Structure and Representations of Q-Groups (Lecture Notes in Mathematics, 1084) 9783540138655, 354013865X

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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1084 Dennis Kletzing

Structure and Representations of Q-Groups

Springer-Verlag Berlin Heidelberg New York Tokyo 1984

Author

Dennis Kletzing Department of Mathematics and Computer Science Stetson University DeLand, Florida 32720, USA

AMS Subject Classification (1980): 20C, 20 E, 13H ISBN 3-540-13865-X Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13865-X Springer-Verlag New York Heidelberg Berlin Tokyo

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© by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach / Bergstr. 2146/3140-543210

Preface

These notes span a decade of research and reflect the contributions of many individuals. There is one person, however, who has influenced this work to the extent that it would not exist were it not for his constant support and encouragement. He is Professor Ernst Snapper of Da.rtrrDuth College. It is with sincere express my appreciation to him for many, many years of mathematical

that I and

unselfish friendship. I also wish to thank Dr. John Rasmussen for many stimulating conversations. Finally, this work was supported by research grants received from Stetson Uni versi ty. It was writ.ten while on sabbatical leave at Da.rtrrDuth

DeLand, Florida July, 1984

Dennis Kletzing

Introduction Notation .

5

1:..

7

Chapter 1.

Basic

2.

Structure of Q-groups having abelian or dihedral Sylow 2-subgroups

of Q-groups

8

and involutory Q-groups

3.

20 24

4.

Solvable Q-groups . . . .

39

5.

The partially ordered set defined by a Q-group

47

Constructions

58

Q-Groups

1.

Wreath products .

59

2.

Semi-direct products

77

3.

App.L:1Ccl.tl_on to the Weyl groups of types

4.

Theory of transversal pernartat ion r-epr-aserrrat.i.ons

Chapter 3.

and D

n

93 97

105

Local

1.

Closed algebras. The local

no

2.

Local idempotents

130

3•

The combinatorics of p-classes

4•

Local restriction and local induction

5•

The local subgroup

6.

Local multiplicities

Chapter

The local induction prin(:iI)le

143 159 161 174

Representations

1.

The local invariants

180

2.

The local character ring rCGV)V

187

3.

Local splitting

207

.

VI

Application to the Weyl Groups of Exceptional

220 221

1.

F

2.

E S

233

3.

E

240

4.

E

4

7

243

8

Appendix . .

2S9

The character table of F4

270

rl*(F

271

.

4)

The character table of F4/Center

272

rl*(F

273

4/Center)

. .

Conjugacy classes of E S

274

The character table of E S

275

rl*CE

27S

.

S)

•.. . .

+

277

ConJugacy classes of E 7

+

The character table of E 7

278

"r\;'(E+) 7

280

•••••

Conjugacy classes of E

8

281

References

283

Index

288

Introduction

A Q-group (frequently called a rational

is a finite group all of whose

ordinary complex representations have rationally valued characters. Although the most; familiar examples of these groups are the symmetric groups, all of the Weyl groups are Q-groups. The purpose of these notes is to present a detailed investigation into the structure and rational representations of Q-groups. The interplay between the structure of a finite group and the representaticns of the group by means of pe:rmutations or linear transfonnations has had, and continues to have, deep consequences for both theories. By imposing certain constraints on the group, such as being abelian or nilpotent, one is able to draw conclusions about its representations. Conversely, restrictions on the representations can lead to specific group structures. Specifically, to what extent does the rationality of character values affect the structure and rational representations of the group? It is within this context that we approach the study of Q-groups. There is no general clasEdfication theory of Q-groups and we do not attempt to develop such a theory in these notes. Instead, we shall concentrate on the following two questions:

1.

II.

What can be said abcut the structure of a Qe-group? Under what circumstances can we conclude that the rationally represented characters of a Q-group are generalized permutation characters? It should be mentioned that being a rationally represented character is a stronger requirement than

a

rationally valued character.

Roughly speaking, Chapters I and 2 concentrate on question I while Chapters 3 and 4

2

deal with question II. It was question II which originally rrotivated these investigations. Let Q stand

for the field of rational numbers and let V be a finite dimensional vector space over Q. A rational representation of a finite group G is a method of representing the elements of G as non­singular matrices with entries in Q or equivalently, as non­singular linear transformations of V. More precisely, it is a horrorrorphism

GL(V) of G

into the general linear group of V. If V has a basis which is permuted by every element of G, then the representation is called a permutation represen1:ation of G. For example, the natural representation of the symmetric group Symvn) obtained by permuting the coordinate entries in Qn = Q x ... x Q is a permutation representation of Sym(n). Another :important example occurs in Galois theory. The Galois group G of a Galois extension KlQ acts on K to define a rational representation of G. The Normal Basis Theorem asserts that K has a Q­basis which is permuted by G and hence this representation is a permutation representation of G. However, apart from specific examples such as these, it is almost never the case that a rational representation is a permutation representation. By reformulating these concepts in terms of characters and enlarging the scope of the problem slightly, we are lead to ask question II. The work of Frobenius and Young shows that the rationally

characters

of the symmetric and hyperoctahedr'al groups are generalized permutation characters ([21J, [59]). The first result of a general nature was obtained by E. Artin in a series of three papers (1923­1931) devoted to the study of L­series ([1], [2], [3J). It is in this work that one finds the result, usually referred to as Artints theorem,

that a rationally represented character of a group may be written as a linear ccmbination of permutation characters with rational numbers as coefficients. If G is a finite group, it foHews from Artints theorem that there is a smallest positive integer y(G) with the property that y(G)X is a generalized permutation character of G whenever X is a rationally represented character of G. Chapters 3 and 4 are a s'tudy of the invariant y(G). Our approach to the study of y is based upon techniques from

The underlying rings are the Burnside ring of the group and the

geometry.

r

of generalized

rationally represented characters of the group. The basic tool of this approach is

3

the concept of a

character. The theory of local characters enables one to define

local invariants which completely characterize y. It is the local invariants which lead to a satisfactory answer to question II. Here is a brief sunmary of the five chapters. In Chapter 1, the basic group theory of Q-groups is established. Some of this material has appeared before, but nost of it is new with the exception of the rros't elementary properties in Section 1 and the exposition of solvable Q-groups in Section 4. One of the most important results of the chapter is the classification of Q-groups which have an irreducible involution (Section 3). An involution is called irreducible if it cannot be factored into a product of two involutions and we show that in a Q-group, the involutions are either all irreducible or all reducible. Those Q-groups which contain an irreducible involution are then classified. It should be remarked that the order of a non-trivial Q-group must be divisible by 2 thus insuring the existence of an involution. In Chapter 2, two constructions for obtaining new Q-groups are discussed. These constructions involve the action of a group on a transversal

set and lead to the concept of a

representation. The basic results can be summarized by saying

that the constructions will produce Q-groups from Q-groups whenever the relevant group action is transversal. A unified treatment of the Weyl groups of types An' B and D n n is then presented from this point of view, showing that they and their Sylow 2-subgroups are all Q-groups. Transversal representations will also

an important rClle

in Chapter 4. Chapter 3 develops the theory of local characters and their arithmetic. The concept of local class is defined and shown to be the appropriate local structure within the group which reflects the local structure of the Burnside V is a local class, then the localization of

and the ring

r.

If

r at V is a finitely-generated module

over the ring of local integers Z(p) and the elements of these modules may be L'1terpreted as functions on V. The term local character is used to describe these functions because they turn out to be the restriction of ordinary characters to the corresponding local class. It will be shown that every local class has associated with it a local subgroup which is unique to within conjugation. Additionally, restriction and induction of characters will be localized thus making it possible to relate local

4

characters of a group to those of a subgroup. The most important result of the chapter is the local

principle which states that every local character of a group

is induced from a local character of the corresponding local

SUbgJC>Q\.lP

Chapter 4 deals with the central topic of these notes,

(Section 5).

the relationship

between the r'at.ional.Iy represented characters and the permutation characters of a group. This

is characterized by the invariant y. The chapter begins with

a historical survey of the work which has been done on y. We shall discuss reasons stemming from a.tgeoraa,c geometry which suggest that there are many Q-grDups all of whose rationally represented characters are generalized permutation characters. Next, a local invariant

Yv

E:

Z is associated with every local class V. The basic relationship

between the local and global invariants is then established: Y is the least common rrail.t ipl.e of all the YV' It is shown that the local subgroup associated with V com-

pletely controls the value of YV' Consequently, the local subgroups hold the key to determining a

answer to question II. At this point the concept of local

splitting is introduced and the main result of the chapter is proved:

If G is a Q-grDUp which is locally on every local class, then y

VS

£

G, o is onto

12

VS

£

G, N(S)/C(S)

Aut(S).

Of

Fur-therrrore , if p is prime, then AuHS)

(N(S)/C(S»

p

p

= (C(S) N(S)p)/C(S) N(S) N(S)

Corollary

p

p

I C(S)

n N(S) p

I C(S) . II p

Let G be a non-trivial Q-group.

(1)

If S

(2)

If P is a prime divisor of IGI, then p-l divides IGI.

(3)

IGI is even. Hence G has a non-trivial Sylow 2-subgroup.

(4)

If G is nilpotent, then G is a 2-group.

(5)

If G is p-nilpotent, then p = 2.

(6)

If G is not a 2-group, then the smallest odd

£

G, then ¢(ISI) divides IGI.

dividing IGI is a Fermat

prime. Proof.

Since [N(S):C(S)] = IAut(S)1 = ¢(ISI), statement (1) is obvious. If S is an

e Ierrerrt of order p, then ¢ ( 1S I)

= p-I

and statements (2), (3) and (6) follow. I f G

is rri.Ipoterrt , then it splits as a direct product of its Sylow subgroups. Since these factors must be Q-groups, they must have even order and therefore G must be a 2-group. This proves (4). If G is p-nilpotent, then G has a quotient group which is a p-group , Since any such quotient is a Q-group , it follows from (3) that p = 2. This proves staterrent ( 5 i. I I

After developing some further properties of Q-groups, it will be shown that staterrent (1) in Corollary 12 can be strengthened to ¢(ISI)

I

[G:Z(G)]. In connection

with statement (6) , it is an interesting and open ouesti.cn as to whether or not every

13

Fermat prime may occur as the smallest odd prime in the order of some Qe-group, Clearly, every Fermat prime p occurs in the order of some Q-group since SymCp) is a Q-group. Whether or not p can be the least odd prime is not so obvious if indeed possible. In Section 3 a o-group will be constructed having order 2 3 52 thus showing that the Fermat prime 5 may occur as the least odd prime in the order of a o-group. Finally, in connection with statement (3) it should be mentioned that there is a long standing conjecture that a Sylow 2-subgroup of a Q-group is also a Q-group. Although we are unable to prove this conjecture, it will be shown that i f G has a tr,m,;versi'31].y embedded 2-subgroup, then a Sylow 2-subgroup of G is a Q-group (Chapter 2, Section 4). On the other hand, for some Q-groups it is easier to construct a Sylow 2-subgroup and use the methods of Chapter 2 to show that it is a Q-group. This questaon will be discussed further in Chapter 2 and a possible line of proof indicated. We now turn our attention to the normalizer and centralizer of a Sylow P-SubgroUP P of a Q-group. A convenient tool for studying N(P) and C(P) is the concept of a pcentral element.

Definition.

Let G be a finite group and let p be a pr-ime. An element S

E:

G is

called p-central if there is a Sylow p-subgroup P of G such that P < C(S) .

Proposition 13.

Let G be a finite group. Then { p-central elements of G}

where the union is over all P primes p if and only if S Proof.

Clearly, S

E:

E:

E:

=

U

CG(P) P Syl (8). Furthennore, S E: G is p-cerrtre.r for all p

Z(G).

CGCP) if and only i f P ::. C(S). Hence UCG(P) is precisely p

the set of p-cerrtr-al, elements of G. Now, the definition of p-cerrtr-al, is equivalent to saying that p

r [G:C(S)].

Hence, S is p-centr-al. for all p i f and only i f C(S)

=G

14

or equivalently, 8 e Z(G). !!

Proposition

(1)

If 8

Let G be a Q-grDup and p a prime. G is p-central, then 181

£

by no prime (2)

{p, m, pm} where (p,m)

1 and m is divisible

1 (nod p).

The p-eentral elements of 8 having order p are precisely the non-identity

elements in Z(P) for any P ( 3)

£

If P

Proof.

£

Syl (G). p

8ylp (G), then Z (P) is an elementary abelian p-group.

£

IS I

Let S be a p-central element and set

(1)

pn m where (pm) = 1. Then

P < C(S) ::. N(S) for some P c Sylp(G) and hence p does not divide [N(S) :C(S) J. Therefore,

1= [N(8) :C(S) ] p

¢(lsl)p

p

( m) p ,

= { p

I t follows that n = 0 or 1 and that ¢ (m)

p

n-l

¢(m)p' n

1. Hence, 181

=

n

o}

:::.1

p , m or pm and q 1: 1

(rrod p) for every prime divisor q of m. (2)

Let P

8ylp(G) and let 8

£

£

Z(P). Then S is p-eentral and has order 1 or p by

statement (1). Therefore, every non-identity element of Z(P) is p-eentral and has order p. Conversely, let S be a p-central element of G having order p. Let P :':. C(S). Then P < Syl (C(S)) and hence some C(S)-eonjugate of S belongs to P. Since C(S)

-

P

centralizes 8, it follows that S (3)

£

P and hence S

E

Z(p) •

This follows immediately from statement (2). !!

Corollary 14.

Let G be a Q-gl70up.

(1)

Every non-identity 2-central element of G is an involution.

(2)

Z(8) is an elementary abelian 2-group.

15

(3)

The abelian Q-groups are precf.se'ly the elementary abelian 2-groups.

(4)

GIG' is an elementary abelian 2-group. In particular", every element of G having

odd order lies in G' .

(6)

If p is an odd prime and P a Sylow p-subgroup of G, then P :::. [p ,GJ. Here

[P

stands for the subgroup generated by the commutators [S,TJ = S T S-lT- 1

S

E:

(7)

P, T

E:

,

G.

G is generated by its 2-elements. To prove statement (1), let S f:. 1 be 2-central. Then

lsi

E:

{2, m, 2m} where

m is odd and no prime divisor of m is '" 1 (mod 2). The only such number m is m Therefore

lsi

= 1.

= 2. Now, the non-trivial elements of Z(G) are p-central for all p and

are therefore involutions by statement (1). Hence, Z(G) is an elementary abelian Statement (3) follows irrnnediately from (2). Statement (4) follows f'rom (3) , some n , and

since GIG' is an abelian Q-group. To prcve (5), observe t.hat [G:G'J

2 therefore 0 (G)

= 0 2(G').

To prcve statement (6), let FocG(P) stand for the focal

subgroup of PinG; FocG(P) is generated by the commutators [S,T], S which lie in P. Let

therefore P IFoc (P) is an abelian Q-group. Since p G

= FocG(P)

P, T

E:

G,

stand for the transfer map into the abelian

section P/FocG(P). It is a fundamental result of transfer' theory that

Hence, P

E:

:::. p(1[P,GJ :::. P and therefore P

t-

T

is onto and

2, it follows that P

= pn[p,GJ.

= Foc G(P) .

Hence P :::. [p,GJ and

statement (6) is proved . Finally, let K be the subgroup generated by the 2-elements of G. Then K 3, q ::: 3,5 (rrod 8) or n, q = 2 or the Janko

group J(ll), or is of Ree type.

Let G be a Q-group with abelian G I t follows from (A) that GIN is a Q-group 2. of odd order for some N O. Let Pi :: 2

:: 0

PI '"

p

m

where PI' ... , p

::

are Fermat primes

< i :. m. Then

;A::OJl Aut(S)

m

; A :: 2

x Z01 x .. , x

z0m .

; A > 2

Since Aut(S) is elementary abelian, it follows that

:: ••,.

:=

::

1 and

A :'::.. 3.

Therefore, the possible values for IS I are: 1, 2, 3, 4, 6, 8, 12, 24. The following Corollaries surnma:rize these conclusions.

28

Corollary 28A.

The order of an involutory Q-gr>::mp has the form

b. 3 All elements

of odd order have order equal to 3. All such groups are solvable.

Corollary 28B.

Every element of an involutory Q-group has order equal to 1, 2, 3,

4, 6, 8, 12 or 24.

I t is easy to see that each of the dihedral groups

involutory Q-group. In oar-t.icul.ar-

H , H , H and H is an 2 4 6 3

it is possible for an involutory Q-group to consist

of only 2-elements. It is also possible for the Sylow 3-subgroup of an involutory Q-group to be non-abelian; the Q-group [3,3,3J (Z2 x Z2) constructed in Example 22B is involutory since the elements of orders 3 and 6 are inverted by some involution but [3,3,3J is non-abelian.

, in view of Corollary 28A, the groups Sym(n) for

n > 5 are not involutory although they are strong. We now study the involutions in a Q-group and begin by

some preliminary

remarks about an arbitrary group G of even order. Let I stand for the set of involu-

in.

tions of G (L

A

of G is any element of G which is not an involu-

tion but may be written as a product of two involutions. Let B stand for the set of bi-involutions of G (L 27 that B

=G

E:

B). I f G is a strong Q-group, it follows from Proposition

- I. It is clear that both subsets I and B are normal although in gen-

eral, neither one of them forms a subgroup. Let

T

be an involution of G. Then

the set I into the following two subsets:

Clearly, T1+ T

= IT+ and

TI

=

=

T'

=

{T I E:

E:

ITT'

E:

I

i I }.

TTl

I

T

splits

29 r: 1T 1

< = T1+ T "T1

The number of involutions in G is odd. Proof.

Let X stand for the character of the permutation representation of G acting

on 1 by conjugation. Then, for any involution

Now, T'

T,

it follows that

if and only if T' commutes with T and T' 1 T (T i

£

since

=1

i

n.

Hence

Since

=

it follows that

on the set IT+ and hence

T

defines a fixed-point free bijection of order two

1+1 • even. IT a.s

. an odd number. Now, a.i.f 1 (T) Therefore X(T ) .i.s

stands for the principal character on the cyclic subgroup (T), then it follows from Frobenius reciprocity that

30

=

and therefore X(1)

=2

»

X(T). Since

and XCd is odd, it follows that X(1)

t

(2). T l' T 2'

Let T be irreducible and suppose that T

=

since T T 2 = TIE 1. This contradicts

Then

for some

= 0.

Hence

cannot be so factored.

(2)

==>

(3).

Assume that

T

cannot be factored as a product of two involutions and

suppose that C(T) contains some involution T' f. T. Then T T' is an involution and T

= (T T') T' is a factorization of T as a product of involutions. This is a contra-

diction and hence no such

T'

exists.

31

Suppose that T is the only involution in C(T). I f T' e

(1).

,thenTT'

is an involution and therefore T and T' commute, That is, T' t:: C(T). Since 1 i I, it follows that TT' ;t. 1 and hence T I ;t. T. This is a contradiction and hence we conclude that I; is empty, Therefore T is irreducible. II

Let G be a Q-group. Then G contains an irreducible involution i f and

Proposition

only i f a Sylow 2-subgroup of G is either Z2 or Q8' Proof.

Suppose that T is an irreducible involution of G and let

be a Sylow 2-

subgroup of G chosen so that T t:: G Then Z(G is a non-trivial elementary abelian 2, 2) 2-group and Z(G < C(T), Since C(T) contains a unique 2)

it follows that

< C(T). Therefore G iE a 2-group corrtairring a 2

lution. It follows that

is either the cyclic group Z2n of order

alized quaternion group Q ([33J, Satz 8.2), If G :: 2n 2

, then n

invo-

or is a gener1 since Z(G is 2)

elementary abelian. Otherwise, G :: Q which has generators A and B with relations 2 2n 2n - l and therefore A :: 1, ¢( !AI )

>

But Q is a Sylow 2-:sul)gl:'Otlp of G. Therefore 2n-3 < n , It follows that n :: 3 and 2n

Conversely, suppose that

is either Z2 or Q and let -1 stand for the unique 8

involution in either one of these groups. If there is an involution T e C(-l) , T ;t. -1, then (T)

X

(-1) is a 2-subgroup of G and hence lies in some

of G But 2.

neither Z2 nor Q has an elementary abelian subgroup of rank 2. Hence T does not 8 exist and therefore -1 is irreducible. II

32 Corollary 32A.

I f G is a Q-group containing an irreducible involution, then all

involutions in G are conjugate and hence all are irreducible. In particular, the involutions of a Q-group are either all irreducible or all reducible. Proof.

Let G be a Q-group containing an irreducible involution

T. I f

is a Sylow

2-subgroup of G containing T, then Z(G 2)

= (T)

Let T* be any involution in G and let

be a Sylow 2-subgroup of G containing T*.

Then G and 2

are conjugate and therefore

only involutions in G and 2

T

and T is the only involution in G 2,

and

are conjugate since they are the

respectively. II

By Corollary 32A, it now makes sense to refer to "the involutions" of a Q-group

as being irreducible or reducible. If the involutions are irreducible, we will say that they have

Z2 or Q8'

Corollary 32B.

Let G be a strong Q-group, I the set of involutions and B the set of

bi-involutions. Let (1)

The involutions of G are irreducible if and only i f 1

(2)

The involutions of G are reducible if and only if 1

Proof.

by def'irri'tdon , it follows that 1

r 2nr = Bnr = Ql

2

B.

= G.

Suppose the involutions of G are irreducible. Then I

2 1 CB. Since B

then

2

2

nI

2

:: 0 and hence 2

= B. Conversely, if 1 = B,

and therefore the involutions are irreducible. Now, suppose

2, 2 the involutions of G are reducible. Then L cr . Since BCI it follows that G

=r

•

+ B c;;;;r

2

and hence that

= G.

Conversely, if

r2

= G,

then I

and the

involutions of G are therefore reducible. II

We now classify the Q-groups whose involutions are irreducible, It was shawn in Proposition 31 that i f a Q-group has a generalized quaternion Sylow 2-subgroup,

33

then the involutions of the group are irreducible. Thus, the

classification

may be regarded as a continuation of the work in Section 2 to Q-grDUpS with generalized quaternion Sylow 2-subgrDups. The simplest case to consider is when Z2 is a Sylow 2-subgrDup of G.

Let G be a Q-grDUp whose involutions are irreducible of type Z2' Then G

= E3

CT) where

is a (possibly trivial) elementary abelian 3-grDup and

an involution which inverts every element of E

3.

T

is

Conversely, every group of this form

is a Q-grDup whose involutions are irreducible and of type Z2' Furthermore, every group of this form is involutory. Proof.

Let G = (T). Then G is abelian and hence G = G (T) by Proposition 21. To 2 2 3

show that G3 is elementa...ry abelian, let S [N(S)2:C(S)2] fore

T

S

T f;

= ¢(ISI)2 = 2. S. Hence

normal subgroup G 3

T

Since IG21

E

G 3' S f l, and set 1S!

= 2,

it follows that C(S)2

= 3a.

Then

(1) and there-

defines a fixed-point free automorphism of order 2 on the

G". It follows from this that G is abelian and therefore ele3

mentary abelian (Propoai.ttion 14(3» and that Conversely, let G G are the elements in

T

inverts all elements of G ([25]). 3

(T) be a group of this form. The only non-involutions in

and these elements have order 3. Since

T

inverts these ele-

ments and inversion is an automorphism of order 2, it follows that G is an involutory Q-grDUp. Finally, C(r )

(T) and hence T is an irreducible involution of G. Therefore

the involutions of G are irreducible of type Z2' II

The structure of those Q-grDUpS whose involutions are irreducible of type QS will be determined by a sequence of preliminary results. Let 0(8) stand for the largest normal odd order subgroup of G.

34 Proposition 34A.

Let G be a Q-group having Q as a Sylow 2-13Ub,gn)up. I f 0(8) 8

I t follows from the BPauer-Suzuki theorem that z(8)

Proof.

= (,)

(1) ,

for some involution

r ([ 2S] ). Therefore, GI( r) is a Q-group having abelian Sylow 2-subgroup Q I (, ) 8 Z2 x Z2' Hence G/(,) = N [Qa/(,)] for some 3-group

GI(r )

N

N

V

/""

N 3

Hence N then N3

= N3

(r )

x (,). Therefore N 3, the space V must admit a non-singular H-invariant bilinear form which is both syrrunetric and sympl.ecti.c. This is impossible and therefore p is either 2 or 3. When the condition of

is dropped, we no longer have the existence

of a non-singular H-invariant sy="tr,ic bilinear form on V although we retain such

44

a symplectic form if p > 3. Consequently, the proof of statement I must be developed in a different direction. The basic approach is to use the following well-known result for fixed-point free representations ([25], [47]):

(6)

Let!i be

finite

!i such that Per has

and let p:H

GL(V) be

representation of

non-zero fixed

when o 'I 1. Then:

distinct primes,

every subgroup

(a)

I f p, q

(b)

Ifp>2,

(c)

The Sylow 2-subgroups of !i

--

--

Sylow p-subgroups

generalized guaternion.

In general, elements of V are centralized by elements lying outside of V and

consequently the representation p is not fixed-point free. In order to get into the fixed-point free situation we proceed as follows. For each non-zero element x

E

V,

let

be the stabilizer of x in H. Let U be a maximal subgroup among the U ' Let x

stand for the fixed space of U. If N(U) is the normalizer of U in H, then N(U) acts . on VU and de f.iaries a representation

p

45

by setting PcrU(x) = po(x). If oU

t-

U, it follows from the maximality of U that PoU

p is

has no non-zero fixed-points. Hence

a representation satisfying the hypothesis

of statement (6) and therefore information about the group N(U)!U is obtained. Urifortunately, N(U)!U is not

E'>th G and H are

Q-groups. Proof. let S

We have already remarked that H is a Q-group. To show that G is a Q-group, E:

G and let m > 1 be an

setting f(y) fon= l(f)

S for every y

r-e Lat ive.Iy prime to lsi. Define f:Y--7 G by

Y. Then l(f)(y) :: f (y) :: S for every y

f. Hence l(f;l)1

Y and there-

lsi. Since G 1 (H,Y) is a Q-group, then= is an element

(h;T) such that

f(y)ill :: Hence S -

for every y

Y.

in G. I t follows that G is a Q-group. II

This resurt is somewhat unfortunate. One would like to build new Q-groups by wreathing certain non-Q-groups around a given Q-group H. In view of Propoai.t ion 63, this 'techni.que will not work; G must be a Q-group. Fur-therrrore , then= must be some type of res'trd.ctdon on the action (H,Y) if G 1 (H,Y) is to be a Q-group. This is

illustrated by considering the group G = Z2 1 (Sym(3),Sym(3»re Sym(3)

g,

when= (Sym(3),

stands for the le:tt··relW:Lar representation of Sym(3). G is not a Q-group.

To see this, let

0

::

(123),

T

(12) and define f:Sym(3) ----7Z2 by setrt.ing

f(S)

fl, S

-Ie

=

Then (f;o) has order 6 and it is easily verified that (f;o) f (f;O)-1 in G. Hence G is not a Q-group. The concept of a transversal permutation represerrtatlon provides an appropriate restr-iction on the action (H, y) •

64

Let (G,X) be a permutation representation of a finite group G and let When the action of G on X is restricted to the cyclic subgroup

0

c G.

one obtains a

(0),

decomposition of X into transitive constituents. Each constituent is the o-orbit of an element x c X and is denoted by 0 (x) . Let 0

I

{O (x)

o

N(o), then TOO(X) = 0a(TX) for every x e X

stand for the orbit space of (0). If T

and therefore T induces a permrtation T'" of 0

defined by T

T;'. To say that T

permrtas each of the orbits 0 (x), x

o

Definition. o

G, ker

(1)

*

(G ,X) is a

0

,

Thus, there is a homorrorphism

lS

equivalent to the statement that T;'

X.

perrrRltation representation of G if, for every

intersects every coset of C(o) in N(o). That is,

o

G,

Cl

G and every coset a

the permutation

n

e N(O)!C(o).==}

(2) (G,X) is a strongly

a

X}

x

Cl

f. 0.

perrrRltation representation of G N(o)!C(o) , there is some element T

IOo(x) has a fixed point for every x c X.

N(o)

C(O)

TC(O)

(

\

ker

*

'\ )

ker

*n

for every Cl

such that

65

Now, a coset a SOllE

o

£:

N(cr)/C(cr) corresponds to an element V such that V a V-I

a

integer n, (n ,

1(1) = 1.

a

a

= an

for

To say that (G,X) is transversal means that for every

G and every Va e N(o), there is

SOllE

element

N(o) such that

T

For a Q-group G to have a transversal representation CG,X), it is required that each pair of generators of a cyclic subgroup not only be conjugate but that each such conjugation be ac=rrplished by an e Ierrerrt which leaves every orbit in the orbit space of the cyclic subgroup invariant. Specifically, for every

every x

0

c G and every A

AutCcr),

X. This is precisely the situation which exists in the symmetric groups:

the natural action of Symvn) on the set!::. = {L, ... , n} is a strongly transversal representation. For suppose that jugator" of

0

0

Ci

l

1

a.ill 1

= 1,

The term "standard con-

means any permutation of the set!::. defined by the scheme

to

a.

i

= Coill) .

c Symf n) and Co)

.•• , s, Where

. .. it)

1 ... 1

=

(jl

jt)'

, ••. , Os are the cycles of

0.

By its definition, a standard

conjugator leaves every cr-orbit invariant. Therefore, (Sym(n), Furthenrore, since so that j I

= i l·

, ... , it}

= {jl'

is transversal.

... , jt}' one may arrange the letters in

In this case, a standard conjugator may be found which leaves a

point in each o-orbi.t fixed. Therefore, the action (Symf n) , !::.) is strongly transversal. Let G and H be finite groups and let

(H,y) be

a permutation representation of H.

66 By using the projection horrorror-phi.sm G

group G 1 1 and (m, 1(1)

and hence r Cx)

X £ F(o),

then

,F(o)

Ffo ) and henoe F(rr) is invariant under N(o). Now, ,(0-1)

(am -Dr = (0-1)

if p does not

1. If

F(o). Therefore

£

+ ••• + a + 1) . Therefore ,Im(o-I)

=

,0-,

om,_,

=

Imfo -L) and it follows

that Im(o-l) is invariant under N(o). This proves statement (I). Now, suppose that p

t 101·

Let x

£

F(O)nIm(O-l), say x

= o Cy)

-y, y e V. Then o'(y)

=x

+ y and, by

induction , it follows that as (y) = sx + y for any integer s > 1. Let s = sx = O. Therefore, x = 0 since p by dimensionality that V

If a

£

= F(o)

G and x c V, let Va

defines a linear transformation

t

1&

s , Hence F(O)nIm(o-I)

z: { O}. It

101

to get

now follows

Im(o-l). / /

= V/(o-l)V

and X

o

=x

+ (o-l)V. Every element T c G

79

, (x + (a-l)V) a

by setting

= ,(x)

+ (, 0 ,-1 -l)V.

Then 'a is a well-defined linear Lsorrorphi.sm.

Proposition 79.

Let (x,a), (x',a')

£:

if and only if there is some element,

VG. Then (x ,c) and (x',a') are conjugate in VG E

G such that the following conditions are

satisfied:

Proof.

Suppose that (y,,)(x,a)(y,T)-l= (x ' ,a') for some y e V, T e G. Then, a

a' and y + LX - , a T- 1 y = x ". Hence rvx) - x'

a'(Y) - Y

= (a'

,-1

-l)y. Therefore,

Conversely, if T e G is such that statements (1) and (2) are satisfied, then r Cx) x'

£:

(a' -l)V. Set r Ix) - x '

=

(a' -l)y. Then (y;r)(x,a)(y,T)-l = (x ' ,a'). II

We are only interesting in the case of Proposition 79 when (x',a') = (x,a)m for m relatively

to s(x,a). In this event, a'

For, (am -1)V = (o-D (am-1 + ••• + a + l)V

and it follows that Va'

c:=: (a-l)V.

some integer n ::: 1 such that aITm = a. Therefore (a-1)V Hence

(JU

-1)V = ( 1. Therefore, -

(rr -J.)x (rr-l

+ ••• +

(rr- l

+ .•• + 0 + l)w

0

+ l)(o-l)x

-L)x e (0 t -1)V(J W = (0 I -l)W. p p

£

w.

-1)(0 ,X)

P

86

Set (a , -l)x p

= (ap ,

-Dw', w'

E:

W. Then

w -

(a , -L)x p

w - (0 , -Dw' p

E:

Set (0

w

P

-1)(0 ,x) p

(0

P

=

(0

=

(o_D[(oB-l + '"

E:

(o-DW.

P

-l>w",

W ll E:

W. If

0

P

= aB ,

B > 1, then

-1)(0 ,x) + (0 , -l)x P p

Hence (o-l)VnW

= (o-l)W

+

rJ

+ Dw"

+ (cf-l + ••• + a + Dw'J

and 'therefore (W,H> is admissible. II

We apply the above results to the following situation. Let p:G ---) GUV) be a rational representation of G. Then p is equivalent to an integral representation of G and therefore V contains a G-invariant lattice L. If P is a

, then L/pL is an

FP-vector space on which G operates. We restrict our attention to the special case where (G ,X) is a permutation representation of G and

87

and

L

Ix

0x

x) Z·

E

If P is prime, set

In order that X G be a Q-group, it is necessary that p

P

= 2;

for, if A

=

A , summed x

over all x EX, then A f: 0 and oA = A for every a E G. Consequently, A is a nontrivial element of order p in the center of X G. Since the center of a Q-group is p

elementary 2-abelian, it fo.l.Lcws that p

PrDpcsition.§2..

=2

if X G is to be a Q-group. p

Let (G,X) be a permutation representation of a Q-group G. Let H be

a Q-subgroup of G and W an H-invariant subspace of X Assume that: 2.

(1)

the restricted action (H,X) is transversal

(2)

(W,H) is an admissible pair.

Then, WH is a Q-group and the pUll-back action (WH,X) defined by (w,o)x = ox is transversal. Proof ,

let

ill

We shoe that the criteria in Corollary 80 are satisfied. Let (w,o)

E

WH and

be a pcsitive .irrteger which is C'elatively prime to s (w ,0). Since p = 2, one

may assume that s (w ,0) is even and hence that

ill

is odd. Thus

transversally on X and therefore there is some element, and ,0 (x) = o

° (x) for all x 0

E

X. I t must be shown that

E

TllWa

w(J' Now, H acts

H such that,

'w(J (w) (J

0,

w(J' Let x

-1

E

consider (,-UA Since r Cx) s 0o(x), there is an integer k > 1 such that ,(x) x' ok(x). Therefore ,A

x

= akA and hence x

= rlV «A) a

x a

)

= aill X and

=

88

Since the Ax form a basis for X it follows that TV 2,

o

IV . Finally, since the pair

o

(W,H) is admissible , it follows that

Wo and therefore TW = o

lw . Hence 0

=

lw0

T (w ;:; w . To shew that the action of WH on X is W0 o) 0

transversal, one need only remark that the equation T (w ) = Wo 0 tence of an element (y,T)

°o (x) = O(

w ,c

E

WH which conjugates (w,o) to

JlW

0

(W,O)ffi.

implies the exisThen (y,T)O(

W,O

)(x)

) (x) by the definition of T. Therefore, the pull-back action

(WH,X) is transversal. I I

Corollary SSA.

Let (G,X) be a transversal permutation representation of a Q-group

G. Then:

(2)

I f a Sylew 2-subgroup G of G is a Q-group and the restricted action (G is 2 2,X)

transversal, then the Sylow 2-subgroups of X are Q-groups. 2G Proof.

Since the pair (X is admissible, it follows that X is a Q-group. Final2G 2,G)

Ly , since X

2G2

is a Sylow 2-subgroup of X and since (G is transversal, it fol2,X) 2G

lows that X is a Q-group. II 2G2

It should be observed that the representation of H on W may be a permutation type representation. That is, Wmay have a basis

although it may not be true that

Bw,

Bw c:= X.

Bw c:= X2 which

is permuted by H

Nevertheless, if H acts transversally on

then WH is a Q-subgroup of X 2G.

Corollary SSB. of G and let Y

Let (G,X) be a permutation representation of G. Let H be a Q-subgroup X be an H-invariant subset of X such that H acts transversally on

89 Y. Then the group Y is a Q-group. 2H

These results give two methods for obtaini.ng Q-subgroups of

If G acts

transversally on X, then X is a Q-group and we asked under what circumstances WH 2G is a Q-subgroup when W c::::::: X and H .:.. G. Proposition 87 regards WH as a subgroup of 2 X and requires that H be a Q-group which acts transversally on X and that the pair 2G (W ,H be admissible. On the other hand, Corollary 88B considers the group WH inde2) pendently of its relationship with X and requires that H be a Q-group and that W 2G have the form Y for some subset Y c::= X on which H acts transversally. 2

Remark,

The wreath and semi-direct products coincide for groups of the form X G. P

To see this, let Z

p

= {I,;,;, •.• , ;,;p-l}. If

;,;s(x), 0 < sex) .:.. p-l, and define ¢:Z

¢(f;cr)

(G ,X)

p

=

(

x e X

p

is a function, let f (x) =

X G by setting p

s(x)A ,cr). X

It is completely straight-forward to show that ¢ is a group-isomorphism. Because of

this Lsomorphi.sm , the wreathing action of (Z ,Z )reg 1 (G,X) on the set Z x X may p p p be transferred to the group X G as follows: p

x

£

X

,cr)

.x)

=

X

=

These two groups are then operator .i.sorror'phi,c ,

Let (G,X) be a permutation representation of a finite group G. The group B(G,X)

= Z2

1 (G ,X) was introduced at the end of Section 1. It follows from the above remark

that B(G,X)

c;

X Because of this, the elements of B(G,X) will frequerrtf.y be written 2G.

90 D(G,X)

I t is clear that D Z be the free Z-mcxJ.ule generated by P. A multiplication is

defined on M[PJ by setting

xy

for x , y

E:

P

E:

P

x,y

(p) p

P and extending linearly to M[PJ. In order to establish the ring-theoretic

properties of M[PJ, one first uses the zeta-function of P to define evaluation mappings on M[pJ. If x

E:

P, define

:M[pJ

--7 Z by setting 1;x(y) = 1;(x,y) where y

E:

P

and extending linearly to M[pJ. The functions 1;x are Z-algebra homomorphisms with the property that sx(a)

=0

for every x

E:

= O.

P if and only if a

It follows from

this that MEPJ is a commutative Z-algebra with identity ([53J); M[PJ is called the P.

The elements e

onal idempotents and

x

E:

x

L= y

E:

P

j..l(y,x) y form a set of pairwise or-thog-

e is the identity element of M[pJ. Now, it follows from P x

the above merrt.icned property of the s

x

that if

a

E

n ker 1;

M[pJ is a Z-closed algebra and it follows that Hornz(M[PJ ,Z)

x

= O.

, then a

= {sx I

x

E

Therefore

p}. We remark

that not every Z-elosed algebra is the Mobius ring of some partially ordered set since M[P] always contains orthogonal idempotents while Z-elosed algebras do not, in general, contain such idempotents.

129

The algebra MQ[PJ

= Q GlZ M[PJ

is called the rational Mobius

of P. I t

may be used to characterize the Burnside algebra BQ(G) = Q 0z B(G) of a finite group G as follows ([56J):

where 8(G) is the set of dered by the relation K(H)

classes of subgroups of G and is Q' Take p

=0

Conversely, let X

in PrQposition 138 to get

E;

X(H) ¢H for every H

B (G) be such that ch(X) Q E

8(G). Therefore, 0

= O.

Now, X =

= ch(X)

H

:L: X ¢H wher X ¢H = H

X(H) ch ¢H = 2 X(H) A W H cyclic

Since the A are linearly independent, it follows that X(m H

=0

whenever H is cyclic.

Hence,

X

Corollary

= ;>

H non-cyclic

(Generalized

X(H) ¢H

Theorem).

E;

(¢H

I H non-cyclic) Q'

II

Let G be a finite group, p a prime. Let

X be a rationally represented character of G which is constant on the p-classes of G, Then there is a positive integer N such that Nx is a generalized permutation character of G and p does not divide N. Proof,

Let X = E X A be the local decornposition of X, the summation being over all V

Q-p-classes V. Since X is constant on each p-cl.ass and is rationally valued, it is also constant on each Q-p-class V. Let XCV) stand for the value of X on V. Then

Now, q; f!.(V)

E;

Bp (G) and hence 'ther'e is some integer N, P tN, such that N q; f!.(V)

B(G) for every V, Ther fore, ch(N q; f!.(V» G. Hence

E;

is a generalized permutation character of

140

is also a generalized permutation character of G since X(V)

E:

Z. / /

Remarks. 1.

As noted in Corollary 138, Solomon was the first person to

of the kernel of the map BQ(G)

this description

The kernel of the character map ch:B(G)

reG) is more difficult to describe although it is relatively

to show

that its rank as a Z-lIlOdule is equal to the number of conjugacy classes of non-cyclic subgroups of G. For, it is clear from the construction of 'the absolute idempotents ¢H that IGI ¢H

E:

B(G) and therefore

is a subrrodu'Ie of kernel (B(G)

reG)) which has the same Z-rank as kernel (B(G)

---7 r(G)). However, these two modules are not generally the same. 2.

Artin I s theorem is obtained from Corollary 13 9 by setting p

represented character of G

o : 6:. rationally

g-linear combination of permutation characters of §..

In fact, it is not difficult to show that these permutation characters may be chosen so that they are induced from cyclic subgroups. For, let X

E

r(G). Then

X

It follows easily from the construction of ¢(s) that the coefficients

=0

for all non-cyclic subgroups K. Therefore, the transitive constituents of ¢ (s) must have cyclic stabilizer. It follows that X is a Q-linear combination of permutation characters induced from to as Artin's theorem.

subgroups. This statement is also , we note that when p 't 0, Corollary 139 cannot be

strengthened so that the permutation characters are induced from

subgroups.

141

The problem is that a

p-class t,(V) generally contains nO!l-cVC.L1.C subgroups

and therefore the transitive constituents in

t:.(V)

do not necessarily have cyclic

stabilizers.

3.

The Corribinatorics of p-Classes

The purpose of this section is to obtain IIDre information about the primes which divide the cardinality of a p-class of a finite group. -Lf X is a

IJ- m(V). Now, suppose that H

If H is cyclic, then H

T

E

= (T)

E MV)

)

are not conjugate . Hence

is not conjugate to (T) for any T

E V.

and p(T) is Q-conjugate to some element of V. Therefore

V and this contradicts the definition of H. Hence H is non-cyclic. It new follows

164

= m(V)

that m(6(V»

+ C. II

A Q-'p-class V is called

= 1.

if m(V)

The following result characterizes

simplicity in 'terms of the centralizer of a p-regular element of V.

Proposition 164.

Let S be a p-regular element of V.

= (1).

(1)

V is simple if and only if C(S)

(2)

If G is a Q-group and V is a simple p-class of G, then Ivl

Proof.

p

p

IGI p .

It should be remarked that statement Cl) is independent of any particular

p-regular element of V because any two such elements generate conjugate cyclic subgroups. Let X

= Kp (S).

It follows easily that V = K(S) if and only if X = K(S). Since

by Proposition lL;lB, it follows that

V is simple

¢=?

X = K(S)

IKp (1 ;C(S) I

¢=?

=1

C(S)p = (1) •

This proves statement (1). To prove (2), let G be a Q-group, V a and Sap-regular element of V. Then C(S) 143 that Ixl

p

= \Gl ' But V p

p

=

(1) 0 such that tv Xv ip MV) tv N X A.q

E

Irn ch

= P(G). Since NX =

integer t > 0 such that tNX

E

V=

chv(Xv

ipMV»' Now, there is

BCG) and p

£

t

Therefore

i: NXA.q, it follows that there is an

P(G) and p

t

t. To show that y(G) IN, let

182

pa be the largest p-poeer- which divides y(G). It follows from the rninimality of y(G) that there is some character X c r(G) such that X i P(G) but pa X e peG) and, if m X c P(G) for some positive integer m, then pa

I m,

From the above discussion it follows that t N X £ peG) for some positive

integer t, p

t

t. Therefore pal t N and hence

IN.

Hence, every prime

factor of y(G) occurs with equal or higher multiplicity in N. Therefore y(G)

I N.

It follows from statements (A) and (B) that y(G)

Corollary 182.

y(G)

=1

if and only if YV(G)

=1

= lcm

YV(G). II

for every prime p and every Q-p-

class V.

Values of the local invariants YV(G) are related to those of subgroups as follows. Let H ::. G, V a Q-p-class in G and W a Q-p-class in H such that W

V. In

Chapter 3, Section 4, we associated with the pair Wc=::: V the local restrictioninduction maps: Res(V;W) - - - - - 4 ) r(H)w

B(G) r:. 0, this is a contradiction since C i. O(mod pZ(p))' II

207

Thus, by choosing the constituent X properly, the local invariant Y\i(G may p V) be computed directly from the local equations on \i.

3.

Local Splitting

The local equation on \i corresponding to the pair 1jJ,p is said to be solvable if a l (S) divides a", (S). It is clear from the results in the previous section ,Xp 'j',p that any criterion for insuring that Y\i(G = 1 must take into account the solvabilV) ity of the local equations. In this section we define the concept of local splitting and show that it provides such a criterion.

Definition.

Let G be a finite group, p a prime, V a Q-p-class of G and Sap-regular

element of V. G is said to be locally split on V if C(S) is a direct factor of -p N(S) ; that is, N(S) P

P

= C(S)

P

x K for some K < N(S) . G is locally split at l2. if G -

P

is locally split on every Q-p-class V. G is everywhere

split if G is locally

split at all primes.

on V is well-defined; that is, if S and S I

The property of being locally are p-regul.ar elements of V, then C(SI) C(S)

P

p

is a direct factor of N(S')

P

if and only if

is a direct factor of N(S) . This follows from the fact that S and S I are Q-

P

conjugate and therefore N(S l) a prime p does not divide

p

1G I ,

-

N(S)

P

and C(S I)

P

C(S) • It is also clear that if P

then G is locally split at p since the Sylow p-sub-

groups are trivial. Therefore, in order to determine those Q-p-classes on which G is locally split, it is sufficient to consider only those primes which divide

Proposition 207.

Let G be a Q-group and let S e G. If 1jJ

£

Irr (S), then

IGI.

208

where

S»

is the cyclic group generated by the root of unity

N(S)

Proof.

=

Let

Now, N(S)

Then

p

p

(Sn). We show that

Aut(Sn). N(Sn). If x

N(Sn) and C(SD) (s(x), Is

P =L

y S y -1

=

Ss(X). Ther-efor-e x SD X-1

that sex)

= sn(X)D = SS(X)D = y

X £

¢==}

I t follows that

a such

= N(S)(l C(SD).

X £

£

N(SD) , let

I t follows from the Chinese Remainder ==

rntx) (mod ISDI) and

Since G is a Q-grDUp 'ther-e is some element y

Next, we claim that

(S). In particular,

£

N(S) such that

Sn y -1. Hence y -1 X c C(SD)

For,

N(S) and

x S X-1

= Sm(x)

= and

mtx)

=

209

and, in particular, Aut(Sn)

c

p

N(S) IN(S)1jJ. I I p p

Let G be a Q-group. Then G is locally split on V i f and only i f

Proposition

is a direct factor of N(S)p for every 1jJ E lrr (S). Proof. --

Suppose that N(S)1jJ is a direct factor of N(S) for every 1jJ p p

be a positive integer which is relatively prime to IS

setting 1jJ(S)

I and

lrr (S). Let m

E

define 1jJ: (S)

I;m where I; is a primitive ISI-th root of unity. Then 1/1

it follows that C(S)

P

Irr (S) and

on V.

is a direct factor of N(S) . Hence, G is locally

P

Conversely, suppose that G is locally split on V and let N(S) some subgroup K. Let 1jJ

E

C by

E

p

= C(S) p

x K for

lrr (S). To shew that N(S)1jJ is a direct factor of N(S) p P

it is sufficient to show that N(S) (lK is a direct factor of K. For, suppose that p

for some L < K. Then

Hence

N(S)

= C(S)

p

P

x K = [C(S)

P

x (N(S)1jJ(lK)] x L P

= N(S)1jJ x L. P

To shew that N(S)1jJ(lK is a direct factor of K we proceed as follows. We have that p

K '" N(S) IC(S) p P

o:

(Aut(S))

P

and, under this isomorphism, the subgroup N(S)1/I(lK

P

corresponds to the subgroup

=

=JIl