 1 and let x be the involution in Z(M ). Then x is in the center of NF (W ) and SOτ , so x ∈ Z(F) by 15.7.1. Set F + = F/x , and observe that, using 6.6.9.2, τ + = (F + , Ω+ ) satisfies Hypothesis 15.5.18. Therefore by induction on the order of τ , F + is a central factor system of SL+ d [m], so by 3.3.16, F is also such a factor system.  Because of 15.7.2, in the remainder of the section (until the final theorem) we assume: Hypothesis 15.7.3. Hypothesis 15.5.18 holds with Z(M ) = 1. Notation 15.7.4. Set F˜ = L+ d [m], T = SM , F1 = SOτ , E1 = T Oτ , F2 = ˜ of NF (W ), E1 = FT (M ), and E = E1 , E2 T . By 5.4.13, the Lie packet τ˜ = (F˜ , Ω) Ld (q) satisfies Hypothesis 15.7.3 for suitable q with (q − 1)2 = m/2, and by 5.2.5, ˜ ∈ M (˜ M τ ) contains the Sylow group S˜ of F˜ . As Z(M ) = 1, we conclude from 4.3.8 ˜ . Therefore, identifying M with M ˜ , we may take M = M ˜ , T = S, ˜ etc. that M = M ˜ Subject to this convention, E2 = FS˜ (M ). ˜ τ˜ . Lemma 15.7.5. (1) E1 = SO ˜ (2) E = F . Proof. By Theorem 2.7.3, T Oτ is uniquely determined by T and Ω, so as ˜ we conclude that (1) holds. From 15.7.4, E2 = F ˜ (M ˜ ), so by T = S˜ and Ω = Ω, S ˜ ) . Finally F˜ = SO ˜ τ˜ , F ˜ (M ˜ ) by 15.7.1 applied to ˜ τ˜ , F ˜ (M (1) we have E = SO S S ˜ ˜ ˜ F, since NF˜ (W ) = FS˜ (M ).  Lemma 15.7.6. For i = 1, 2, Ei  Fi . Proof. By construction, E2  F2 . As T is Sylow in E2  F2 , we have T  S,  so E1 = T Oτ  SOτ = F2 by Theorem 5 in [Asc11]. Theorem 15.7.7. Assume Hypothesis 15.7.3. Then F ∼ = L+ d [m]. Proof. We show that the tuple F1 , F2 , E1 , E2 satisfies Hypothesis 1.4.7 in [Asc19]. Then we appeal to 1.4.8 in [Asc19] to conclude that E  F. Hence F = F ◦ = E, so the theorem follows from aI.5.2. Hypothesis 1.4.1 in [Asc19] is satisfied by 15.7.1. By 15.7.6, Ei  Fi for i = 1, 2. Hence condition (1) of 1.4.7 in [Asc19] holds. Next NF (T ) ≤ NF (W ) = F2 , so as E2  F2 , we have AutF (T ) ≤ Aut(E2 ). Similarly, setting E = ZS , Oτ  NF (E), so AutF (T ) ≤ Aut(E1 ). That is condition (2) of 1.4.7 in [Asc19] holds.

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Condition (3) follows from the definition of E as E1 , E2 , and finally E is saturated by 15.7.5.2, so condition (4) holds. This completes the verification of Hypothesis 1.4.7 in [Asc19] and hence also the proof of the theorem.  Theorem 15.7.8. Assume τ = (F, Ω) is a quaternion fusion packet such that F = F ◦ and μ(τ ) ∼ = Weyl(An ) for some n ≥ 4. Assume the Extended Inductive Hypothesis is satisfied. Then for z ∈ ZS we have |Ω(z)| = 1, and either (1) W ∈ W (τ ) is normal in F, or (2) F is a central factor system of SL+ n+1 [m]. Proof. Let d = n + 1, W ∈ W (τ ), and M ∈ M (τ ). Then μ = M/W ∼ = Sd . Let K ∈ Ω and z = z(K). Then a = dK is a transposition in μ and D = aμ is the set of all transpositions. As μ ∼ = Sd with d ≥ 5, Va = {a}, so Ω(z) = {K} as for each J ∈ Ω(z), dJ ∈ Va . Therefore 15.1.1.1 is satisfied. By hypothesis the Extended Inductive Hypothesis is satisfied, so 15.1.1.4 holds, and as μ(τ ) ∼ = Sn+1 , conditions (2) and (3) of 15.1.1 hold. Hence Hypothesis 15.1.1 is satisfied, so as μ∼ = Weyl(An ), also Hypothesis 15.5.1 holds. Define c ∈ W0 as in 15.5.9. If c = 1 then conclusion (2) of our theorem holds by 15.5.12.2, so we may assume c = 1. Similarly if W (τ  )  T then conclusion (1) of our theorem holds by 15.5.16, so we may assume otherwise. Hence Hypothesis 15.5.18 is satisfied. By 15.7.2 we may assume that Z(M ) = 1. Therefore Hypothesis 15.7.3 holds, so conclusion (2) of our theorem holds by Theorem 15.7.7. This completes the proof. 

15.8. Φ = En In this section we assume the following hypothesis: Hypothesis 15.8.1. Either ˜7 [m], (1) τ = (F, Ω) is the Lie packet with F isomorphic to E6+ [m], E7 [m], E or E8 [m], or (2) Hypothesis 15.2.1 holds with F = F ◦ and μ = μ(τ ) ∼ = Weyl(Φ), where Φ is of type En for some n ∈ {6, 7, 8}. Notation 15.8.2. We adopt Notation 15.1.3, and as usual let SM be Sylow in M. Lemma 15.8.3. Assume n = 7. Then (1) μ ∼ = Z2 × Sp6 (2). (2) Let d = dK∩W . Then Cμ (d) = d⊥ = d × Dd , with Dd ∼ = W eyl(D6 ). (3) μ(τz ) ∼ = Weyl(A1 D6 ). (4) W0 /Z(M ) is centralized by Z(μ) and is the natural module for μ/Z(μ) ∼ = Sp6 (2). ˜7 [m]. (5) If Z(M ) = 1 then Z(M ) = Z(F) and either W  F or F ∼ =E ∼ W eyl(E7 ), so (1) holds. Let σ generate Z(μ) and Proof. By hypothesis, μ = ν = E(μ). Then d = σ · δ, where δ ∈ ν is a transvection, so Cν (d) = Cν (δ) ∼ = W eyl(D6 ). Further for a ∈ Dd , a = σα for some transvection α ∈ Cν (d), so Dd ∼ = Cν (d) with Dd ∩ Cν (d) = Cμ (d)∞ of index two in each of the two subgroups. As

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δ ∈ Cμ (d)∞ , it follows that σ ∈ d⊥ , and then that (2) holds. Then (2) implies (3). ¯ be the universal group ω Let M ¯ (E7 , m) defined in 4.3.1. By 4.3.8 there is a ¯ → M with ker(π) ≤ Z(M ¯ ) = Z. ¯ By 4.3.7.2, W ¯ 0 is surjective homomorphism π : M ¯ ) of order 2. Then as the natural module of rank 7, and by 5.5.5 there is Z¯0 ≤ Z(M for Sp6 (2) is its only nontrivial 6-dimensional module, we conclude that Z¯ = Z¯0 and (4) holds. Finally suppose Z(M ) = 1. Then π is an isomorphism and we identify M with ¯ via that map. Let c generate Z = Z(M ). From (4), M has three orbits on M W0# with representatives c, z, cz. As c = Z(M ), c ∈ Z(NF (W )). We show c is in the center of Ft for t ∈ {z, cz}, so that Hypothesis 14.1.11 is satisfied and then c ∈ Z(F) by 14.1.18. Set F + = F/c . By 6.6.9.2 and induction on the order of τ , either W +  F + or F + ∼ = E7 [m]; then (5) follows from 3.3.16. It remains to show Ft ≤ Fc . In 15.8.1.1 this follows as c ∈ Z(F), so assume 15.8.1.2 holds. By (3), μ(τt ) ∼ = Weyl(A1 D6 ), so by 15.1.5 and 15.1.9, τt◦ = τ1 ∗ τ2 , where τ1 = (T1 , K) and τ2 = (T2 , Ω − {K}) with μ(τ1 ) ∼ = Z2 and μ(τ2 ) ∼ = Weyl(D6 ). Let M2 ∈ M (τ2 ). Then M2 = ω ¯ (D6 , m) is the universal group with Z(M2 ) = z, c . If Z(M2 ) = Z(T2 ) then as c is weakly closed in W , we have c ∈ Z(Ft ). Thus we may assume Z(T2 ) = t , so T2 is not by 15.1.9, T2 ∼ = Spin13 [m], Spin− 14 [m], or ˆ AE13 , But then, as in the proof of 11.5.17.2, c is fused to ct in T2 , contradicting c weakly closed in W .  In light of 15.8.3.5, during the remainder of the section we assume: Hypothesis 15.8.4. Hypothesis 15.8.1 holds with Z(M ) = 1 if n = 7. ˆ + (2), for n = 6, 7, or 8, Lemma 15.8.5. (1) μ ∼ = O6− (2), Z2 × Sp6 (2), or O 8 respectively. (2) Z(μ) centralizes W0 and W0 is the natural module for μ/Z(μ). (3) If n = 7 then M is transitive on W0# . (4) If n = 6 or 8 then M has two orbits on W0# with representatives z and t, with z nonsingular and t singular in the orthogonal space W0 . (5) If n = 6 then μ(τz ) ∼ = Weyl(A1 A5 ) and μ(τt ) ∼ = Weyl(D5 ). (6) If n = 8 then μ(τz ) ∼ = Weyl(A1 E7 ) and μ(τt ) ∼ = Weyl(D8 ). (7) If n = 7 then F = NF (W ), Fz . (8) If n = 6 or 8 then F = NF (W ), Fz , Ft . Proof. As μ ∼ = Weyl(En ), (1) holds. Suppose n = 7. Then Z(M ) = 1 by 15.8.4, so (2) and (3) follow from parts (3) and (4) of 15.8.3. Now nS = 7 is odd (cf. 15.8.6.2), so F = E(τ ) by 3.4.6. By 3.4.11, E(τ ) = SO(τ ), NF (O(ξ), NF (W ) : ξ ∈ Ξ , where Ξ consists of the subsets ξ of ZS with NS (ξ) transitive on ξ and O(ξ) ∈ F f . Now SO(τ ) ≤ Fz , and from 15.8.6.2, NF (O(ξ)) ≤ Fz if ξ ∈ Ξ is of order 1 or 2. Finally if |ξ| = 4 then NF (O(ξ)) ≤ NF (WS ) = N , and by 1.3.2 in [Asc19], N = SCF (WS ), NN (T ) where T is Sylow in CF (WS ). But SCF (WS ) ≤ Fz and NF (T ) ≤ NF (W ), so (7) holds. Therefore we may assume n is 6 or 8. ¯ = μ/Z(μ) ∼ Let d = dK∩W ∈ D ⊆ μ. By (1), μ = On (2) where (n, ) = (6, −1) or (8, +1). By 3.1.20.3, d¯ induces a transvection on W0 with center z. By 4.3.7.2,

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¯ 0 ) = n, so as the natural module is the only nontrivial n-dimensional module m(W for μ ¯, it follows that (2) holds. Further as z is the center of d, z is nonsingular in the orthogonal space W0 ; then (4) follows from (2). As d¯ is a transvection in μ ¯, we conclude that Cμ (d) ∼ = Z2 × S6 , E4 × Sp6 (2) for n = 6, 8, respectively, so μ(τz ) ∼ = Weyl(A1 Σ) for Σ = A5 , E7 , respectively. Similarly by (2) and (4), Cμ (t) is the parabolic S5 /E16 or S8 /E27 , so μ(τt ) is Weyl(D5 ) or Weyl(D8 ), completing the proof of (5) and (6). If remains to prove (8). If n = 8 then (8) follows from (4) and 14.2.4, so we may assume n = 6. Then m(WS ) = 3 by 15.8.5.3, so Hypothesis 11.3.8 is satisfied, and then (8) follows from Theorem 11.3.10.  Lemma 15.8.6. (1) WS is a maximal totally isotropic subspace of the symplectic or orthogonal space W0 , so m(WS ) = m(W0 )/2. (2) If n = 7 then ZS = WS# , so nS = |ZS | = 7. (3) If n = 6 or 8 then ZS = WS −E0 , where E0 is the totally singular hyperplane of WS , so  nS = 4 or 8. (4) s∈ZS s = 1. Proof. For a, b ∈ ZΔ , b ∈ D(a) iff [dO(a)∩W , dO(b)∩W ] = 1 iff a is orthogonal to b in W0 . Therefore WS is totally isotropic, so m(WS ) ≤ m(W0 )/2. As nS is the width of the 3-transposition group μ, we have nS = 4, 7, 8, respectively, so  m(WS ) ≥ m(W0 )/2. The lemma follows. Lemma 15.8.7. (1) τz◦ = τK ∗ τ  is a central product of packets, where τK = (O(z), K) and τ  = (T , Ω − {K}) with μ(τ  ) ∼ = Weyl(Φ ) and Φ is A5 , D6 , E7 for n = 6, 7, 8, respectively. (2) Z(T ) = z . (3) One of the following holds: (i) W  ∈ W (τ  ) is normal in T . (ii) n = 6 and T ∼ = SL+ 6 [m]. (iii) n = 7 and T ∼ = HSpin+ 12 [m]. ˜7 [m]. (iv) n = 8 and T ∼ =E Proof. By 15.8.3 and 15.8.5, μ(τz ) ∼ = Weyl(A1 Φ ), where Φ is A5 , D6 , or E7 . Then in 15.8.1.2, (1) follows from 15.1.5 and 15.1.9, while in 15.8.1.1 it follows from 5.5.3.1.  By 15.8.6.4, z = p where p = z =z∈ZS s. Therefore z ∈ T , so z ∈ Z(T ). Let M2 ∈ M (τ  ). If n = 7 then from the proof of 15.8.3.5 and 15.8.4, we have Z(M2 ) = z , so (2) holds in this case. If n = 6 or 8 then by 5.7.2.6 and 15.8.3.4, ¯ 2) ∼ Z(M = Z2 , so again (2) holds. In 15.8.1.1, (3) follows from 5.5.3.1, so assume 15.8.1.2 holds. Assume (3i) does not hold. If n = 6 or 8 then (3) holds by (1), (2), and 15.1.5.4, so take n = 7. Then + − by 15.1.9.2, T is Spin+ 12 [m], HSpin12 [m], Spin13 [m], or Spin14 [m], or m = 8 and ˆ 12 , H AE ˆ 12 , or AE ˆ 13 . As Z(M2 ) ∼ T is AE = Z2 , we conclude that T is HSpin+ 12 [m] ˆ or H AE12 , and we may assume the latter and it remains to derive a contradiction. Let z = z1 ∈ ZS and E1 = z, z1 ; by 15.8.6.2, z2 = zz1 ∈ ZS . Let Ki ∈ Ω(zi ). ˆ 12 . Now Fz◦ is constrained, so it has a model Gz and Gz = K ∗ H where H ∼ = H AE By (2), z = Z(H), so there is X of order 3 in H with CKK1 K2 (X) = KE1 . Let R = O2 (Gz )CS (O2 (Gz )) and G0 a model for NF (R). By 15.8.5, AutF (E1 ) = GL(E1 ), so there is X1 of order 3 in G0 with CKK1 K2 (X1 ) = K1 E1 . Then X1 acts on Gz , a

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427

contradiction as H has no group X1 of automorphisms with CK1 K2 (X1 ) = K1 E1 . This completes the proof of (3) and the lemma.  Lemma 15.8.8. Assume n = 6 or 8. Then (1) τt◦ = (Y, Ω) with μ(τt◦ ) ∼ = Weyl(Dk ) and (n, k) = (6, 5) or (8, 8). (2) t = u(τt◦ ) ∈ Z(Y). (3) One of the following holds: (i) Wx ∈ W (τx ) is normal in Fx for x ∈ {z, t}. + ∼ (ii) n = 6, T ∼ = SL+ 6 [m], and Y = Spin10 [m]. ∼ ∼ ˜ (iii) n = 8, T = E7 [m], and Y = HSpin+ 16 [m]. Proof. Part (1) follows from 15.8.5. In 15.8.1.1, (2) and (3) follow from 5.5.6, augmented by arguments in the proof of 15.1.6. Therefore we may assume 15.8.1.2 holds. By 15.8.6, for each s ∈ ZS , st ∈ ZS , so t = u(τt◦ ) by 15.1.6.1. Thus (2) holds. By 15.1.9.2, one of the following holds: (a) Wt ∈ W (τt ) is normal in Y. + − (b) Y ∼ = Spin+ 2k [m], HSpin2k [m], Spin2k+1 [m], or Spin2k+2 [m]. ˆ 2k , or AE ˆ 2k+1 . ˆ 2k , H AE (c) m = 8 and Y is AE Let X = CF (z, t )◦ and ξ = (X , Ω). Then ξ is a quaternion fusion packet such that ξ = CFz◦ (t)◦ , so from 15.8.7.3, ξ = τK ∗ τ0 ∗ ζ, where t = zz0 with z0 ∈ ZS , K0 ∈ Ω(z0 ), τ0 = (O(z0 ), K0 ), and ζ = (D, Ω − {K, K0 }) with μ(ζ) = Weyl(Σ) and Σ = A3 or D6 for n = 6 or 8, respectively. In particular if 15.8.7.3.i does not hold + ◦ then D ∼ = SL+ 4 [m] or Spin12 [m], respectively, so as ξ is the packet for CY (z) , we + + conclude that (b) holds with Y ∼ = Spin2k [m] or HSpin2k [m]. If n = 6 then k = 5 is odd, so the half spin system does not exist, and hence (3ii) holds in this case. If n = 8 then from 15.8.6, WS ∼ = HSpin+ = E16 , so Y ∼ 16 [m] and therefore (3iii) holds in this case. Therefore we may assume that 15.8.7.3.i holds, so W   T . Let J ∈ Ω − {K} and j = z(J); then J ∩ W = J ∩ W   CG (WS ). But if α ∈ A(z0 ) with z0 α = z then W α = W and WS α = WS with Kα ∈ Ω − {K}, so also K ∩ W  CF (WS ). Hence (a) holds, completing the proof.  Theorem 15.8.9. If Wz ∈ W (τz ) is normal in Fz◦ then W  F. Proof. The proof is much the same as that of 15.5.16. If n = 6 or 8 let {x, y} = {z, t} and define Wx as in 15.8.8.3.i, so that Wx  Fx . If n = 7 let (x, y) = (z, z0 ) with z = z0 ∈ ZS , let α ∈ A(y) with yα = x, and let Wy = Wx α−1 . Then Wx  Fx and Wy  CFx (y). Observe next that, in any event, W = Wx Wy and Wy  CFx (y). Hence as y ∈ Wx , we have W = Wx Wy  CFx (Wx ), so W ≤ O2 (CFx (Wx )) and therefore W  Fx . Then by parts (7) and (8) of 15.8.5, W  NF (W ), Fr : r ∈ {x, y} = F, completing the proof.  Given Theorem 15.8.9, in the remainder of the section we assume: Hypothesis 15.8.10. Hypothesis 15.8.1 holds, Wz ∈ W (τz ) is not normal in Fz , and if n = 7 then Z(M ) = 1.

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Lemma 15.8.11. Let ZS = {zi : 1 ≤ i ≤ nS }. Then for 1 ≤ i ≤ nS there exists SL2 [m] ∼ = Oi = Ozi ≤ F with Sylow group O(zi ) such that (1) Oi = O(zi ) if m > 8. (2) Oτ = O1 ∗ · · · ∗ OnS  NF (WS ) and NF (WS ) is transitive on Λ = {Oi : 1 ≤ i ≤ nS }. (3) Oz  Fz . Proof. The proof is much like that of 15.5.24, and as such is omitted.



Lemma 15.8.12. Let x ∈ {z, t}. Then (1) W  NFx (Wx ). (2) Fx = CS (x)Oτ , NFx (W ) . (3) In aJ.1.1, F = SOτ , FS (M ) . Proof. The proof of (1) and (2) is much like that of 15.5.25. Define x and y as in the proof of 15.8.9, and let Sx = CS (x), X = Sx Fx◦ if x = z, X = Sz Oz T if x = z, and T Sylow in Fx◦ . By 1.3.2 in [Asc19], Fx = X , NFx (T ) , so as Wx is weakly closed in T unless x = z and m = 8, we conclude: (a) Fx = X , NFx (Wx ) . We next observe that: (b) X = Sx Oτ , NX (Wx ) . + + + ∼ ˜ For T ∼ = SL+ 6 [m], HSpin12 [m], or E7 [m], while Y = Spin10 [m] or HSpin16 [m]. Therefore (b) follows from 11.5.3 if T or Y is a spin system, from 15.6.3 if T ∼ = ∼ ˜7 [m]. [m], and by induction on the order of τ if T E SL+ = 6 Observe that (1), (a), and (b) imply (2), so it remains to prove (1). As in the proof of 15.8.9, W = Wx Wy . Let C = CFx (Fx◦ ); as in the proof of 15.5.25: (c) W  Sx C. Let Mt be the 2-fusion system of M (τx ); as in the proof of 15.5.25, NFx (Wx ) = Sx Mx , NFx (T ) , with W  Sx Mx and NFx (T ) = Sx C, so (1) follows from (c). Finally we prove (3), so assume that 15.8.1.1 holds. We show F = G, where G = NF (W ), Fz . Then as NF (W ) = FS (M ) and as Fz = SOτ , NFz (W ) by (2), the lemma holds. If n = 7 then F = G by 15.8.5.7, so take n = 6 or 8. Then F = G, Ft by 15.8.5.8, so it suffices to show that Ft ≤ G. But this follows from (2).  Notation 15.8.13. Set F˜ = En [m], or more precisely E6+ [m] if n = 6. Set T = SM , F1 = SOτ , E1 = T Oτ , F2 = NF (W ), E2 = FT (M ), and E = E1 , E2 T . ˜ Ω) ˜ of En (q) satisfies Hypothesis 15.8.10 By 5.5.2 and 5.5.3, the Lie packet τ˜ = (F, ˜ ∈ M (˜ for suitable q, and by 5.2.9 and 5.2.10, M τ ) contains the Sylow group S˜ of ˜ We conclude from 4.3.8 that M = M ˜ . Therefore, identifying M with M ˜ , we F. ˜ , T = S, ˜ etc. Subject to this convention, E2 = F ˜ (M ˜ ). may take M = M S ˜ τ˜ . Lemma 15.8.14. (1) E1 = SO ˜ (2) E = F . (3) For i = 1, 2, Ei  Fi . Proof. See the proofs of 15.7.5 and 15.7.6.



Theorem 15.8.15. Assume Hypothesis 15.8.10. Then F ∼ = En [m] if n = 7 or 8 and F ∼ = E6+ [m] if n = 6.

15.8. Φ = En

Proof. See the proof of 15.7.7.

429



Theorem 15.8.16. Assume τ = (F, Ω) is a quaternion fusion packet such that F = F ◦ and μ(τ ) ∼ = Weyl(En ) for some n ∈ {6, 7, 8}. Assume the Extended Inductive Hypothesis holds. Then for z ∈ ZS we have |Ω(z)| = 1, and either (1) W ∈ W (τ ) is normal in F, or ˜7 [m], or E8 [m]. (2) F ∼ = E6+ [m], E7 [m], E Proof. Let K ∈ Ω and z = z(K). Arguing as in the proof of 15.7.8, Hypothesis 15.8.1 is satisfied. If n = 7 and Z(M ) = 1 then the theorem follows from 15.8.3.5, so we may assume otherwise. Therefore Hypothesis 15.8.4 holds. Similarly if Wz  Fz◦ then (1) holds by 15.8.9, so we may assume Hypothesis 15.8.10 is satisfied. Now Theorem 15.8.15 completes the proof. 

Theorem 1 and the Main Theorem

CHAPTER 16

Proofs of four theorems 16.1. The proof of Theorem 1 In this section we prove Theorem 1. Thus we assume the hypotheses of that theorem: τ = (F, Ω) is a quaternion fusion packet such that F = F ◦ is transitive / O2 (F). on Ω. Let K ∈ Ω and z = z(K) ∈ F f . Set μ = μ(τ ). We may assume z ∈ Further, proceeding by induction on the order of τ : Lemma 16.1.1. τ satisfies the Extended Inductive Hypothesis. Lemma 16.1.2. Assume F is isomorphic to FS (G) for some group G of Lie ˜ Ω) ˜ be the Lie packet of type and odd characteristic. Set F˜ = FS (G) and let τ˜ = (F, ˜ so τ is isomorphic G. Then there exists an isomorphism δ : F → F˜ with Ωδ = Ω, to a Lie packet. Proof. Assume first that μ = Weyl(Φ), where Φ = An for some n ≥ 2, or ˜ ∈ M (˜ Φ = Dn for some n ≥ 4, or Φ is E6 , E7 , or E8 . Let M ∈ M (τ ), M τ ), and ¯ M =ω ¯ (Φ, m). Sift through the proof below for the lemma (Lemma X) where F ∼ = F˜ arises; the isomorphism is established in a theorem (Theorem Y) proven earlier in ¯ /U ¯ ∼ ˜ for the paper. Inspecting the proof of Theorem Y, we find that M ∼ =M =M ¯ ≤ Z(M ¯ ). Now M ¯ /U ¯ is a model for a system M where ρ = (M, Γ) is a certain U the quaternion fusion packet described in 4.3.7.5. We have isomorphisms α : M → ¯ /U ¯ and β : M ˜ →M ¯ /U ¯ such that Ωα = Γ = Ωβ. ˜ ˜ is M Hence γ = αβ −1 : M → M ˜ Then the proof of Theorem Y shows that γ extends an isomorphism with Ωγ = Ω. ˜ and of course Ωδ = Ω. ˜ to an isomorphism δ : F → F, So assume μ is not Weyl(Φ). Then from the lemmas below, μ is elementary abelian. Hence from 16.1.3, and as μ is abelian, we get |Ω(z)| = 1 or 2. Suppose |Ω(z)| = 2. Then as μ is abelian, we conclude from 16.1.8 that F is P Sp4 [m] or L− 4 [m]. But then Z(S) = z and Ω is the pair of Sylow groups of the factors O 2 (Fz ) ∼ = SL2 [m] ∗ SL2 [m]. So we may take |Ω(z)| = 1. Then F is one of the conclusions in 16.1.11. Hence F makes its appearence in Theorem 13.6.6. Tracing through the proof of that theorem, we find that in lemmas 13.2.21 and 13.2.23, an isomorphism δ : S → S˜ is ˜ and δ is shown to define an iomorphism δ; F → F˜ in constructed mapping Ω to Ω, 13.6.2.  Lemma 16.1.3. If |Ω(z)| > 2 then Ω = Ω(z) is of order 4 and F ∼ = Ω+ 8 (2) or

P Ω+ 8 [m].

Proof. This is a consequence of 16.1.1, Theorem 10.2.13, and the assumption  that z ∈ / O2 (F). 433

434

16. PROOFS OF FOUR THEOREMS

By 16.1.3 we may assume: Lemma 16.1.4. |Ω(z)| = 1 or 2. Lemma 16.1.5. A(τ ) = ∅. Proof. Assume otherwise. By Theorem 7.2.3, as F = F ◦ , F is not transitive on Ω, contrary to the hypothesis of Theorem 1.  Lemma 16.1.6. D is a set of 3-transpositions of μ. Proof. This follows from 16.1.5 and 4.2.10.3.



Lemma 16.1.7. Assume (1) |Ω(z)| = 2, and (2) Z ∩ O(z) = {z}. Then μ ∼ = S4 and F ∼ = L+ 4 [m]. Proof. We first observe that Hypothesis 9.3.1 is satisfied. For example A(τ ) = ∅ by 16.1.5 and 9.3.1.4 holds by 16.1.1. Now appeal to Theorem 9.3.24 to complete the proof.  Lemma 16.1.8. Assume |Ω(z)| = 2 and D∗ (z) = D(z). Then F ∼ = P Sp4 [m] or

L− 4 [m].

Proof. Observe that Hypothesis 9.4.1 is satisfied and appeal to Theorem 9.4.33.  Lemma 16.1.9. Assume |Ω(z)| = 2 and D∗ (z) = D(z). Then μ ∼ = W eyl(Dn ) for some n ≥ 3 and one of the following holds: − (1) F ∼ = (P )Ω+ 2n [m], Ω2n+1 [m], or Ω2n+2 [m]. + ∼ ˇ (2) (n, m) = (4, 8) and F = Ω8 (2). (3) (n, m) = (3, 8) and F ∼ = Sp6 (2). Proof. If Z ∩ O(z) = {z} then the lemma follows from 16.1.7 since A3 = + D3 and L+ 4 [m] = P Ω6 [m]. Therefore we may assume that Z ∩ O(z) = {z}, so Hypothesis 12.3.1 is satisfied. Now the lemma follows from Theorem 12.3.50.  Lemma 16.1.10. We may assume that Ω(z) = {K} is of order one. Proof. Assume otherwise; then by 16.1.4, |Ω(z)| = 2. We will show that one of the conclusions of Theorem 1 holds in this case: namely we will show that either ˇ + (2) or Sp6 (2). If D∗ (z) = D(z) this follows from 16.1.8 τ is a Lie packet or F is Ω 8  and 16.1.2, while if D∗ (z) = D(z), it follows from 16.1.9 and 16.1.2. Lemma 16.1.11. Assume μ is abelian. Then F ∼ = (P )Sp2n [m], (P )SL− 2n [m], (1) or L2 [2m] .

L− 2n+1 [m],

Proof. Using 16.1.1 and 16.1.10, Hypothesis 13.1.1 is satisfied. Then Theorem 13.6.6 completes the proof.  Lemma 16.1.12. We may assume that D∗ (z) = D(z). Proof. If not, μ is abelian, so Theorem 1 holds by 16.1.11 and 16.1.2.



16.2. PROOFS OF THE MAIN THEOREM AND THEOREMS 6, 7, AND 8

435

Lemma 16.1.13. We may assume that Hypothesis 15.2.1 holds with μ isomorphic to Weyl(Φ) where Φ is An for some n ≥ 4, Dn for some n ≥ 3, or En for some n ∈ {6, 7, 8}. Proof. By 16.1.1, 16.1.10, and 16.1.12, Hypothesis 15.2.1 holds. Then by Theorem 15.2.29, μ ∼ = Weyl(Φ) where Φ is An , Dn , or En . As μ is nonabelian, Φ is not A1 . If Φ is A2 then Theorem 1 holds by Remark 15.2.30. Similarly as μ is nonabelian we have n ≥ 3 if Φ is Dn . As D3 = A3 , we may take n ≥ 4 if Φ is An . Therefore the lemma holds.  Lemma 16.1.14. If Φ = Dn for some n ≥ 3 then Theorem 1 holds. Proof. If n ≥ 5 this follows from Theorem 15.3.10. If n = 4 it follows from Theorem 15.4.30. Finally if n = 3 it follows from Theorem 15.3.14.  Lemma 16.1.15. If Φ = An for some n ≥ 4 then Theorem 1 holds. Proof. See Theorem 15.7.8.



Lemma 16.1.16. If Φ = En for some n ∈ {6, 7, 8} then Theorem 1 holds. Proof. This is a consequence of Theorem 15.8.16.



Observe that we have completed the proof of Theorem 1. Namely by 16.1.13, μ = Weyl(Φ) where Φ is An for some n ≥ 4, Dn for some n ≥ 3, or En for some n ∈ {6, 7, 8}. In each case, Theorem 1 holds by one of 16.1.14-16.1.16. 16.2. Proofs of the Main Theorem and Theorems 6, 7, and 8 In this short section we complete the proofs of the Main Theorem and of Theorems 6, 7, and 8. First, Theorem 6 follows from Theorem 1 and the discussion in section 12.4. Namely from section 12.4, Theorem 6 holds under the extra assumption that the Extended Inductive Hypothesis holds. But now that we have established Theorem 1 in section 16.1, this extra assumption is indeed satisfied, completing the proof of Theorem 6. Similarly Theorem 7 follows from Theorem 1 and Remark 13.6.7. We next prove Theorem 8, so assume the hypothesis of that theorem. We claim that Hypothesis 15.2.1 holds. By the hypothesis of Theorem 8, parts (1) and (2) of 15.1.1 hold, and as μ is nonabelian, 15.1.1.3 holds. By Theorem 1, 15.1.1.4 holds, completing the proof of the claim. By the claim and the proof of 16.1.13, the conclusion of 16.1.13 holds. Then, as in the proof of 16.1.14-16.1.16, Theorem 8 holds. Finally here is the proof of the Main Theorem. Assume the hypothesis of that theorem. Then there is an involution z in S and a subnormal subsystem E of Fz isomorphic to SL2 [m] for some m such that z is contained in the Sylow group K of E. Then E is a component, or a solvable component if m = 8, of Fz , so, setting Ω = K F , τ = (F, Ω) is a quaternion fusion packet by 2.4.1. As Ω = K F , F is transitive on Ω, and by the hypothesis of the Main Theorem, F = [K]F , so F = F ◦ .

436

16. PROOFS OF FOUR THEOREMS

Therefore τ satisfies the hypothesis of Theorem 1, so by that theorem, τ satisfies one of its conclusions. Suppose that z ∈ Z(F). Then E is subnormal in Fz , so as F = F ◦ is transitive on Ω, it follows that F = E. Hence τ is the Lie packet of SL2 (q) for suitable q, so the Main Theorem holds in this case. Therefore we may assume that z ∈ / Z(F). Thus if z ∈ O2 (F) then the hypotheses of Theorem 2 are satisfied, so τ satisfies one of its conclusions. This contradicts SL2 [m] ∼ = E subnormal in Fz . Therefore z ∈ / O2 (F), so τ satisfies one of conclusions (2), (3), or (4) of Theorem 1. Now if conclusion (2) holds then so does the Main Theorem, so we may assume conclusion (3) or (4) holds. But this contradicts SL2 [m] ∼ = E subnormal in Fz , and this contradiction completes the proof of the Main Theorem.

16.3. Lie fusion packets In this last section we tie up a loose end. From Theorem 1, if F is the 2fusion system of a group G of Lie type and odd characteristic, and τ = (F, Ω) is a quaternion fusion packet with F = F ◦ transitive on Ω, then τ is isomorphic to a Lie fusion packet. But is τ the Lie packet for G? The main result of this section shows that this is almost always the case. Theorem 16.3.1. Let τ = (F, Ω) be a Lie fusion packet. Then (1) F is tamely realized by some group G of Lie type and odd characteristic. (2) Let ρ = (F, Γ) be the Lie packet of G. Then there exist α ∈ Aut(F) with Γα = Ω. (3) There exists β ∈ Aut(G) with α = β|S . (4) Either Ω = Γ or G = G2 (q) with q = 3e and β interchanges long and short root subgroups of G. Thus Γ = {K} where K = S ∩ Ll for some long fundamental subgroup Ll of G and Kα = CS (K) = S ∩ Ls , where Ls is a short fundamental subgroup of G with O 2 (CG (z)) = Ll ∗ Ls . The proof of Theorem 16.3.1 involves a short series of reductions. Assume τ = (F, Ω) is a Lie fusion packet. Lemma 16.3.2. (1) F = FS (G) for some group G of Lie type and odd characteristic tamely realizing F. (2) Let ρ = (F, Γ) be the Lie packet of G. Then Γ = {S ∩L : L ∈ L and S ∩L ∈ Syl2 (L)}, where L is the set of fundamental subgroups of G. (3) There exists α ∈ Aut(F) with Γα = Ω. (4) α = β|S for some β ∈ Aut(G). Proof. As τ is a Lie packet, F = FS (G) for some group G of Lie type and odd characteristic. By 3.5 in [AO16] we may choose G to tamely realize F; that is (1) holds. Then (2) follows directly from the definition of “Lie packet” in the introduction. From Theorem 1, all quaternion fusion packets ζ = (F, Σ) with F = [Σ]F and F transitive on Σ are isomoprhic so (3) holds. Then (1) and 2.22 in [AO16] implies (4).  Lemma 16.3.3. If β permutes the long root subgroups of G then Γα = Γ.

16.3. LIE FUSION PACKETS

437

Proof. Let U denote the set of long root subgroups of G, and suppose β acts on U. Then β also acts on the set P of pairs (U, V ) such that U, V ∈ U and U and V are opposites. Hence β permutes L. Therefore for K ∈ Γ, K = S ∩ L ∈ Syl2 (L) for some L ∈ L, so Kα = Kβ = (S ∩ L)β = S ∩ Lβ ∈ Γ, completing the proof.  Lemma 16.3.4. If Γα = Γ then G = G2 (q), where q = 3e , and β interchanges long and short root subgroups of G. Proof. By 16.3.3, the root system of G has two root lengths and β interchanges long and short root subgroups. Then the lemma follows from 2.5.1 in [GLS98].  Observe that Theorem 16.3.1 follows from lemmas 16.3.2-16.3.4, together with the fact that G2 (q) has one class z G of involutions and O 2 (CG (z)) = Ll ∗ Ls where Ll is a long fundamental subgroup and Ls is a short fundamental subgroup.

References and Index

Bibliography Michael Aschbacher, Radha Kessar, and Bob Oliver, Fusion systems in algebra and topology, London Mathematical Society Lecture Note Series, vol. 391, Cambridge University Press, Cambridge, 2011. MR2848834 [Alp67] J. L. Alperin, Sylow intersections and fusion, J. Algebra 6 (1967), 222–241, DOI 10.1016/0021-8693(67)90005-1. MR215913 [AO16] Michael Aschbacher and Bob Oliver, Fusion systems, Bull. Amer. Math. Soc. (N.S.) 53 (2016), no. 4, 555–615, DOI 10.1090/bull/1538. MR3544261 [AS04] Michael Aschbacher and Stephen D. Smith, The classification of quasithin groups. I, Mathematical Surveys and Monographs, vol. 111, American Mathematical Society, Providence, RI, 2004. Structure of strongly quasithin K-groups. MR2097623 [Asc73] Michael Aschbacher, A condition for the existence of a strongly embedded subgroup, Proc. Amer. Math. Soc. 38 (1973), 509–511, DOI 10.2307/2038941. MR318308 [Asc77] Michael Aschbacher, A characterization of Chevalley groups over fields of odd order. II, Ann. of Math. (2) 106 (1977), no. 3, 399–468, DOI 10.2307/1971063. MR498829 [Asc80] Michael Aschbacher, On finite groups of Lie type and odd characteristic, J. Algebra 66 (1980), no. 2, 400–424, DOI 10.1016/0021-8693(80)90095-2. MR593602 [Asc86] Michael Aschbacher, Finite group theory, Cambridge Studies in Advanced Mathematics, vol. 10, Cambridge University Press, Cambridge, 1986. MR895134 [Asc87] Michael Aschbacher, Chevalley groups of type G2 as the group of a trilinear form, J. Algebra 109 (1987), no. 1, 193–259, DOI 10.1016/0021-8693(87)90173-6. MR898346 [Asc94] Michael Aschbacher, Sporadic groups, Cambridge Tracts in Mathematics, vol. 104, Cambridge University Press, Cambridge, 1994. MR1269103 [Asc97] Michael Aschbacher, 3-transposition groups, Cambridge Tracts in Mathematics, vol. 124, Cambridge University Press, Cambridge, 1997. MR1423599 [Asc08a] Michael Aschbacher, Normal subsystems of fusion systems, Proc. Lond. Math. Soc. (3) 97 (2008), no. 1, 239–271, DOI 10.1112/plms/pdm057. MR2434097 [Asc08b] Michael Aschbacher, Standard components of alternating type centralized by a 4group, J. Algebra 319 (2008), no. 2, 595–615, DOI 10.1016/j.jalgebra.2006.09.033. MR2381797 [Asc10] Michael Aschbacher, Generation of fusion systems of characteristic 2-type, Invent. Math. 180 (2010), no. 2, 225–299, DOI 10.1007/s00222-009-0229-z. MR2609243 [Asc11] Michael Aschbacher, The generalized Fitting subsystem of a fusion system, Mem. Amer. Math. Soc. 209 (2011), no. 986, vi+110, DOI 10.1090/S0065-9266-2010-00621-5. MR2752788 [Asc15] Michael Aschbacher, Classifying finite simple groups and 2-fusion systems, ICCM Not. 3 (2015), no. 1, 35–42, DOI 10.4310/ICCM.2015.v3.n1.a5. MR3385504 [Asc19] Michael Aschbacher, On fusion systems of component type, Mem. Amer. Math. Soc. 257 (2019), no. 1236, v+182, DOI 10.1090/memo/1236. MR3898993 alia Castellana, Jesper Grodal, Ran Levi, and Bob Oliver, Subgroup [BCG+ 05] Carles Broto, Nat` families controlling p-local finite groups, Proc. London Math. Soc. (3) 91 (2005), no. 2, 325–354, DOI 10.1112/S0024611505015327. MR2167090 [BLO03] Carles Broto, Ran Levi, and Bob Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), no. 4, 779–856, DOI 10.1090/S0894-0347-03-00434-X. MR1992826 [Cra11] David A. Craven, Normal subsystems of fusion systems, J. Lond. Math. Soc. (2) 84 (2011), no. 1, 137–158, DOI 10.1112/jlms/jdr004. MR2819694 [AKO11]

441

442

[Dic58]

[Foo97]

[Gla66] [GLS98]

[GLS99]

[Gol74] [Gol75] [Lyn14] [McL69] [Pui09]

BIBLIOGRAPHY

Leonard Eugene Dickson, Linear groups: With an exposition of the Galois field theory, with an introduction by W. Magnus, Dover Publications, Inc., New York, 1958. MR0104735 Richard Foote, A characterization of finite groups containing a strongly closed 2subgroup, Comm. Algebra 25 (1997), no. 2, 593–606, DOI 10.1080/00927879708825876. MR1428800 George Glauberman, Central elements in core-free groups, J. Algebra 4 (1966), 403– 420, DOI 10.1016/0021-8693(66)90030-5. MR202822 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1994. MR1303592 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1994. MR1303592 David M. Goldschmidt, 2-fusion in finite groups, Ann. of Math. (2) 99 (1974), 70–117, DOI 10.2307/1971014. MR335627 David M. Goldschmidt, Strongly closed 2-subgroups of finite groups, Ann. of Math. (2) 102 (1975), no. 3, 475–489, DOI 10.2307/1971040. MR393223 Justin Lynd, The Thompson-Lyons transfer lemma for fusion systems, Bull. Lond. Math. Soc. 46 (2014), no. 6, 1276–1282, DOI 10.1112/blms/bdu083. MR3291264 Jack McLaughlin, Some subgroups of SLn (F2 ), Illinois J. Math. 13 (1969), 108–115. MR237660 Llu´ıs Puig, Frobenius categories versus Brauer blocks, Progress in Mathematics, vol. 274, Birkh¨ auser Verlag, Basel, 2009. The Grothendieck group of the Frobenius category of a Brauer block. MR2502803

Index

ˆ 3 (2)/E64 , 130 L ˆ 6 (2), 10 Sp ˆ Ω(τ ), 63 ˆ + (2), 10 Ω 8 A(V1 , V2 ), 60 A(τ ), 14, 65 A(z), 14, 65 B(z), 7 D(τ ), 59 D(z), 67 D∗ (τ ), 59 D∗ (z), 67 Dc , 7 Dm (z), 205, 255 Do (z), 205, 255 E(τ ), 16, 72, 173 F ◦, 7 Fz , 39 O(τ ), 14, 68 O(z), 14, 43, 54 OK , 14, 40, 54 Z(τ ), 7, 39 Z m (τ ), 341 ZS (τ ), 14, 39 ZΔ (τ ), 14, 57 Zin , 173 Zout , 173 ω(Cn , m), 127 ω(Dn , m), 90 Chev(p), 4 Chev∗ (p), 4 τ (K), 159 τ ◦, 7 τz , 66 dV , 14, 61 e(O), 259 nS , 269 z(K), 4 z ⊥ , 67 3-transpositions, 80 Ad , 80 D-subgroup, 81 Dd , 80

2¯ ω (Cn , m), 128 2ω(Cn , m), 128 AEn , 96, 101 Dη , 14, 61 E(t, Q1 ), 209 E6 [m], 10 E7 [m], 10 E8 [m], 10 F4 [m], 10 G2 [m], 10 ˆ n , 100, 101 H AE L2 [2m](1) , 10, 22 L3 (2)/23+6 , 10 L3 (2)/E8 , 10 L3 (2)/E64 , 10 (1) , 10 L+ 3 [m] Ln [m], 10 M (τ ), 15, 69 M12 , 10 O(τ ), 14, 54 O(z), 14, 41, 54 SL2 [m], 4, 40 SL2 [m](2) , 22 SL12 [m/2], 40 Spn [m], 10 W (τ ), 14, 57, 69 WI , 270 WS , 14, 273 WΔ , 269 Z ∗ -Theorem, 3, 48, 177 Δ(τ ), 14, 57 Δ(t), 14 Δ(z), 54 Ω(z), 7, 14, 39 Ω2n+1 [m], 10 Ω2n [m], 10 ω ¯ (Cn , m), 127 ˇ n , 96, 101 AE ˇ 3 (2)/23+6 , 131 L η(τ ), 7, 14, 57 ηS (τ ), 14, 57 ˆ n , 100, 101 AE ˆ 3 (2)/23+6 , 131 L 443

444

Vd , 80 Wd , 80 D(D), 80 d⊥ , 80 width, 81 Alperin data, 40 Alperin’s Fusion Theorem, 11, 22, 24, 26, 45, 46, 48, 149 Alperin, J., 3 Alperin-Goldschmidt Fusion Theorem, 13, 16, 73, 76, 228, 237, 291, 298, 376 central product of subpackets, 8 complementary graph Bc , 7 conjugation map cy , 10 Craven’s Theorem, 25, 42 Craven, D., 12 Dickson, L., 157 Extended Inductive Hypothesis, 15, 17, 169, 205, 305, 325, 338, 341, 374, 385, 389, 402–404, 410, 424, 429, 433, 435 Foote, R., 117 fundamental subgroup, 4 fusion system, 3, 10 CF (U ), 11 NF (U ), 11  Op (F ), 12 Op (F ), 13 A(P ), 12 Eα∗ , 10 F -invariant, 12 F c , 11 F e , 13, 72 F f , 11 F r , 11 F f rc , 11 FS (G), 3 Comp(F ), 27 Comp+ (F ), 27 Sol(F ), 27 Alperin data, 11 centric, 11 conjugates P F of P , 11 constrained, 13 essential subgroup, 13 exotic, 3 factor system F /T , 12 fully centralized, 11 fully normalized, 11 model, 13 morphism, 10 normal closure [Ω]F , 4 normal closure [Σ]F , 12 normal subgroup, 13 normal subsystem, 12

INDEX

product system T F0 , 12 radical, 11 saturated, 3 solvable component, 27 strongly closed, 12 subnormal closure sub(F , T ), 13 subnormal series subi (F , T ), 13 subsystem ΣT generated by Σ, 11 Sylow group, 11 weakly normal, 12 Glauberman, G., 3 Goldschmidt group, 117 Goldschmidt, D., 3 Holt’s Theorem for Fusion Systems, 24 induced map α∗ , 10 Inductive Hypothesis, 15, 168, 195, 215, 223, 240, 255, 263, 269, 278, 286, 288, 291, 305, 385 Lie fusion packet, 4 Lynd transfer, 26 Lynd, J., 26 order of τ , 15 Puig, L., 3, 11 quaternion fusion packet, 4 reduction red(F ) of F , 25 saturated fusion system, 3, 11 subnormal closure s(K), 159 subnormal series subi (F , T ), 159 Thompson group μ(τ ), 7, 9, 14, 64 tightly embedded subsystem, 16 universal group ω ¯ (Φ, m), 15, 90 Weyl group Weyl(Φ), 15

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CONM

765

ISBN 978-1-4704-5665-8

9 781470 456658 CONM/765

Quaternion Fusion Packets • Aschbacher

Let p be a prime and S a finite p-group. A p-fusion system on S is a category whose objects are the subgroups of S and whose morphisms are certain injective group homomorphisms. Fusion systems are of interest in modular representation theory, algebraic topology, and local finite group theory. The book provides a characterization of the 2-fusion systems of the groups of Lie type and odd characteristic, a result analogous to the Classical Involution Theorem for groups. The theorem is the most difficult step in a two-part program. The first part of the program aims to determine a large subclass of the class of simple 2-fusion systems, while part two seeks to use the result on fusion systems to simplify the proof of the theorem classifying the finite simple groups.