Characters of finite groups (Mathematics lecture notes)

:(Q)f

} tll1lite G;n:iloups

Walter Feit

CHARACTERS OF FINITE GROUPS

MATHEMATICS LECTURE NOTES

Paul J. Cohen stanford University

Set Theory and tile Continuum Hypothesis

Walter Feit Yale University

Characters of Finite Groups

Marvin Greenberg Northeastern University

Lectures on Algebraic Topology

Serge Lang Columbia University

Algebraic Functions

Serge Lang Columbia University

Rapport sur la Cohomologie des Groupes

Jean-Pierre Serre College de France

Algebres de Lie semi-simples complexes

Jean-Pierre Serre Coll~ge de France

Lie A 1gebras and Lie Groups

CHARACTERS OF FINITE GROUPS

WALTER FEll' Yale University

W. A. BENJAMIN, INC.

New York

Amsterdam

1967

CHARACTERS OF FINITE GROUPS

Copyright © 1967 by W. A. Benjamin, Inc. All rights rese rved Library of Congress Catalog Card Number 67-20769 Manufactured in the United States of America

The manuscript was put into production on September 20, 1966; this volume was pUblished on October I, 1967

W. A. Benjamin, Inc. New York, New York 10016

PREFACE These notes contain the material covered in a course I gave at Yale University during the academic year 1964-1965. A knowledge of linear algebra, Galois theory, and elementary group theory is the only background required of the reader. The primary aim of this book is to familiarize the reader with some of the methods which have proved fruitful in current research in that aspect of group theory which uses the theory of characters. These notes are not meant to replace any of the textbooks on the subject but rather to supplement them. Some overlap with the many group theory books in circulation is. inevitable but I have tried to keep it to a minimum. In Chapter I representations and characters are defined and their basic properties are developed. Nowadays, this is usually done by way of the theory of algebras. However, these topics are here approached from a more classical point of view. This is done partly to illustrate the elementary nature of the material but mainly to derive the basic properties of characters as rapidly as possible. Chapter II centers about R. Brauer's fundamental theorem concerning the character ring of a finite group and some of its generalizations. Several applications are given including some concerned with splitting fields and the Schur index. Chapter III contains various criteria for a group to be nonsimple. Included are several classical results of Burnside and Frobenius. P. Hall's characterization of solvable groups, and J. G. Thompson's criterion for a group to have a normal p complement for an odd prime p. Many of the results in this chapter and their generalizations are of basic v

vi

PREFACE

importance for any further study of the structure of finite groups. As is well known, most of these results can be proved by using the transfer in place of character theory. The material in Chapter IV is mainly of recent origin. Several disconnected topics are discussed. These are meant to provide a random sample of some of the work that has recently been done in this area. Most of the proofs in this chapter utilize the concept of a trivial intersection set whose importance for character theory was first recognized by R. Brauer and M. Suzuki. Some generalizations of this concept are also treated together with the related concept of coherence. Several people who listened to these lectures made helpful comments which were incorporated in these notes. I especially want to express my thanks to G. Seligman and F. Veldtkamp for their suggestions. I am also greatly indebted to Mr. Leonard L. Scott whose critical scutiny of the material prevented several errors from appearing in print. Walter FeU New Haven, Connecticut March, 1967

CONTENTS Chapter I

91. §2. §3. §4. §5. §6. §7. §8. §9. §10.

Representations Characters Complex Representations Integral Representations The Center of the Group Algebra Some Properties of Characte~s Character Tables Induced Representations Induced Characters M-Groups

1 10 18 23 27 32 41 43 45 58

Chapter II

§11. §12. §13. §14. §I5. §16. 917.

The Schur Index A Combinatorial Result and Some Consequences Rational Valued Characters ;;-Elementary Groups The Character Ring Schur Indices and Splitting Fields Equations in Groups

61 67 69 71

78 85 87

Chapter III

§18. §I9. §20. §21. §22.

Criteria for Solvability Quotient Groups Nonsimplicity Criteria Normal 'IT-Complements Thompson Subgroups vii

93 98 103 113 117

CONTENTS

viii

Chapter IV §23. §24. §25. §26. §27. §28. §29. §30. §31. §32. §33.

T.!. Sets Characters of Relatively Small Degree Frobenius Groups An Excursion into Number Theory CN Groups Nonsimplicityof Certain Groups of Odd Order Properties of Involutions Groups with Quaternion S2-Subgroups Coherence A Class of Doubly Transitive Groups Isometries

123 125 133 139 142 147 152 156 158 167 172

Notation

181

References

183

Index

185

CHARACTERS OF FINITE GROUPS

CHAPTER § 1.

REPRESENTATIONS

Let N be a group and let If be a field. An If-representation ~ of (~is a homomorphism of (\) into the group of nonsingular linear transformations of some finite-dimensional vector space 'U over If. 'U is called the underlying vector space of ~. The dimension of 'U is the degree of ~. An Ifrepresentation is faithful if it is an isomorphism. Two Ifrepresentations if l ' 32 of N are similar if they have the same underlying vector space'U and if there exists a nonsingular linear transformation S of'U such that i'ij(G) S-l ij2 (G)S for all G E: 6). ~ representation of ® is an ij -representation of (~for some field ~. If N is a group and If is a field, then the group algebra If(N) of (~) over If is the ring consisting of all elements of the form LNa(G)G, where a(G) E: If for all G E: (~ and where addition and multiplication are defined in a natural way as follows; L;a(G)G + :Eb(G)G "" L; {a(G) + b(G)} G N

(\)

6(.. a(G)GI L;("lb(H)H ., •

'6

(\)

= L;

G,HE:N

I

L; a(GH- 1 )b(H) G

GE:('! HE:N

a(G)b(H)GH

2

CHARACTERS OF FINITE GROUPS

If ~ is an ~-representation of N with underlying vector space '0, then ~ can be extended to a ring homomorphism of ~'N) into the ring of linear transformations of '0 by defining ~{~(,\a(G)G} ~Na(G) ~(N). In this way '0 has a unital ~(N)-module structure imposed on it. Conversely a finitely generated .unital ~«~)-module gives rise to a uniquely determined ~-representation of ~. Thus the study of ~-repre­ sentations of N is equivalent to the study of finitely generated unital ~(~)-modules. The module point of view has been treated extensively in the literature, see for instance the books by Curtis and Reiner or M. Hall. In these notes we will however adhere to the representation approach. If '0 is an n-dimensional vector space over the field ~, then the group of nonsingular linear transformations of '0 is isomorphic (in many ways) to the group of nonsingular n x n matrices with coefficients in ~. Thus an ~-representation of degree n of (\\ could equally well have been defined as a homomorphism of ~ into the group of nonsingular n x n matrices with coefficients in 5=. It will often be convenient to consider it as such. Let ~ be any field. The following are some examples of ~ -representations of a group N. (i) The unit ~-representation ~ of N of degree 1 is defined by ~(G) = 1 for all G E: ()). (ii) If il is a homomorphism of ~ onto a group of permutations on the n symbols 1, ... , n, then il may be interpreted as an 5=- representation in the following manner. Let '0 be a vector space over 5= with basis {VI' ... , vn}. Define vi il(G) vi ~(G) for G E: N, i 1, ... , n. Such an ~-repres entation is called a permutation ~ - representation of (~. (iii) The (right) regular ~-representation ~ of ~ is the permutation ~-representation of (';\ arising from the (right) regular representation of N. More explicitly let '0 be a vector space over ~ with basis {vHI H E: ('I}. Define vH9{(G} vHG for all G,H E: (~. (iv) If ill' il2 are ~-representations of (~ with underlying vector spaces'O,'W respectively. Then ~'I ® ~2' ill ® il2 are ~-representations of N with underlying vector spaces '0 ® 'W, '0 ® 'W respectively.

REPRESENTATIONS

3

(v) If I" is an 9=-representation of (~ define the contragredient representation 1"* of il by 3* (G) = I1(G-l Y for G E: (\), where the prime denotes transpose. Let '0 be the underlying vector space of an 9=-representation Ii of N. An invariant subspace of '0 is a subspace 'W such that'W I"(G) c 'W for all G E: N. If 'W is an invariant subspace of '0 then I" defines l'F-representations of (~with underlying vector spaces 'Wand 'O/'W. These representations are called constituents of Ii. An l'F - representation I" of N or its underlying vector space '0 is l'F -irreducible if 0, '0 are the only invariant subs paces of '0. I" or 'U is l'F - reducible if it is not I" -irreducible. I" or '0 is completely reducible if '0 = '0 1 •• " ~k be a maximal set of pairwise nonsimilar absolutely irreducible \i-representations of 6.l. Such a set exists by (1.10). Let ~s(G) = (afj (G» for G E: (~\. Let X be the extension field of ~ generated by all afj (G). The finiteness of (~ implies th.,.at Ix: 5] is finite since each afj (G) is algebraic over 5. If I~ is a Xirreducible X- representation of (~~ then in ~, ~ is similar to a direct sum of various ~j' Thus if ~(G) (aij (G» for G E: (~ then all is an 5 linear combination of the a~j . Thus by (1.9) ~ is similar to some ii s in X. Hence X is a splitting field of N.

10

CHARACTERS OF FINITE GROUPS

Assume that char 5=YI("'1 . Let ~ be an 5=-representation of ("'. If G E: and thus b E:: &Oi. Hence 11 = &Oi as required. Throughout this section we will adhere to the notation just introduced. (4.1) may be used to give an alternative proof of (2.18). We will here generalize that result.

(4.2) Z(",) .

Proof. If suffices to show that 1("':.8l/x(l) is a local integer at every prime p. Choose a prime p. By (4.1) there exists a 1>-representation ~ of (») which affords the character X. Let ~(G) = (aij(G» for G E:: (~i. If Z E:: .8 then ~(Z) = E(Z)I, where E (Z) is a root of unity. Thus for GE::(l},ZE::.8

Let G1 ,

•••

tatives of

.B

,G

m

be a complete system of coset represen-

m

m

2] all (G. Z) all (Z-l G. -1) i=1

.B

in N. Hence for Z E::

1

=E

i=1

1

Therefore by (1.9)

: : I(»):.BI X (1)

all (G.) all (G. -1) 1

1

26

C H A RAe T E RS 0 F FIN I T E G R0 UPS

Since a l l (G) E: [) for G E: (\) this implies that /(\\:.8I/x(l) E: [) completing the proof. For a E: [) let a* denote the image of a in :D/~ == :1)* . Assume that p l I(~ I. Let ~, ~ be :I)-representa(4.3) tions of (\) which are absolutely irreducibie ~-representa­ tions of 6J. Then ~ * , ~ * are absolutely irreducible '!'* representations of (\). Furthermore ~ is similar to ~ in ~ if and only if ~ * is similar to ~* . Proof. If ~ * is not absolutely irreducible then replacing by an extension field if necessary it may be assumed that 1* is reducible. Hence there exists a matrix T with coeffi~

cients in :D* such that T-l!* (G)T

==

((Sl~G) <S2~G»)

for all

G E: (\). Let S be any matrix with coefficients in !D such that S* = T. Then det S ;: det T ~ O(mod~). Thus S-1 has coefficients in

1).

Hence {S-11 (G)S}*

Thus it may be assumed that 1* (G) G

E: (~.

Let 1 (G)

= (aij (G»

=

= S*-1 I*(G)S* .

(£I~G) <S2~G») for

and let n be the degree of

~

.

By (1.9)

(I~n I) *

= L; a * (G) a * (G) = 0 @

In

nl

Thus 1~l/n ;: O(mod p) contrary to assumption. Hence ~* and ~* are absolutely irreducible. If ~ * is similar to ~* then, as above, it may be assumed that ~* = 5B*. Let >B(G) = (b .. (G» for G E: . Let e be an irreducible character of < P, G >. Then 6(1) 1 and 6(PG) = 6 (P)6(G). Furthermore 6 (P) is a pm-th root of unity for some m and so 6 (P) 1(mod ~). The result follows. (Solomon [2]) Let sr l , ... , sr'k be the conjugate classes of ®. Let G . e::: .tt J.• Then E~ 1 X(G.) is a nonJ - - J= J negative rational integer for any irreducible character X of ®. (6.5)

Proof. Let ! be the permutation representation of ® on the elements of @' defined by H!(G) = G-l HG for G, H e::: ®. Let 6 be the character afforded by !. Since H~(G) = H if and only if HG = GH, the number of elements fixed by ~(G) is IC(G)I = 1@1/I·stl, where Sl is the conjugate class of @ containing G. Thus by (2.3) e(G.) . k J I®I/I~.I. Let 6 = E. 1 a. X· where Xl' ... , X are the k J 1= 1 1

irreducible characters of ®. Then each a. is a nonnegative integer and 1 a. = 1

I~I L;6(G.)I·~.lx.(G.) \'Y. J J1J J

=

L; x.(G.) .1J J

s 0 M E PRO PER TIE S 0 F C H A RAe T ERS

35

L. Solomon and J. G. Thompson have pointed out the fact

~hat in contrast to (6.5) 1:;~::::1 xi(G) need not be a nonnegative mteger. A character e of ® is a linear character if linear character is necessarily irreducible. (6.6)

e(1)

1. A

Let X., X·, X be irreducible characters of ®. The 1 J multiplicity of X in X. X. is equal to the multiplicity of X. 1 l 1 in XX.. If furthermore X is linear then the multiplicity of J X in X· X. is 0 or 1 . 1 l-

Proof. The multiplicity of X in Xi X j and the multiplicityof Xi in XXj are both equal to

1/INI

1:;(\\

Xi(G)Xj(G)x(G).

If X is linear then XXj is irreducible. Thus Xi is a con-

stituent of XX j if and only if Xi:::; XX j in which case the multiplicity of Xi in XX j is 1. (6.7) Let ~ be a representation of ® which affords the character e . Let S; be the kernel of eo Then (i) le(G)1 :5 e(1) for G E: ®. (ii) e(G) ;:: e(l) if and only if G E: .po (iii) If le(G)1 e(l) then G.\) is in the center of ®/S). If e is irreducible thenconversely III (G}I :::; e(1) for G.n in the center of ®/~). Thus in particular {Gle(G) :::: e(l)} 0, i

1, ... , n. Then

@ con-

tains an element G of order p;l ... pan . n

b· b·+l Proof. Let p.111@1 and p.l 11@1. Then a. --

1

b.-a.h

5. be the field of I@\/p.l 1

1

1

~

b .. Let

1

-th roots of unity over

1

~.

Since X(G ) ¢ 5 i there exists an element 0i E: S ~ 1@1/5 . i 1 such that 6.(x(G.)) :r= X(G.). On the other hand o.(X(G.» 1 1 1 1 J X(G.) for j :r= i. If the result is false then for G E: @, J ci.(x (G» = X (G) for some i depending on G, 1 ~ i ~ n. 1

n

Therefore n =l (1 - 0i)(x(G»

0 for all G E: @. Hence i n~=l (1 - 0i)(x) ::; O. Expanding this product we get that

41

CHARACTER TABLES

x + :6 (\

OJ' (X ) + •••

i<j

:6 0. (X) i

:6

+

i<j

G )

1 and s

;t!

=1

if and only if

n,19.(G.) 11

= 0

(1, ... ,1). Since O. vanishes on p~ -ele-

n

I

l

ments it may be assumed that in the definition of s, G.

1

ranges over the p. -elements of @. Since 11. (G) ) 0 for any 1

p. - element G. it follows easily that s 1

1

1

=1

if and only if

condition (i) is satisfied. For any n-tuple (G , ... ,G ) define HI' G ... G. for 1 n I l i 1, ..• , n - 1. Thus

C H A RAe T ERS 0 F FIN I T E G R0 UPS

50

1

x

6

H , ... ,H 1

n-l

X. (H)··· X. (H- l ) J 1 J n-l 1 n

Applying (6.0 n - 1 times this yields that

1

By the Frobenius reciprocity theorem s = 1 if and only if for j

;;t!

n1;:l

c h ::; 1. Thus

1 there exists i such that c .. = O. 1J

In other words s = 1 if and only if condition (ii) is satisfied. Thus the statement that s = 1 is equivalent to both statements (i) and (ii). It was first pointed out by P. Hall that condition (i) of (9.7) is satisfied in case @ is solvable. This can be proved by induction in a straightforward manner. He also conjectured that

INDUCED

51

CHARACTERS

the converse is true. The proof of this conjecture has recently been given by J. Thompson. However, it is far beyond the scope of these notes. (9.8) (Mackey) Let Xl,tI be subgroups of ®. Let 0,11 be complex valued class functions on ~), tI respectively. Let ()G l .\!; •.• ,'£)G m tI be all the (,u, tI) double cosets in ® and let G)i :: ~)Gi for i :: 1, ... , m. Then

n \'

«() *,11 *)®

m ( G'1

:: . ~

1-1

()

)

1®.,1111U 1

\'Y.

1

<M

i

Proof. By definition (f)

*,11*)Ci}

1 11M

1-';1

m I®I

1

1 1 [\II I®I

l.u! If MN-l

HGi L for H E:

and

E

f)MN"l (G)11 (G)

.E

E

N E: G) M,G E: ®

E

N"l (G)l1(G)

f)M

M,N,GE:®

,u, L E: tI then OHGiL ::

=

GE:®

E

G L f) i (G)l1(G)

GE:®

E

1

f)G i (G)l1 L - (G)

GE:®

E

f)G i (G)l1(G)

GE:®

=

E G E: G)i

N

(G)11 (G)

f)

f)G i (G)11(G)

f)

Gi L

52

C H A RAe T ERS 0 F FIN IT E G R0 UPS

Thus

m

=

1 11 - E E IRII i\il IGI i = 1 M E: ()~

E

NE:@ MN-1 E: £'>G. \1 1

The result now follows since

1.\'1 Ittl =

I~Gi ~I

I®/

(9.9) Let e be the character afforded by a transitive permutation representation ty of ev. Let ~) J:>e the subgroup of ()~ consisting of all elements G such that ~(G) leaves a given object fixed. Then 11911@ is equal to the number of domains of transitivity of ~). Thus in particular iT is doubly transitive if and only if lIell@ 2. Proof. By (8.3)

* Let Sj = \? and is).

(J

(J

= 1]

=

1,\, in

(9.8). Then in the notation of (9.8)

Thus He II@ is the number of double cosets of ,\) in implies the result. Suppose that ,\' <j

@

@

which

Let {} be a complex valued class

INDUCED CHARACTERS

53

function on .\). Then OG is also a class function on ~) for G €: ®. Define the inertial group J(£1) of IJ by ",(e) ::: {G IG

Clearly

.("1

€: @,

£1 G ::: £1}

C ...'(9) and £1 G ::: £1 H if and only if GH-1

€: ...'(9).

(9.10) Let~) 1. Let S8 be a subgroup of I with >l\ ~ I. If ~ = (B) then Q'~ (B) = III and if >8 ~ (B) then o~ (B) O. Thus if {3 = .t Q'~ where S8 ranges over all subgroups of 9{ with 5B ~ 'tl, then {3 (A) l'tl I if (A) ;It 't( and (3(A) 0 if (A) = ~. Hence Q'! III 11 - {3 and the result follows by induction. (13.2) (Artin) Every rational valued character of 6} is a rational linear combination of characters of the form where !( ranges over the cyclic subgroups of ~.

1i'

Proof. Let 'l) be the vector space over Q consisting of all rational linear combinations of rational valued characters of ~ and let n be the dimension of 'l). By (12.3) there exist n pairwise nonconjugate cyclic subgroups 11"'" In of ~. Let Q'i::: a ~l i be defined as in (13.1). Then Q't vanishes outside the Q-conjugate class of (~; which contains a generator of Ii. Thus Q'~ is a basis of 'l) as required.

at .... ,

(13.3) (R. Brauer) There exist cyclic subgroups ~j of (~ and nonprincipal linear characters of ~lj such that p(\~ = 1 N + Z; aj t'j, where each aj is a rational number.

C H A RAe T ERS 0 F FIN I T E G R0 UPS

70

Proof. For a cyclic group ~ let a be defined as in 1l (13.1). Let 131l = qJ(l1l:l)p ~ - 01l:' where (p is the Euler fupcti~m. If ~ is an irreducible character of 1l: (A) then ~(Al) E l~ where E is an I~tlth root of unity. Therefore :0 ~(Ai) a~ i (AI) = I~I Tr~ 11ll/tJ (E). Thus I(~, O''ft)'' I ~ (p (11l1). This implies that

(t, 13,,) ~l

(~,

= qJ (l1l/ )

0' ~)11

~ 0

Since (p®, 1®) = 1 it now suffices to show that P( n 2: 1. Then T-1.P" T = .P" = .P". Thus

(T, pP) is an abelian group. Since I(T, P) : (T, pPi I

= p,

(T, PP)

O.

Proof. If () is an irreducible character of N then (i) - (iv) are clearly satisfied. Assume that (i) (iv) are satisfied. By (15.3) (J :E a i X., where each X. is an irreducible charl l acter of (\\ and each a i is a rational integer. By (iii) 1 = ~ ai 2 •

Thus there is exactly one nonzero a = ±1. Hence i for some i. By (iv) 0 = X. as required.

(J

= ± Xi

1

We will prove several lemmas and then use them to give a proof of (15.1) and {15.2L Throughout the remainder of this section n denotes the exponent of Nand .• By (18.7) IX'I c c 1::: I PI 1 P2 2. Thus by (18.8) @ is not simple. Let 1 ~!(
==

I S\ 11~(Hi} 12

II t II~ = 1

Thus condition (iii) is verified and by (15.4) ducible character of ®.

e is an irre-

®o is a normal complement of .\1 over .\)o in ®.

(19.11)

Proof. Let 1:; l ' 1:; 2' ••• be all the irreducible characters of . Let 8 i = e~i be defined as in (19.10). Then the definition of e i yields that Bi (G) == e i (1) for all i if and only if G E: @o. Hence by (19.10) ®o - .\}o)G =

U~1l)\

and l'lo has the required form.

n remains to verify conditions (19.1) - (19.4).

104

CHARACTERS OF FINITE GROUPS

Let

~

be a subset of G-l

~,

~o.

f

)8

~G c ~

If

for G e:: 6;

then )8 S ~ n ~ ~o and so G e:: ~. If ~ is a Sp-subgroup of ~ for some p e:: 11 this implies that Nl~(~) S ~ and so )8

~

= {H}

C~.

is a Sp-subgroup of

This verifies (19.1). If

~o we get that C(~\(H) :::: C~(H).

for some He::.\)

Furthermore if H' e:: ~ and H' is conjugate to H in 6\ then H' is conjugate to H in ~. Thus in particular (19.2) and (19.3) are proved. Since

16\1 : : ~?1= 0 1(;'\1. 1 and I~I = ~?1= 0 I'1.'1 ..... '

show that I~il

= IN: ~11~il

for i

>0

it suffices to

in verifying (19.4).

If .\1 is a conjugate class of (~\ with .\1 c Ni for some

n

i > 0 then we have just shown that .\1 ~ is a conjugate class and .\1' ~ ~ ~Q' Thus for H e:: .11

n s

1·111 = 16\:C~(H)1 = Il~i:~II~:C.\1(H)1 :::: IN:~II·l1n ~I This implies that I(~\il = IN: ~IINil for i > 0 and completes the proof. Due to its importance in many applications we state a special case of (20.1) explicitly. As will be seen later more direct proofs can be given of this theorem. (20.2)

(Frobenius) Let

~

that ~ ~G = \ 1) for G e:: anormal complementOt ~ in

n

be a subgroup of @ -

~

(,i.

(~\.

Assume

Then there exists .

As an application of (20.1) we will prove a special case of the Frobenius conjecture mentioned in section 17. See Feit [1], Wielandt [1] . (20.3)

IW :10 I

W be a subgroup of N and let W1) . Thus by (20.10) there exists a normal complement IMo of S> over ~o' Then

of ~ with .t> "* @ satisfies (21.4). Then ~ satisfies (21.4). Proof. Let ~l and ~2 be Sp-subgroups of (~I. The proof is by induction on 1~1: ~l n ~21. If 1i3 1 : i31 n '13 21 : : : 1 then ~l ::::; ~2 and (21.4) is satisfied with G 1. Given ~l and '.\3 2 let l' = ~l n ~2' If '1:' : : ; (I) the result is trivial. Suppose that (1) "* l' ~1' Thus N{\j(!') 1'. For i = 1, 2, let ('Ii = ~i n N(\) (1'), let 9l i be a Sp-subgroup of N(\\ ('1:') with (~\ ~ iRi and let ~i be a Sp -subgroup of N with 9li ~ ~i' Since Ni S; 'l3i n ~i it follows by induction ~hat there exist Gi E: C(~ (~i n ~i) ~ C(~ ('t» such that ~~l ::::: ~i for i = 1, 2. By assumption there exists H E: C(;l ('1:') such that 9l~ : : : 9l z • Since 9l z f: ~1 H n ~2' induction yields the existence of K E: C(\l (~IH (') ~z) f: CN (1') such that ~1 HK ~2' Consequently

'*

(21.8)

(Frobenius) Let p be a prime and let '13 be a

'*

THOMPSON SUBGROUPS

117

Sp-subgroup of 0), Assume that for every subgroup ~ of '.f3, NO} (S»/C(~l (~) is a p-group. Then (~} contains a normal p -complement. Proof. The proof is by induction on 1@ I. If I @! = 1 there is nothing to prove. The assumption of the theorem is clearly satisfied by every subgroup of @. Hence by induction it may be assumed that @ has no p-factor group ;r. {I}. If (I) ~ 1 S) 1112

CHARACTERS OF RELATIVELY SMALL DEGREE 127

Proof. Let Xo = 1@, Xl' ... ~e all the irreducible characters of @. Define ai = (Xi' XX), b i = (Xii f), 1~) for all i. The Frobenius reciprocity theorem implies that b i = (Xi' 1~ ). Thus by the Frobenius reciprocity theorem

Since.\) is abelian X(1) ~ (24.2) implies that

!1 XI f) II~.

As a o

= bo = 1 = X(1)2 - 1

1~ 11/2 Since X(1)

'*

1 this yields the required result.

(24.4) Suppose that for some prime p the Sp-group 'V of @ is abelian and a T.1. set in @. Assume that @ has a faithful character X with X(1) ~ p1/2 - 1. Then 'V =: {G I ah ::: 1}. (iii) lris a in N with N(" «%) = (% ~ Nand I and 6;/(> isomorphic to
(25.7) Let (~l be a Frobenius group with Frobenius kernel .p. Let IZ be a complement of .p in (~l. Then a Sylow group of l! is either cyclic or a quaternion group. Every abelian subgroup of (.! is cyclic. Proof. By (25.6) a Sp-subgroup of (.! has a unique subgroup of order p for p E: 1T ( i.f). This is known to imply the first statement. The second statement is an immediate consequence of the first. By using (25.5) it is possible to give a complete classification of groups l! which can occur as the complement of a Frobenius kernel .p in a Frobenius group (\l. The conditions in (25.6) are in general not sufficient. However by using (20.16) it is easily shown that they are necessary and sufficient in case all Sylow subgroups of (.! are cyclic. Much less is known about which groups .p can be the Frobenius kernel of a Frobenius group. Before looking at this question we prove another consequence of (25.5).

138

CHARACTERS OF FINITE GROUPS

(25.8)

The Frobenius kernel of a Frobenius group is

unique. Proof. Let .i) and .i)l be Frobenius kernels of the Frobenius group (~L It may be assumed that .i)l 4: .i). Thus there exists p E: 1f(.i)1)' p Ef 1f(.i». Let 13 be a Sp-subgroup of .i)l' By (25.3) .i) ~ .i)l' By the Sylow theorems Iill: .i)l IIIN(~ ('V) I. Thus by (25.3) there exists a complement lj of .i)l with G ~ N(,,; (~). Hence ~ 'V is a Frobenius group with Frobenius kernel 'V. Thus if X is an irreducible character of (% 'V then by (25.4) either 1~ :; XI' or 1 ~ !; Xlj' This contradicts (25.5) since ljlJ.i) is a Frobenius group with Frobenius kernel .i). (Burnside) Let N be a Frobenius group with Frobenius kernel .i). If I N: ~ I is even then .i) is abelian. (25.9)

Ht : :

Proof. Let J be an involution in ()j. Suppose that Hil Hil for HI, H2 E: .i). Then H2 Hil :::: (H 2 Hil )J. Therefore HI = H2 . Thus there are I.i) 1 distinct elements of the form H-I HJ in .i). Consequently if G E: .i) then G H-l HJ for some H E: .i). Thus GJ ::: H- J H :::: G-I. Therefore if Gu G2 E: ~ then

Hi

-I)J -_ G -J G -J_ GI G2 -_ (G-1 2 GI 2 I - G2 GI as required. It is possible to give an elementary proof of (23.2) and thus of (25.2) in case Ilj I is even by elaborating on the above argument. See Burnside p. 172. The following result is deeper.

(25.10) (Thompson) The Frobenius kernel of group is nilpotent.

Frobenius

Proof. Let (~; be a Frobenius group with Frobenius kernel ~. Induction on \.i) I. If Z(~) "* (1) then Z(.i» does not have a normal p-complement. Choose p 2 if possible. Let ~ be a Sp-subgroup of (\I;. If p :::: 2 then ~ and so T(~) . Suppose that p 2. By the Sylow theorems I(\1;: ~. liNN (~)!. Hence there exists a complement (! of .t> in (\\ with CZ ~ N~ (~). CS = C.t>(Z(~» .t> since Z(~) :::: (I). By (25.3) CZCS is a Frobenius group and by induction (£ has a normal p-complement. By (22.2)' N'.t>(T(~» does not have a normal p-complement. Hence T( 13) by induction. Thus in any case T(~) is a S-subgroup of @ and ffi.t'> f: CC@ (0) 2 there exists a subgroup l.p 13 > g.

If g

~ and

Proof. Induction on inequality in (29.6)

I ~ I.

Let n

.p of

1~ II m.

~

with

By the second

g :s; cn2 + (b - c)n ::: cn (n - 1) + bn For some j with 1 :s; j :s; s, b j ::::: b :s; s !;i=l 1/ ci we get that n :s; Cj' Thus g :s; CCj (Cj - 1) + Cj (Cj - 1)

Cj

= (c

1. Since lin :::

+ 1 )Cj (Cj - 1)

If Cj:S; C then g:s; c 3• As c::::: IC~(Gi)1 for some strongly

real element with Gi ;# 1 it follows that C~ (Gi ) ;# ® as required. If C < c j then I @ I < cj and we are done unless Cj g. In that case G :; : Gj is an involution in the center of ~. If g 4 the result is trivial. If g ;# 4 and g/2 is even then by induction there exists .p such that .pI (G) ;#

156

CHARACTERS OF FINITE GROUPS

[Sj/ (G) [3 > I @/(G) 1 3. This implies that I Sj 1 > I @ I as required. If finally g/2 is odd then by (20.14) @ has a normal 2-complement ~ and I Sj I = g/2 > ~ as required.



U ml

160

CHARACTERS OF FINITE GROUPS

(31.2) Suppose that (31.1) is satisfied. Then 90 (S) "* 0 and S is coherent. Furthermore l' is uniquely determined unless k 1, n1 ::: 2 and ill = i 12 • Proof. Since n 1 ~ 2, 9 0 (s) "* O. Induction On k. If k = 1 then S is coherent either by assumption or by (23.3). If iu = lIS for s ::: 1, ... , n 1 then the uniqueness of l' follows from (23.3) in case n 1 ~ 3. If ils > ill for some s then ~;1 is uniquely determined by I(~Z., ilS~;;' - ~is)1 > 1. Thus ~'[s = liS ~'[1 - (lIs ~ll - ~lS)1' and so l' is uniquely determined in this case also. Assume that k > 1. Let ::I = U~;l Si' Then ::I is coherent by induction. Thus there exist irreducible characters Xit of N such that ~it ± X it defines a linear isometry l' on 9(::1) which extends l' defined on 90 (::1). Since iit~~(l)­ ~tt (1) ::: 0 for ~it E: ::I it follows that ~lt EXit for ~it E: ::I where E = ±1. Since ~~;1 iis ~ 3 the first part of the result implies that l' is uniquely determined on ::I. For ~it E: S let 0it = (iit ~ll - ~it )1'. Thus 0it ::: E(iit Xu - Xit) for ~it E: :J. Define the integer a by (Xu, ilkl ) = £ (l.kl a). If ~it E: :J, ~it '" ~ll then

::

(Xit ' ilkl )

:::

(iit Xu, ilkl ) - (iit Xu - Xit' ilkl )

= lit (Xu, il ) kl

::: EiOt (£ 1

kl

de it ,

- a) - €loti 1

il

kl

)

kl

- Edit Furthermore II ilkl 112::: l.lu + 1. As .g(:J 1') .go (s 1') = .go (:J 1') it follows that ilkl 4.g0 (:J 1') since lk \11 - ~kl 4 90 ( :J). Hence there exists an irreducible char~cter X of (\) such that X ~ :J l' and (Okl ' X) "* O. Therefore

n

COHERENCE

161

or equivalently k-1 ni

U kl a + a 2

:E :E

~it

::; 0

i=1 t=1

If a*-O then, (31.1) (iv) yields that

Hence a 2 < a. This is not the case as a is an integer. Thus a O. Consequently ek1 ::: e.(lkl Xu - Xkl) where Xk 1 i~ a generalized character of N which is orthogonal to ~ • Hence II Xkl 112 = 1. Since 8k1 (1) = 0 this yields that Xk1 is an irreducible character of N. If nk =:; 1 setting 'T

~ kl

= e. Xkl

completes the proof. Suppose that nk > 1. Then in any case Sk is coherent by (23.3). Thus there exist irreducible characters xict of

(\j

and E.k ::: :1:1 such that (lkt ~kl - lkl ~kt>7 = E.k(i'·kt Xk1 1 k~ Xkt). If Xu ::: Xict for some t then choosing s -:I t we get that

o :::

(e 12 , (f kt ~ks - fks ~kt

f)

::: -EEk{l12fks + (X 12' Xks)lkt} which is impossible as (X12 , Xks) ~ o. If nk = 2 and tkl lk2 choose E. k =:; c.. Thus Xk1 = Xkl (6 kl , (~kl ~kt)7) -1. Hence also Xk2 = X'k2 and the proof is complete in this case. If n l ~ 3 or l k2 > I kl it follows that E.Xkl E k X'kl since (b k1 , (l kt ~kl - I kl ~kt)7) = -£ kt for all t. Thus E E.k and 7 defined by ~kt ::: EXkt is the required isometry.

162

C H A RAe T E RS 0 F FIN I T E G R0 UPS

In case S contains some reducible characters the situation is a good deal more complicated though there is a similar criterion available. See Feit-Thompson [2, Theorem 10.1]. The remaining results in this section are concerned with conditions under which (31.2) may be applied. See Feit-Thompson [2, Section 11] and Feit [2, Section 2]. If S is a set of characters of a group 9l and ~ al :"'0 1tJ: -Pa i . Since 9) is nilpotent this implies that cp(l )211 tJ: -P21. Let b be the square free part of 1-p 1 : ~21 and let c :::: (a, b). Then the square free part of I-P: -P21 is ab/c z. Thus cp(1)z II-P: -P2 i c2/ abo Every character in Si is a constituent of some character of 9l induced by an irreducible character of -P. Hence if ~ E: Si ~(1) :"'0 19l: -P Ivll-P: -P21 c2/ab. Thus (31.4) implies that I

~1:

-P21 - I

~1:

-P I

:"'0

21'i d2

2d 191: -P I

:"'0

VI

-P: !2 I c

Therefore I-P: -P212 - 21-P: -P21 + 1 :::: (I-p: -Pl)1 - 1)2

:"'0

4d2 I-P: !J I c

2

Since ab/c2 is a common denominator this yields that

Thus

•. I.'0.

c:;. 'VI

1_ 2

4e2 + 1 then S is coherent by (31.3). Assume that I~: ~' I ~ 4e2 + 1. If n ;; 1 then (11) holds. Suppose that n ~ 2. If Pi is odd then I~i: ~i I : :; 1 (mod 2e). Hence if 1T(~) contains at least two odd primes (31.6) implies that I~:~' I ;: : (2e + 1)2 - 1 > 4e2 + 1 contrary to assumption. Thus it may be assumed that n 2, PI :::; 2 and P2 = P is an odd prime. Let I~1: ~~ I : :; 2s and I ~2 : ~~ I : :; pt. Let 2s - 1 :::; mi e, pt - 1 :::; 2m2 e. Thus 4ell + 1 ~ I~: ~' I Hence m 1 pt

:::;

>

2ml mil ell + 1

mil = 1. Consequently

2e + 1 :: 2S +1

-

1 (mod 4)

COHERENCE

165

Thus t is odd. Hence p! a where a square free part of 1.\1: ~' I. If (p - 2)e2 < p then e 2 < p/(p - 2):=; 3 which is not the case. Thus (p 2)e2 ?,: p and so pe 2 ?': 2e2 + p. Thus

Hence

1.\1: .\1'1

>

2

4e 2

4e + 2 p

a

+ 2

and so S is coherent by (31.3) as required. For the next result it is necessary to know more about the original mapping 7" defined on do (S). Generalizations of this result may be found in Feit-Thompson [2] Section 10. Let ~ be a T.I. set in @ with 91 = N@ (.\1). As(31. 7) sume that 91 is a Frobenius group with Frobenius kernel .\1 and e = 91:.\1 I. Let S be the set of irreducible characters of 91 which do not have .\1 in their kernel. For Ci E: do (S) let = Ci • Assume that (S, 7") is coherent. Then there exists a rational integer c such that if t E: S then

(7" (H)

=

{(H) + (1) c

e

for H E: .\1*. If furthermore X is an irreducible character of @ such that ±X ¢ S7" then X is constant on .\1*. Proof. Let S {(i} where ~l (1) = e. Let (~191 = ~j asj ~j + ~ s where ;\ s is a character of 91/.\1. By the Frobenius reciprocity theorem, (23.1) and the coherence of $ we get that

166

CHARACTERS OF FINITE GROUPS

~j (1)

-e- aS1 -a·= sJ "

= ( ~s'

* *)

~j (1) ~1 -e-

-

~j ®

Thus

Since As (H) ;::: As (1) for H exists an integer c s with

"

~s 1.\1#

€.:

,p this implies that there

~s l,p# + C s

By (23.1)

~s (1) " - e - ~11,p

, , _ (s (1)

- ~s l,p - - e - (11,p - ~s I~

and so C s : : : (~s (1)/e) c 1 proving the first statement. Let Xl9l = ~bj (j + A where A is a character of 9lj,p. The Frobenius reciprocity theorem, (23.1) and the coherence of S yield that

~j(1) b e

1

b. J

~j

-_ ( X, -(1) -~

e

* J*)_ ® - (.

1

- 0

c LAS S

0 F DO U B L Y T RAN S I TI V E G R0 UPS

167

Therefore

A(1) - bl completing the proof.

§32. A CLASS OF DOUBLY TRANSITIVE GROUPS A group ® is a Zassenhaus group if it has a faithful permutation representation which is doubly transitive and in which no nOnidentity permutation leaves 3 or more letters fixed. If ® is a Zassenhaus group the follOwing notation will be used: h + 1 is the degree of the defining permutation representation. 9l is the subgroup of @ consisting of all elements leaving a given letter fixed in the defining permutation representation. I.! is the subgroup of 9l consisting of all elements leaving a second given letter fixed in the defining permutation representation.

I IS! : :;

e

I~ll : :;

eh

I @ I : :;

eh (h + 1)

If e :::; 1 it is easily seen that @ is a Frobenius group whose Frobenius kernel has order h + 1. If e :f 1 then 91 is a Frobenius group whose Frobenius kernel has order h. In this case Sj will denote the Frobenius kernel of m, Thus e I (h 1). It follows directly from the definition that ~ is aT.!. set in @ and '9l :::; N® (.f)). Zassenhaus groups have been completely classified. See Zassenhaus [1], [2], Feit [2], Ito [3] and Suzuki [4]. In this section we will only give an intermediate step in this classification to illustrate how the results of Section 31 can be applied, The following will be proved. (3~.1)

Let ® be a simple Zassenhaus group, Assume

168

C H A RAe T E RS 0 F FIN I T E G R 0 UPS

that e is odd. Then either .p is a nonabelian p-group for some prime p with 9,):.p' I ::s 4e2 + 1 or e (h - 1)/2. Thus in any case .p is a p-group for some prime p. The proof will be given in a series of short steps. Until further notice it is assumed that @ satisfies the hypotheses of (32.1) and e < (h -1)/2. Observe that the simplicity of 6; implies that e > 1. (32.2) ~ is cyclic and a T.1. set in ®. N® (\!) is a Frobenius group with Frobenius kernel ~ and IN@(~): ~I

= 2.

Proof. If E E: ~# then E leaves exactly two letters fixed in the defining permutation representation. Thus if ~ '" ~G for some G E: ® and E E: ~# It ~G then E leaves more than two letters fixed and E '" 1 contrary to the definition. Thus l5' is a T.I. set in ®. Hence N@ «(.! )/(.! is a permutation group on the two letters fixed by all the elements in (.! # and so I N@ «(.!) ! ::s 2e. By (25.7) every Sylow group of N@ (\!) is cyclic. By (20.15) N@ «(I)' is cyclic and N® (\!)/N® «!)' is cyclic. Suppose q is an odd prime and q II N@ (e-1 h

170

CHARACTERS OF FINITE GROUPS

Thus t - 2 .~ e as required. If h is odd then (29.7) and (32.3) imply that

2

t

>-

-

eh(eh + 1) - 1

I@I

eh + 1 _ 1 h + 1 (e - 1)

-

1

- e - h+1

>

e - 2

Thus t - 2 ~ e - 1. Since .p contains no involutions the result follows in this case also. Let S be the set of irreducible characters of 9l which do not have .p in their kernel. For a E: So (S) let aT a *. By (23.1) T is a linear isometry. If S is not coherent the result follows from (31.5). Thus it may be assumed that S is coherent. Let S = {~) and choose the notation so that ~l (1) = e. Let ~ = ~l'

II ~* II@ = e

(32.5)

+ 1 and ~* (H)

= ~(H)

for H

E:

.p#.

Proof. If H E: .p# then ~* (H) ~(H) by (23.1). Since (1) = (h + 1)e and ~* (G) = 0 if G is not conjugate to an element of .p this implies that ~*

II ~* jI2

=

l'e

1 @

-

1

- I@

{e I

e(h + h

2

(h + 1)2 +

2

(h

~ .E I ~(H) 12} he .p#

+ 1)2 _ e2 1@1+ I ® I 1; I ~(H) 12} he

1) _ e

h

+

II ~ W 9l

he .p

e + 1

c LA SS

0 F DO UB L Y T RAN SIT I V E G R0 UPS

171

q:::; q

(32.6) + ~i (l)/e r where r is a sum of irreducible characters none of which are in ± Proof. If X is a nonprincipal irreducible character of @ which is not in ± S'T then ~ is not in the kernel of X and so by (31. 7) (~*, X) :f; O. By (32.4) there exist at least e such characters. By (31. 7) and (32.5) there exists e E: ± S'T such that (~*, e) :f; O. Thus by (32.5)~* e + r where r is a sum of irreducible characters of @ which are not in ± S'T. If I s I : :; 2 then 'e + 1 :::; I.\): ~I I and ~ is a p-group for some prime p since (h -l)/e > 2. Thus it may be assumed that I S I 2: 3. There exists E :::; ±l such that (32.7)

*

~i (1) *

~i - -e- ~

=E

Il 'T

\~i

~i (1) - -e~ J 'T\

If E -1 then (~[, ~*) :f; 0 for i > 1. Hence e : :; ~::::; ~; which is impossible. Thus E :::; 1 and so e = ~'T. Now (32.6) follows from (32.7). Let 1}1' ... , 1}e-l be all the irreducible nonprincipal characters of 91/(! where 1}j :::; 1}j + {e 1)/2 for j 1, ... , {e - 1)/2. Then 1}j (l) :::; 1 for all j. It follows easily from (32.2) that

(32.8)

1}:' (E) :::;

(E) + 1}. (E) for E J ~ (H) :::; 1 for H

J

II

1}.

J

J

E: ij if E: ~ #

By (32.8) 11} r (G) I ~ 11m (G) I for all G E: @ and 11}91 (E)l < 2 11;1 (E)I for some E E: (!. Hence J I'll jw < 1/191112. As @ is doubly transitive on the cosets of 91, 111;1112 ;: : 2 by (9.9). Thus II q 112 1. Hence (32.8) implies that {IJ; I j 1, ... ,(e 1)/2} is a set of pairwise distinct irreducible characters of @. Furthermore 1@ + ~ where ~ is an irreducible character of 1M with ; (H) 0 for H E: 6)#. Thus ~ $: ±I'T and I/j if: ±Ir by (31.7) and (32.8). Since Ii> = 1(\j + ~ + :6 (e'l)/2 21) itfollows that (~ 4>' loP) :::;

j

191 : :;

I

r

172

C H A RAe T E RS 0 F FIN I T E G R0 UPS

(~T , 1~)

==

O. The Frobenius reciprocity theorem and

(32.6) imply that (~r91' ~i) Hence in particular (32.9)

~

T

==

0 for i

> 1.

Thus ~r9l

==

~.

(1) ::: e

The proof of (32.1) can now be given by applying an argument due to Brauer. By (32.8) e 1 ::: 1, ... ' - 2 Thus ~2(1) e

1"T ':>

*

11 j +

1"T ':>2

=

)"T '>2

*

11 j

+ ~2(1) e

1"T ':>

r

Hence ~T S ~T 1/ and so 11 j S ~T ~T by (6.6). Since l1j(1) = h + 1 and 1(}J C ~T~T(32.9) implies that

Thus e + 1 ;::: (h + 1)/2 or e ;::: (h -1)/2 completing the proof of (32.1).

§SS. ISOMETRIES Let 11 be a subset of (}J with ~ ~ N(}J (~) == 91. Let d ("n) be the set of all generalized characters of 91 which vanish on 9l - ~. If ~ is a T.I. set in @ with ~( = ~# then by (23.1) the map T defined by aT = a* is a linear isometry from d (~) into the character ring of @ with aT (1) == 0 for a C d (,.). For some purposes the assumption that ~ is a T.I. set in (}J is too restrictive. In Feit-Thompson (2) Section 9 such a map T was constructed under weaker hypotheses. Dade [2] has simplified and generalized that construction and his method will be presented in this section.

ISOMETRIES

173

The following notation will be used: is a fixed set of primes. For G E: @, G1T is defined by G = G1T G1T , = G1T , G1T where G1T , G1T , is a 1T-element, 1T' -element respectively. ft is a subset of @ consisting of 1T-elements. 91 is a subgroup of @ such that ft s; m ~ N@(ll). 11(11), e(~) is the set of generalized characters, complex valued class functions respectively, of m which vanish on m- ft. The following assumptions are relevant. 1T

(33.1) (i) If two elements in ft are conjugate in @ then they are conjugate in m. (ii) If A E: ft then C@ (A) = Cm(A),p(A) where Cm(A) n ,p(A) = (1), ,p(A) C@ (A) and ,p(A) is a S1T' -subgroup of C@(A).

Assume that (33.1) holds. If a a T (G)

=a

E:

e(ft) define aT by

(A) if G is conjugate to A 11

E:

I.

= 0 otherwise

By (33.1) (i) aT is well defined. Observe that if 71 is a T.1. set in @ with " = 1# and 91 ::: NO) (ft) then (33.1) is satisfied with 1T 1T(~;). By (23.1) aT a * for a E: e(I). The main purpose of this

section is to prove Assume that (33.1) is satisfied. If a, f3 E: e(l) then (aT, f3 T )@ = (a, (3)m' If furthermore a E: u(l) then aT is a generalized character of @. (33.2)

Since a *(1) = a (1), (33.2) implies that the mapping T satisfies the assumptions needed in Section 31 provided that 1 = 1#. In case (33.1) holds let .11 1 , Sl:i!1 ••• be all the conjugate classes of m which lie in t. Let @i = {G I G E: @, G1T is conjugate to an element of .Il i }.

174

CHARACTERS OF FINITE GROUPS

As an immediate consequence of (33.2) one gets the following analogue of the Frobenius reciprocity theorem for T. See FeU-Thompson [2, Lemma 9.4}. (33.3) Assume that (33.1) is satisfied. Let a E: e(!l). If 6 is a class function on ® such that for all A E: !l, 6 is constant on the coset A.p (A) then

(a T, 6)® = (a, 619l)9l Proof. Let K,. E: ~'i' Then 6(G) ;:: 6(Ki) for G E: ®i' Thus if 60 is the class fUnction on 9l defined by 6 0 (A) = 6(A) for A E: ,. and 6 0 (N) = 0 for N E: 9l !l then 8{G) ;:: 8~(G) for G E: U®i' Hence T _ 1 T-(a , 6)® - ~ ~ a (G)8(G)

since aT vanishes outside U®.. Similarly 1 1 __ (a, 619l)9l = ~ a (G) 6(G)

Til

= (a,

80 )9l

The result follows from (33.2). Suppose that S is a set of irreducible characters of 9l with ~o (s) .s; e(!l). Assume further that (S, T) is coherent. Let S = {~i}' It is useful for many applications to

ISOMETRIES

175

have an analogue of (31.7) available for the mapping I defined in this section. By using the theory of modular characters it can be shown that ~! is constant on A~(A) for all 1 A E: t( then (33.3) can be usee to prove su.ch an analogue of (31. 7). The proof of (33.2) will be given in a series of steps. Assume that (33.1) holds. (33.4)

If a, (}

Proof. Let ~

! @i I = I @: = \ @:

E:

e(t() then (ai, (}I)®

E:

·R'i' Then by (33.1)

C® (~)\! {a 1 a

E: @,

C@ (~ )1\

E:

{a I a

=

(a,

Mm.

an = ~}!

~(Ki)} I

= I@: C@(~) 1\ ~(~) \ \@\

= .--::\c:::-@-;-(~-=-.-:-): :......,~-:-(~-• .,..-,-)I

I®I Therefore

1

=

I'~i\

= TmT Hence

--

IC91(Ki ) I a(~)(}(~) --

a (~)(}(~)

176

CHARACTERS OF FINITE GROUPS

= (a, (3)91

(33.5) If ~ is a 'If' -group and G is a 'If-element with G E: N® (~) then

G~ = U {GC.\'> (G)}H .\'> Proof. Since ~ (G)}H ~ G~. Suppose that L E: G~ then L'If is conjugate to G in (G).\'> by the Schur-Zassenhaus theorem. Thus L'If = GH for some H E: .p. Hence L E: L'If C ~ (Ln) = {GC~ (G)}H as required. If SB is a nonempty subset of I let .p(SB) = n SB.\'>(B).

E:T.

(33.6) For any nonempty subset 5B of I N91 (5B) ~ N® (.p(5B» and 14''.» (SB) n .p(SB) = (1).

Proof. By (33.1) ~(1;\) (B)

(1)

For any class function a of N9l (SB) define (33.7) a SB (NH) = a (N) for N E: N9l (SB), H E: ~(SB). By (33.6) a is a class function of N91(t\)~(t\). Further5B more at\ is a generalized character of N (t\).p (5B) in case

m

a is a generalized character of N91 (5B). (33.8) If Ci E: e(l) then aT -

- -

2: 5B

(-1)1 SBI

I ~l:

Nm (SB)

I

0'*

SB

177

ISOMETRIES

where

~

ranges over the nonempty subsets of !.

Proof. Define

a*

~

where ~ ranges over all the nonempty subsets of ft. Thus y is a class function and for G c ® (-1) I

y(G) ;: -

5l\1 0' (GM)

~ ~ I ~l: Nm (~)IIN:(~)4.1(~) I

where for each ~, M ranges over all elements of ® with G M c N (~)4.1(~). m By (33.6) INm(~)4.1(~)1 ::: IN91(~)II~(t\)I. If N c N~l (5l\) is the unique element such that GM c N4.1(~) then 0' 5l\ (GM ) ;: 0' (N) by (33.7). Thus 0' ~ (GM ) ::: 0' (N) ::: 0 unless N c ft. Hence (33.9)

_ y(G) - -

1

~

Ti1

( -1) 15l\ I0' (N)

I ~(5l\) I

~,M,N

where for each Sl\, (M, N) range over all ordered pairs such that N c _Nm (~) n ft and GM c N4.1(~). Thus by (33.5) N is conjugate to G1T if it occurs in the above summation. Thus y(G) = 0 = O'T (G) if G ¢ U®i. Hence it may be assumed that G c U®i and by changing notation that G c ®l' Since y is a class function it may further be assumed that G1T c "ll' If GM c N 4.1(~) then by (33.5) G1T is conjugate to N in (N) 4.1. Thus 0' (N) = 0' (G1T ) and N c .n 1 • Interchanging the order of summation in (33.9) implies that 0' (G1T )

_

y(G) -

-

I 911

~ ~

Nc.5l 1

M,~

(_1)1~1 I 4.1(~)

I

178

C H A RAe T E RS 0 F FIN I T E G R0 UPS

where for each N in .n l ' (5l.\ M) ranges over all pairs such that N E:: Nm(t\) and GM E:: N .\'>(51.\). The inner sum is independent of Nand l·n 1 1 = 191: Cm(G u ) I. The number of conjugates of Gu in Gu .\'>(t\) is I.\'>(t\): C.f)(t\) (Gu)1 by (33.5). Thus (33.10)

y(G) = _

~

a(Gu )

I C91 (Gu ) I

(_1)1t\1 M,5I.\ I C.\'>(5I.\) (Gu ) I

where (t\, M) ranges over all pairs such that Gu E:: N91 (58), GM E:: Gu .\'>(t\) and M E:: C® (G ). Gu E:: Nm(5l\) if and only if Gu E:: N91 ('8 U {Gu}) and GrJ E:: Gu .\'>(58) if and only if GM E:: Gu ~(5l.\ U {G.}). Thus (33.10) implies that Ci

(33.11)

(Gu )

ICm(Gu ) I ~

y(G) = -

(-1 14)(G ) I

M

+

¥

,

(_1)11:1

u

(_1)!1'U{Gu }1 })

II C .\1(1') (Gu ) I + IC.\1(t)U {GuH(Gu>l

where M ranges over C® (Gu ) and 1: ranges over all nonM empty subsets of I - {Gu } with G E:: Gu .\1(1'), Gu E:: N91 ('1:'). Since '\>('1) is a .' -group it follows that C.\>(~) (G u ) = C® (G.)

= .\1(Gu ) n

n

.\1(1')

.\>(t')

= .\>(t' U {G'/I'}) = C.\>('I'U {G } )(Gu )

u

Thus (33.11) implies that

ISOMETRIES

179

where M ranges over C® (G1T ). Thus

completing the proof of (33.8). (33.12)

where ~ ranges over a complete system of representations of equivalence classes of nonempty subsets of " under the action of m by conjugation. Proof, The number of distinct subsets of" of the form t\N with N E: ~l is I ~l: N91 (t\) I for any nonempty subset t\ of ft, Since 0' ~ == 0' ~ N the result follows from (33.8). Now (33.2) is an immediate consequence of (33.4) and (33.12).

NOTATION All groups are assumed to be finite unless explicitly stated otherwise.

II r is the cardinality of ft. 1# = ! - {l}

= B-IAB [A,B J = A-I AB A-I B-1 AB It' = {A B !A E: ft, B E: ti} AB

(A,B ... ) is the group generated by A,B, ...

[1,tiJ = ([A,B]lA

E:

ft, B

E: ~).

= [@,@J

@'

N @Ul) is the normalizer of ft in @ . C@(7l) is the centralizer of ft in @ .

Z(@) is the center of the group @.

.p l> @ means that .p is a normal subgroup of @: If ft 1> @, ft c t' 1> (\;, then till is a factor of @. If t\1 ~ is a minimal normal subgroup of

chief factor of

@

If 5.A/I is a factor l~, then C® (til!)

all B If

1T

E:

@/I, then it is a

{G I (G,B]

E:

I for

5.A}.

is a set of primes, then

1T'

set. 181

denotes the complementary

NOTATION

182

Generally {p} will be identified with p for p any prime. n11 is the largest integer dividing the integer n all of whose prime factors are in 11. (\j

is a 11-group if r~ t11' = [~r.

G is a 11-element if (G) is a 11-group

.p .p

is a Hall 11-subgroup or a S11-subgroup of ~ if

!.p I

= I~ 111

is a Hall-subgroup or a S-subgroup if S';) is a S11-subgroup

for some s,et of primes 11(~)

11.

is the set of primes dividing I~ I.

An involution is an element of order 2.

SL(2,q) = SLa (q), PSL(2,q) PSLa (q) is the unimodular, projective unimodular group respectively of degree 2 over the field of q elements

fJ is the field of rational numbers. fJ n is the field of nth roots of unity over fJ.

s:. ac is an extension field of s: then Tr 3\Js: is the trace of over s:. If ac is a Galois extension of s: then 9 xiir is the

char If

s:

is the characteristic of the field

ac Galois group of ac over S:.

REFERENCES R. Baer [IJ Math. Z. 71, 454-457 (1959). R. Brauer [IJ Ann. oj Math. 42, 926-935 (1941). R. Brauer [2] J. Math. Soc. Japan 3, 237-251 (1951). R. Brauer (3) Ann. oj Math. 5'7, 357-377 (1953). R. Brauer [4J "Proc. International Congress 1954," Vol. 1., pp. 1-9. R. Brauer [5J Proc. A.M.S. 15, 31-34 (1964). R. Brauer [6J "Lectures on Modern Mathematics," Saaty, Vol. 1., pp. 133-175, New York (1963). R. Brauer [7] Math. Z. 83, 72-84 (1964). R. Brauer and K. A. Fowler [IJ Ann. oj Math. 62, 565-583 (1955). R. Brauer and M. Suzuki [1] P.N.A.S. 45, 1757-1759 (1959). R. Brauer and J. Tate [11 Ann. oj Math. 62, 1-7 (1955). E. C. Dade [1] J. oj Algebra 1, 1-4 (1964). E. C. Dade [2] Ann. oj Math. '79, 590-596 (1964). W. Feit [IJ Proc. A.M.S. '7, 177-187 (1956). W. FeU [2J Rl. J. oj Math. 4,170-186 (1960). W. FeU [3J "Symposia in Pure Mathematics," Vol. 6 pp. 6770, (1962). W. Feit [4J Trans. A.M.S. 112, 287-303 (1964). W. Feit, M. Hall, Jr., and J. G. Thompson [1J Math. Z. '74, 1-17 (1960). W. FeU and J. G. Thompson [1 J Pac. J. Math. 11, 1257-1262 (1961). W. Feit and J. G. Thompson [2J Pac. J. Math. 13, 775-1029 (1963). P. Fong [1] Rl. J. Math. '7, 515-520 (1963).

183

184

REFERENCES

P. Fong and W. GaschUtz (1] J. Reine Agnew. Math. 208, 73-78 (1961). P. X. Gallagher [1] J. London Math. Soc. 39, 720-722 (1964). L. K. Hua and H. S. Vandiver (1] P.N.A.S. 35, 94-99 (1949). N. Ito (1] Nagoya Math. J. 3, 5-6 (1951). N. Ito [2] Nagoya Math. J. 5, 75-78 (1963). N. Ito [3] Rl. J. Math. 6, 341-352.(1962). P. Roquette [IJ J. Reine Agnew. Math. 190, 148-168 (1952). P. Roquette [2] Arch. Math. 9, 241-250 (1958). L. Solomon [1] J. Math. Soc. Japan 13, 144-164 (1961). L. Solomon [2] Proc. A.M.S. 12, 962-3 (1961). L. Solomon [3] Math. Z. 78,122-125 (1962). M. Suzuki [1] Amer. J. Math. 77, 657-691 (1955). M. Suzuki [2] Proc. A.M.S. 8, 686-695 (1957). M. Suzuki [3] J. Math. Soc. Japan 15, 387-391 (1963). M. Suzuki [41 Ann. of Math. 75, 105-145 (1962). J. G. Thompson [1] P.N.A.S. 45, 578-581 (1959). J. G. Thompson [2] Math. Z. 72, 332-354 (1960). J. G. Thompson [3] J. oj Algebra 1, 43-46 (1964). A. Weil [1] Bull. A.M.S. 55, 497-508 (1949). H. Wielandt [1] Math. Nachrichten 18, 274-280 (1958). H. Zassenhaus [IJ Abh. Math. Sem. Hamburg 11, 17-40 (1936). H. Zassenhaus [2J Abh. Math. Sem. Hamburg 11, 187-220 (1936).

Books W. Burnside, "Theory of Groups of Finite Order," Cambridge University Press, England, 1911. C. W. Curtis and 1. Reiner, "Representation Theory of Finite Groups and Associative Algebras," Interscience, New York, 1962. M. Hall, Jr., "The Theory of Groups," Macmillan, New York, 1959. W. R. Scott, "Group Theory," Prentice Hall, New York, 1964. H. Zassenhaus, "The Theory of Groups," 2nd ed., Chelsea, New York, 1949.

INDEX Absolutely irreducible, 6 Algebraically conjugate characters, 14 Associated permutation representation, 44

Frobenius group, 133 Frobenius kernel, 133 Frobenius reciprocity theorem, 47 Generalized character, 78 Group algebra, 1

Character, 10 Character ring, 78 Character table, 41 CN group, 142 Coherence, 158 Complement, 56 Completely reducible, 3 Constituents of ff', 3 Contragredient representation, 3

Index of ramification, 53 Induced ff'-representation, 44 Induced function, 46 Inertial group, 53 Integral representations, 23 Irreducible character, 10 Kernel of a character, 13

Degree, 1 Dihedral group, 63

M-group. 58 Minimal simple group, 145 Monomial character, 44 Monomial ff'-representation, 44 Multiplicity, 13 Multiplicity free, 13

ff'-conjugate, 67 ff'-conjugate classes, 67 ff'-elementary, 71 ff' -elementary with respect to the prime p, 71 ff'-normalizer, 81 ff'-reducible, 3 ff'-representation, 1 faithful, 1 similar, 1 Frobenius complement, 135

Normal complement, 98 Normal complement of ~ over .po in~, 98 Normal 7T-complement, 98 Orthogonality relations, 16

185

186 1I"-factor, 98 11"1 -factor, 98 1I"-section, 105 Permutation character, 44 Principal character, 11 Quaternion group, 63 Real conjugate class, 68 Real element, 68 Regular ff-representation, 2 Schur index, 61

I ND EX Splitting field, 9 Splitting field of a character, 13 Thompson subgroup, 118 Transitive permutation ff-representation, 44 Trivial intersection set, 123 Unit ff-representation, 2 Z-group, 111 Zassenhaus group, 167

BENJAMIN BOOKS Of RELATED INTEREST

LECTCRES ON AI.CEBRAIC TOPOLOCY MARVIN J. GREENBERG / Nod/leas/ern l/Ilil'('rsilv This text is int('nded for a ~raduat(' courst' ill al~('hruic topology. Th e four main parts of the book tn'at elementury homotopy theory, singuiur homology theory, orientation and dualitv on manifolds, and producls and iI,e l.e fsclwtz fix f'd point theorem. SET THEORY A[\J) THE CONllNUU;\1 HYPOTHESIS PAUl. J. COHEN / SICln{ord Uni.versily Thi s graduate level text-sllpplement contains a d etail ed expositi on of the author's research on the inderH'ndence of iI" , continuum hypoth esis from the axio ms of set th eo ry. RAPPORT SUR I.A COHOMOLOGIE DES GROUPES SERGE I.ANG / Columbia University Thesf' advanced notes. in French, provide a ddailed introduclion to the Artin-Tate notes on class field tht'ory, for graduate student s interested in homolo gical algebra and co homology of groups. ALGEBRES DE LIE SEMI·SIl\1PLES COl'vlPLEXES JEAN ·PIERRE SERRE / ('ollefje de France This set of notes, in French, is a compact pn'sentation of Ihe str ucture of semi-simple complex l.ie alg('bras and their representation th eo ry. LIE ALGEBRAS AND LIE GROUPS JEAN·PIERRE SERRE / College de France Covers the main general theorems on lie algebras. witl, some results on fr('e lie algebras. plus an introduction to groups and their cnrrespondence to lie theory.

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