Representations of Solvable Groups 9780521397391, 0521397391

Citation preview

Chapter 0 PRELIMINARIES

For this manuscript, all groups will be assumed finite. If G is a group and T an (arbitrary) field, an ^[G]-module V will mean that V is a right T[G]-module and that V is finite dimensional over T. Recall that V is completely reducible if V is the sum of simple T[G]-modules. In this case, V is actually a direct sum of simple modules. Indeed, if V ^ 0 is completely reducible, then V = V\ © • • • © Vi where V%\ ^ 0 is the direct sum of simple isomorphic f[G]-modules and if W{ and Wj are simple submodules of Vi and Vj (resp.), then W{ £ Wj (as T\G\-modules) if and only if % = j . Then V{ are called the homogeneous components of V and are unique (not merely up to isomorphism, but the Vi are unique submodules). Now V\ = U\ ©• • -®Ut for isomorphic ^[G]-modules C/j. While t is unique, the Ui are unique only up to isomorphism. Because solvable groups have an abundance of normal subgroups, we begin by recalling Clifford's Theorem: 0.1 Theorem. Suppose that V is an irreducible J-[G]-module and N < G. Then (a) VN is completely reducible and so Vjy = V\ © • • • © V\ where the Vi are the homogeneous components of VN; (b) G/N transitively permutes the Vi by right multiplication; (c) If Wi and Wj are irreducible N-submodules of Vi and Vj (resp.), then dim(W^) = dim(W}) for all i, j ; and (d) If I = {g E G | Vig = Vi] is the inertia group in GofVi, then Vi is an irreducible I-module and V == Vf* (induced from I to G).

2

PRELIMINARIES

Chap. 0

Proof. This is Hauptsatz V, 17.3 of [Hu].

•

0.2 Proposition. Suppose that V is an irreducible ^[G^module, N < G and VJV is not homogeneous. (i) If C < G is maximal such that Vc is not homogeneous, then G/C faithfully and primitively permutes the homogeneous components of Vc. (ii) There exists N < D < G such that VD = Wx © • • • © W3 for Dinvariant Wi that are faithfully and primitively permuted by G/D (s > 1). Furthermore, whenever N < L < D with L By maximality of C, C — K, proving that G/C acts faithfully on the homogeneous components of VcThis action is primitive by the first paragraph and choice of C. (ii) Since G transitively permutes ft = {V 1? ..., Vi}, we may write ft = Ai U • • • U As with s > 1 and G primitively permuting { A i , . . . , A 5 }. In other words, VJV = W\ © • • • © Ws where s > 1 and each Wi is a sum of some homogeneous components of VN and such that G primitively permutes the

Chap. 0

PRELIMINARIES

3

W{. Let D be the kernel of the permutation action of G on {Wi,..., Ws}. Then VD = W\ © • • • 0 W3 for £)-invariant Wi that are faithfully and primitively permuted by G/D. Furthermore, whenever N < L < D with L < G, each (VFI)L is a sum of homogeneous components of Vi,, by the first paragraph. Since 5 > 1, Vi is not homogeneous. •

The structure of solvable primitive permutation groups is well-known and discussed below in Section 2. In particular, a nilpotent and primitive permutation group has prime order (see [Hu, Satz II, 3.2]). 0.3 Corollary. Suppose that V is an irreducible G-module, N < G and Vjsr is not homogeneous. If G/N is nilpotent, there exists N < C < G with \G : C\ = p, a prime such that Vc = V\ © • • • © Vp for homogeneous components Vi ofVc0.4 Proposition. Suppose that V is an irreducible T[G\-module and that K, is an extension field of T. (i) If char (F) ^ 0, then V ®r >C = Wi © • • • © W\ for non-isomorphic irreducible fC[G]-modules Wi. (ii) IfK, is a Galois extension of T, then V O^r K = e(Vi © • • • © Vi) for a positive integer e and non-isomorphic irreducible JC[G]-modules Vi. Furthermore the Vi are afforded by representations Xi that are conjugate under Gal(/C : T). Indeed {Xi,... ,Xt} is a single orbit under Gal (X : T).

Proof. See [HB, Theorems VII, 1.15 and VII, 1.18 (b)]. The /C[G]-module V ®T JC is denoted by VK in [HB] and by VK in [Is]. • Suppose V is a faithful irreducible ^[GJ-module for some field T. If K is an extension field of T, then G has a faithful irreducible /C[G]-module W by Proposition 0.4. By choosing /C to be algebraically closed, G has a faithful absolutely irreducible representation X: G —» M n (X) for some n. Then the

4

PRELIMINARIES

Chap. 0

centralizer in M n (X) of X{G) consists of scalar matrices. If G is abelian, then G must be cyclic and n = 1. We thus have the following well-known result which is of particular importance to the structure of quasi-primitive linear groups. 0.5 Lemma. If an abelian group A has a faithful irreducible module W (over an arbitrary field T), then A is cyclic. If furthermore W is absolutely irreducible, then dim^-(W) = 1.

The following lemma is sometimes referred to as Fitting's lemma, although [Hu] credits Zassenhaus. 0.6 Lemma. Suppose G acts on an abelian group A by automorphisms and (|G|, |;4|) = 1. Tien A = [G, A] x CA(G). Proof. See [Hu, Satz III, 13.4].

•

We use Irr (G) to denote the set of the ordinary (i.e. complex) irreducible characters of the group G and let char(G) denote the set of all ordinary characters of G. Of course, char(G) C cf (G), the set of class functions of G, and we let [%, 0] denote the inner product of %> 0 G cf (G). For N < G and 0 e Irr (TV), we let Irr(G|0) = {x € Irr(G) | [XN,0] ^ 0}. By Frobenius reciprocity, Irr(G|0) is the set of irreducible constituents of the induced character 0G. Let T be a field of characteristic p such that T contains a |G|-th root of unity. Then J7 is a splitting field for all subgroups of G (i.e. every irreducible ^"-representation of every subgroup of G is absolutely irreducible). It is customary to choose T so that T is a quotient ring of an integral domain of characteristic zero. This is often done via p-modular systems, as in Section 3.6 of [NT]. A slightly different approach is given in Chapter 15 of [Is]. We should point out here that Chapter 15 of [Is] is only intended as an introduction to modular theory and as such is not complete. Recall that each g £ G

Chap. 0

PRELIMINARIES

5

has a unique factorization g = gpgp' = gp'gp where gp is a p-element and gpi is p-regular (i.e. p \ o(gpt)). Each irreducible ^"-character x of G can then be lifted to a complex-valued function ) is a constituent of (pG. When G/N is a p'-group, we get the converse and more. 0.7 Proposition. Suppose that G/N is a p1-group, that ip G IBr^iV) and 6 G IBrp(G). Then the multiplicity oftp in Ojy equals the multiplicity of 0 in yG.

Proof. Let T be a splitting field for N and G in characteristic p. Let V be an (irreducible) Jr(G)-module affording 6, and W an (irreducible) ?r(N)module affording ip. Now Vjq is completely reducible by Clifford's Theorem. Since G/N is a p'-group and W an irreducible iV-module, indeed WG is completely reducible (see [HB, VII, 9.4]). With both VN and WG completely reducible and T a splitting field for N and G, it follows from Nakayama reciprocity ([HB, VII, 4.13]) that the multiplicity of W as a composition factor of VN equals the multiplicity of V as a composition factor of WG.

6

PRELIMINARIES

The proposition now follows.

Chap. 0

•

0.8 Theorem. Let N < G and

-> ipG is a bisection from lBip(I\ip) onto IBrp(G\(p). Proof. For ordinary characters, this is Theorem 6.11 of [Is]. More generally, a similar proof works here. Let \ € IBr^GI^). Clifford's Theorem 0.1 (d, a) shows that \ — A*G f° r some /i G IBij,(J|^) and that %/ = // + A for a (possibly zero) Brauer character A of / with no irreducible constituent of A lying in IBrp(I\(p). Let \j) G TBvp{I\ \\)G is a 1-1 and onto map from IBrp(/|v?) onto IBr^GV). • Theorem 0.8 applies to ordinary characters too; just choose p so that P\\G\0.9 Lemma. Suppose that N < G, (p G IBr^G) and (p^ is irreducible. Then a —> a(p is a one-to-one map from IBrp(G/N) onto IBrp(G\(pisr). Proof. By [HB, Theorem VII, 9.12 (b,c)], note a

aip is one-to-one. Let //GlBri,(G|(^iv)It suffices to show /i = f3ip for some /? G lBrp(G/N). We mimic the proof of [HB, Corollary VII, 9.13]. Let T be an algebraically closed field of characteristic p and V an irreducible .FfG]-module affording (p. Since /i G IBTP(G\IPN), Nakayama reciprocity implies that // is a constituent of Irr(C) where C = CG(A) such that (i) If A is a p-group and \ € I™A(G),

then X^(G, A) is the unique

P e Irr(C) satisfying \xc,P] # 0 (mod p). (ii) UT(CG(T), A/T).

Proof. See [Is, Theorem 13.1].

•

By "uniquely defined" above, we mean there is only one such map (indeed, else (ii) would be meaningless) and this map is independent of choices made in the algorithm implied by the theorem. This map is known as the Glauberman correspondence. If A acts on G with (|A|,|G|) = 1, but A not solvable, then \G\ is odd. Isaacs [Is 2] has exhibited a "uniquely defined" correspondence whenever A acts on G, (|^4|, \G\) = 1, and \G\ is odd. Moreover, this agrees with the Glauberman correspondence when both are defined [Wo 2]. The combined map is thus referred as the GlaubermanIsaacs correspondence. The following appears in [Is, Theorem 13.29] and has a couple of uses in this section alone. 0.16 L e m m a . Suppose A acts on G with A solvable and (\A\,\G\) = 1. Suppose N < G is A-invariant,

x £ lxrA(G)

[XJV, 0] ^ 0 if and only if [Xp(G, A)NnC,

and 0 G lrrA(N).

Then

Op(N, A)] ^ 0.

Much of the following lemma is a consequence of Glauberman's lemma above. In fact, no more is required for G solvable or for parts (a), (b) and (c) in the general case. For G non-solvable, parts (d), (e) and (f) also employ the Glauberman correspondence and more. All parts appear somewhere (possibly as exercises) in Chapter 13 of [Is]. Due to its importance here (particularly when G is solvable), we give a sketch. 0.17 Lemma. Suppose that N < G ), so that H < I < T and / = (IC\G)H. Now If)G = IG((p) is ^-invariant. Also £ —» £G gives a bijection from Irr(/fl G\} (i.e. bo = Irr (M\ip) U IBrp(M\(p)). It suffices to show there is a unique p-block of G covering bo. By Corollary 0.27, we may assume that G/M is a p'-group. By Lemma 0.18, there exists 9 G Irr(60) such that p\ 0(1). Let a, /? G Irr(G|0). By Proposition 0.26, it suffices to show that a and /3 necessarily lie in the same p-block of G. To this end, it suffices to show that the algebraic integer

1

fa(g)

V

f3(g)\

)

is divisible by p whenever C is a conjugacy class of G and g E C; see [NT, Theorem 3.6.24] or [Is, Definition 15.17]. Since a, /3 G Irr(G|0) with 0 G Irr(M), M < G, indeed a(g)/a(l) = p(g)//3(l) for all g E M. Thus we assume that g fi M > Opip(G). By Lemma 0.19, g does not centralize Optp(G)/Op(G). Thus Co(g) does not contain a Sylow p-subgroup of G, i.e. p | \C\. Because p\\G: Af |0(1), we have that p f a(l) /?(1). Thus p does divide |C|(-^ifj — frfy)- Hence a and /3 lie in the same p-block of G. D Let D be a p-subgroup of G. Brauer's First Main Theorem states that i > bG is a bijection from {6 E D1(NG(-£*)) | D is a defect group for 6} b— onto {B E.h\(G)\D is a defect group for B). (See [NT, Theorem 9.2.15] or [Is, Theorem 15.45] and note [Is, 15.44] should read PC(P) < H < N(P).) This is called the Brauer correspondence. Given Bo E bl(G), the Brauer correspondent of BQ is uniquely defined up to conjugation: if Q is a defect group for B, then the Brauer correspondent of B is the unique b0 G H ( N G ( Q ) ) with defect group Q satisfying 6^ = B. The G-conjugates of Q form the set of defect groups of B.

Chap. 0

PRELIMINARIES

21

0.29 Theorem. Let G be p-solvable, K = Op,(G) and P £ Sylp(G). Assume that ip £ Irr (K) is G-invariant and B £ bl(G) covers {}• Set C = CK(P) and let /J, = c = e/x + pA for a (possibly zero) character A of C and a p'-integer e. For x £ N G ( P ) , (G)). Theorem 0.29 describes the Brauer correspondent of B provided \x is G-invariant. One loose end needs to be tied up: namely how the Brauer correspondence works when fi is not G-invariant, i.e. what is the relationship between the Brauer correspondence and the correspondence in Lemma 0.25. 0.30 Corollary. Suppose that K