Finite Group Theory

I. Martin Isaacs

Finite Group Theory

I. Martin Isaacs

Graduate Studies

in Mathematics

Volume 92

American Mathematical Society

Providence, Rhode Island

Editorial Board

David Cox (Chair) Steven G. Krantz

Rafe Mazzeo Martin Scharlemann

2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n . P r i m a r y 20B15 , 20B20 , 20D06 , 20D10 , 20D15 , 20D20 , 20D25 , 20D35 , 20D45 , 20E22 , 20E36 .

F o r a d d i t i o n a l i n f o rma t i on a n d updates on th is book , v i s i t w w w . a m s . o r g / b o o k p a g e s / g s m - 9 2

Library of Congress Cataloging-in-Publication Data Isaacs, I. Martin, 1940-

Finite group theory / I. Martin Isaacs. p. cm. (Graduate studies in mathematics ; v. 92)

Includes index. ISBN 978-0-8218-4344-4 (alk. paper) 1. Finite groups. 2. Group theory. I. Title.

QA177.I835 2008 512.23dc22 2008011388

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permissionaams.org.

' 2008 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights

except those granted to the United States Government. Printed in the United States of America.

' The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

Visit the AMS home page at http://www.ams.org/

10 9 8 7 6 5 4 3 2 1 13 12 11 10 09 08

T o D e b o r a h

Contents

Preface ix

Chapte r 1. Sylow Theory 1

Chapte r 2. Subnormal i ty 45

Chapte r 3. Spli t Extensions 65

Chapte r 4. Commutators 113

Chapte r 5. Transfer 147

Chapter 6. Frobenius Act ions 177

Chapte r 7. The Thompson Subgroup 201

Chapte r 8. Permutat ion Groups 223

Chapte r 9. More on Subnormal i ty 271

Chapter 10. More Transfer Theory 295

Append ix : The Basics 325

Index 345

Preface

This book is a somewhat expanded version of a graduate course in finite

group theory that I often teach at the Univers i ty of Wiscons in . I offer this

course in order to share wha t I consider to be a beautifu l subject w i th as

many people as possible, and also to provide the sol id background in pure

group theory tha t my doctora l students need to carry out their thesis work

in representation theory.

The focus of group theory research has changed profoundly in recent

decades. Start ing near the beginning of the 20th century w i th the work of

W . Burnside , the major problem was to find and classify the finite simple

groups, and indeed, many of the most significant results in pure group theory

and in representation theory were directly, or at least peripherally, related to

this goal. The simple-group classification now appears to be complete, and

current research has shifted to other aspects of finite group theory inc luding

permutat ion groups, p-groups and especially, representation theory.

It is certainly no less essential in this post-classification per iod that

group-theory researchers, whatever their subspecialty, should have a mas

tery of the classica l techniques and results, and so wi thou t at tempting to

be encyclopedic, I have included much of tha t mater ia l here. B u t my choice

of topics was largely determined by my pr imary goa l in wr i t ing this book ,

which was to convey to readers my feeling for the beauty and elegance of

finite group theory.

Given its or igin , this book should certainly be suitable as a text for a

graduate course like mine. Bu t I have t r ied to write it so that readers would

also be comfortable using it for independent study, and for tha t reason, I

have t r ied to preserve some of the informa l flavor of my classroom. I have

t r ied to keep the proofs as short and clean as possible, bu t wi thou t omi t t ing

ix

X P r e f a c e

details, and indeed, in some of the more difficult material , my arguments are

simpler than can be found in pr in t elsewhere. F ina l ly , since I f i rmly believe

tha t one cannot learn mathematics wi thou t doing it, I have included a large

number of problems, many of which are far from routine.

Some of the mater ia l here has rarely, i f ever, appeared previously in

books. Jus t in the first few chapters, for example, we offer Zenkovs mar

velous theorem about intersections of abelian subgroups, Wie landt s "zipper

lemma" in subnormal i ty theory and a proo f of Horosevskii s theorem that

the order of a group automorphism can never exceed the order of the group.

Late r chapters include many more advanced topics that are hard or impos

sible to find elsewhere.

Mos t of the students who attend my group-theory course are second-year

graduate students, w i th a substantia l minor i ty of first-year students, and an

occasiona l well-prepared undergraduate. A lmos t al l of these people had

previously been exposed to a standard first-year graduate abstract algebra

course covering the basics of groups, rings and fields. I expect that most

readers of this book w i l l have a s imi lar background , and so I have decided

not to begin at the beginning .

Mos t of my readers (like my students) w i l l have previously seen basic

group theory, so I wanted to avoid repeating that mater ia l and to start w i th

something more excit ing : Sylow theory. B u t I recognize that my audience

is not homogeneous, and some readers w i l l have gaps in their preparation ,

so I have included an appendix that contains most of the assumed mater ia l

in a fair ly condensed form. O n the other hand , I expect that many in my

audience w i l l already know the Sylow theorems, bu t I am confident that even

these well-prepared readers w i l l find mater ia l that is new to them w i th in the

first few sections.

M y semester-long graduate course at Wisconsin covers most of the first

seven chapters of this book, start ing w i th the Sylow theorems and cul

minat ing w i th a purely group-theoretic proo f of Burnsides famous p a q b -

theorem. Some of the topics along the way are subnormal i ty theory, the

Schur-Zassenhaus theorem , transfer theory, coprime group actions, Frobe-

nius groups, and the norma l p-complement theorems of Frobenius and of

Thompson . The last three chapters cover materia l for which I never have

t ime in class. Chapte r 8 includes a proo f of the s impl ic i ty of the groups

P S L ( n , q ) , and also some graph-theoretic techniques for studying subdegrees

of pr imi t ive and nonpr imit ive permutat ion groups. Subnormal i ty theory is

revisited in Chapte r 9, which includes Wie landt s beautifu l automorphism

tower theorem and the Thompson-Wie land t theorem related to the Sims

P r e f a c e x i

conjecture. F ina l ly , Chapter 10 presents some advanced topics in trans

fer theory, inc luding Yoshidas theorem and the so-called "pr incipa l idea l

theorem" .

Fina l ly , I thank my many students and colleagues who have contr ibuted

ideas, suggestions and corrections while this book was being wr i t ten . In

part icular , I mention that the comments of Yakov Berkovich and Gabr ie l

Navarro were invaluable and very much appreciated.

C h a p t e r 1

Sylow Theory

1 A

It seems appropriate to begin this book w i th a topic tha t underlies v i r tua l ly

al l of finite group theory: the Sylow theorems. In this chapter, we state and

prove these theorems, and we present some appl ications and related results.

Al though much of this mater ia l should be very famil iar , we suspect tha t

most readers w i l l find tha t at least some of the content of this chapter is

new to them .

Al though the theorem tha t proves Sylow subgroups always exist dates

back to 1872, the existence proo f tha t we have decided to present is tha t

of H . Wie landt , published in 1959. Wie landt s proo f is sl ick and short, bu t

i t does have some drawbacks. It is based on a t r ick tha t seems to have

no other appl icat ion , and the proo f is not really constructive ; it gives no

guidance abou t how, in practice, one migh t actual ly find a Sylow subgroup.

B u t Wie landt s proo f is beautiful, and tha t is the pr inc ipa l mot ivat ion for

presenting i t here.

Also , Wie landt s proo f gives us an excuse to present a quick review of the

theory of group actions, which are nearly as ubiquitous i n the study of finite

groups as are the Sylow theorems themselves. We devote the rest of this

section to the relevant definitions and basic facts abou t actions, al though

we omi t some details from the proofs.

Le t G be a group, and let ft be a nonempty set. (We w i l l often refer to

the elements of ft as "points".) Suppose we have a rule tha t determines a

new element of ft, denoted a-g, whenever we are given a poin t a e ft and

an element g e G. We say tha t this rule defines an act ion of G on ft i f the

fol lowing two condit ions hold .

1

2 1 . S y l o w T h e o r y

(1) a - l = a for a l l a G ft and

(2) { a - g ) - h = a-fo/i) for a l l a G ft and a l l group elements g , h e G .

Suppose tha t G acts on ft. It is easy to see tha t i f g G G is arbitrary ,

then the function ag : ft -> ft defined by - a-$ has an inverse: the

function C T 5 _ I . Therefore, ag is a permutat ion of the set ft, which means tha t

og is both injective and surjective, and thus a g lies in the symmetr ic group

Sym(ft) consisting of a l l permutations of ft. In fact, the map g -> ag is

easily seen to be a homomorphism from G into Sym(ft) . ( A homomorphism

l ike this , which arises from an act ion of a group G on some set, is called a

permutat ion representation of G.) The kerne l of this homomorphism is,

of course, a norma l subgroup of G, which is referred to as the kerne l of the

action . The kerne l is exactly the set of elements g e G tha t act t r i v ia l ly on

ft, which means tha t a - g = a for a l l points a e ft.

Generally , we consider a theorem or a technique tha t has the power

to f ind a norma l subgroup of G to be "good" , and indeed permutat ion

representations can be good in this sense. (See the problems at the end of

this section.) B u t our goa l in introducing group actions here is not to f ind

norma l subgroups; i t is to count things. Before we proceed in that direct ion ,

however, it seems appropriate to mention a few examples.

Le t G be arbitrary , and take ft = G. We can let G act on G by right

mult ip l icat ion , so tha t x - g = x g for x , g G G. Th is is the regular act ion of

G, and it should be clear tha t it is faithful, which means tha t its kerne l is

t r i v ia l . It follows tha t the corresponding permutat ion representation of G is

an isomorphism of G into Sym(G) , and this proves Cayleys theorem: every

group is isomorphic to a group of permutations on some set.

We continue to take ft = G, bu t this t ime , we define x - g = g ~ l x g . (The

standard notat ion for g ~ x x g is x.) It is t r i v ia l to check tha t x l = x and that

{ x 9 ) h = X 9h for a l l x , g , h G G, and thus we t ru ly have an action , which is

called the conjugation action of G on itself. Note tha t x 9 = x i f and only i f

x g = gx, and thus the kerne l of the conjugation action is the set of elements

g e G tha t commute w i th a l l elements x e G. The kernel, therefore, is the

center Z ( G ) .

Aga in let G be arbitrary . In each of the previous examples, we took

ft = G , bu t we also get interesting actions i f instead we take ft to be the set

of a l l subsets of G. In the conjugation act ion of G on ft we let X - g = X 9 =

{ x 9 \ x e X } and in the r ight-mul t ip l icat ion act ion we define X - g = X g =

{ x g | x 6 X } . O f course, in order to make these examples work, we do not

really need ft to be a l l subsets of G. For example, since a conjugate of a

subgroup is always a subgroup, the conjugation action is wel l defined i f we

take ft to be the set of a l l subgroups of G. A lso , both right mul t ip l icat ion

1 A 3

and conjugation preserve cardinality , and so each of these actions makes

sense i f we take ft to be the col lection of a l l subsets of G of some fixed size.

In fact, as we shal l see, the t r ick in Wie landt s proo f of the Sylow existence

theorem is to use the right mul t ip l icat ion act ion of G on its set of subsets

wi th a certain fixed cardinality .

We mention one other example, which is a specia l case of the right-

mult ip l icat ion act ion on subsets that we discussed in the previous paragraph .

Le t H C G be a subgroup, and let ft = { H x \ x G G } , the set of r ight cosets

of H in G. If X is any right coset of H , i t is easy to see that X g is also a

right coset of H . (Indeed, i f X = H x , then X g = H { x g ) . ) Then G acts on

the set ft by r ight mul t ip l icat ion .

In general, i f a group G acts on some set ft and a G ft, we write Ga =

{ g G G | a - g = a } . It is easy to check tha t Ga is a subgroup of G ; it

is cal led the stabilizer of the poin t a . For example , in the regular act ion

of G on itself, the stabil izer of every poin t (element of G ) is the t r i v ia l

subgroup. In the conjugation action of G on G, the stabil izer of x G G

is the centralizer C G ( x ) and in the conjugation act ion of G on subsets, the

stabil izer of a subset X is the normalizer N G ( X ) . A useful genera l fact abou t

poin t stabil izers is the following, which is easy to prove. In any action , i f

a - g = /?, then the stabilizers Ga and G p are conjugate in G, and in fact,

{ G a ) 3 = Gp.

Now consider the action (by right mul t ip l icat ion ) of G on the right cosets

of H , where H C G is a subgroup. The stabil izer of the coset H x is the

set of a l l group elements g such that H x g = H x . It is easy to see tha t g

satisfies this condi t ion if and only i f x g G H x . (This is because two cosets

H u and H v are identica l i f and only i f u G H v . ) It follows tha t g stabilizes

H x i f and only i f g G x ~ l H x . Since x ~ l H x = H x , we see that the stabil izer

of the poin t (coset) H x is exactly the subgroup H x , conjugate to H v ia

x . It follows tha t the kerne l of the act ion of G on the r ight cosets of H in

G is exact ly f | H x . Th is subgroup is called the core of H in G, denoted x e G

c o r e G ( H ) . The core of H is norma l in G because i t is the kerne l of an action ,

and , clearly, it is contained in H . In fact, i f AT < G is any norma l subgroup

that happens to be contained in H , then N = Nx C H x for a l l x G G , and

thus N C co reG ( t f ) . In other words, the core of H in G is the unique largest

norma l subgroup of G contained in H . (It is "largest" in the strong sense

tha t it contains a l l others.)

We have digressed from our goal, which is to show how to use group

actions to count things. Bu t having come this far, we may as wel l state the

results that our discussion has essentially proved. Note tha t the following

theorem and its corollaries can be used to prove the existence of norma l

subgroups, and so they might be considered to be "good" results.

4 1 . S y l o w T h e o r y

1.1. Theorem . L e t H C G be a s u b g r o u p , a n d l e t ft be t h e set of r i g h t cosets

o f H i n G. T h e n G / c o v eG { H ) i s i s o m o r p h i c t o a s u b g r o u p o /Sym( f t ) . I n

p a r t i c u l a r , if t h e i n d e x \ G : H \ = n , t h e n G / c o r eG ( H ) i s i s o m o r p h i c t o a

s u b g r o u p of Sn, t h e s y m m e t r i c g r o u p o n n s y m b o l s .

Proof . The action of G on the set ft by right mul t ip l icat ion defines a

homomorphism 6 (the permutat ion representation) from G into Sym(ft) .

Since ker(0) = co reG ( t f ) , it follows by the homomorphism theorem tha t

G / c o r e G { H ) 0 ( G ) , which is a subgroup of Sym(G) . The last statement

follows since i f \ G : H \ = n , then (by definit ion of the index) |ft| = n , and

thus Sym(ft) = Sn. U

1.2. Corol lary . L e t G be a g r o u p , a n d suppose t h a t H C G i s a s u b g r o u p

w i t h \ G : H \ = n . T h e n H c o n t a i n s a n o r m a l s u b g r o u p N of G such t h a t

\ G : N \ d i v i d e s n \ .

Proof . Take N = c o r e G ( H ) . Then G / N is isomorphic to a subgroup of

the symmetr ic group Sn, and so by Lagranges theorem, \ G / N \ divides

\ S n \ = n \ .

1.3. Corol lary . L e t G be s i m p l e a n d c o n t a i n a s u b g r o u p of i n d e x n > 1.

T h e n \ G \ d i v i d e s n l .

Proof . The norma l subgroup N of the previous corol lary is contained in H ,

and hence it is proper in G because n > 1. Since G is simple , N = 1 , and

thus \ G \ = \ G / N \ divides n l . U

In order to pursue our ma in goal, which is counting , we need to discuss

the "orbits" of an act ion . Suppose tha t G acts on ft, and let a G ft. The

set Oa = { a - g | g G G } is called the orbit of a under the given action . It is

routine to check tha t i f 3 G Oa, then O p = Oa, and i t follows that dist inc t

orbits are actual ly disjoint. A lso , since every poin t is in at least one orbit,

it follows tha t the orbits of the act ion of G on ft par t i t ion ft. In part icular ,

i f ft is finite, we see tha t |ft| = where in this sum , O runs over the

ful l set of G-orbi ts on ft.

We mention some examples of orbits and orbi t decompositions. F i rs t , i f

H C G is a subgroup, we can let H act on G by right mul t ip l icat ion . It is

easy to see tha t the orbits of this act ion are exactly the left cosets of H in

G . (We leave to the reader the problem of realizing the r ight cosets of H

in G as the orbits of an appropriate act ion of H . B u t be careful: the rule

x - h - hx does n o t define an action.)

Perhaps it is more interesting to consider the conjugation act ion of G

on itself, where the orbits are exactly the conjugacy classes of G . The fact

1 A 5

tha t for a finite group, the order \ G \ is the sum of the sizes of the classes is

sometimes cal led the class equation of G.

How b ig is an orbit? The key result here is the fol lowing .

1.4. Theo rem (The Fundamenta l Count ing Pr inc ip le) . L e t G a c t o n Q,

a n d suppose t h a t O i s one of t h e o r b i t s . L e t a G O, a n d w r i t e H = Ga> t h e

s t a b i l i z e r of a . L e t A = { H x | x G G } be t h e set of r i g h t cosets of H i n G.

T h e n t h e r e i s a b i j e c t i o n 6 : A -> O such t h a t 9 { H g ) = a-g. I n p a r t i c u l a r ,

\ 0 \ = \ G : Ga\.

Proof . We observe first tha t i f H x = H y , then a-x = a-y. To see why this

is so, observe tha t we can wri te y = hx for some element h s H . Then

a-y = a - { h x ) = { a - h ) - x = a-x ,

where the last equality holds because h G H = Ga, and so h stabilizes a .

Given a coset H x G A , the poin t a-x lies in O, and we know tha t i t is

determined by the coset H x , and not jus t by the part icular element x . It is

therefore permissible to define the function 9 : A O by 9 { H x ) = a-x, and

it remains to show tha t 9 is bo th injective and surjective.

The surjectivity is easy, and we do tha t first. If 0 G O, then by the

definit ion of an orbit , we have 0 = a-x for some element x G G. Then

H x G A satisfies 6 { H x ) = a-x = 0, as required .

To prove that 9 is injective, suppose tha t 9 { H x ) = 9 { H y ) . We have

a-x = a-y, and hence

a = a-1 = { a - x ) - x - 1 = ( a - y ) - x " 1 = a - i y x 1 ) .

Then yx1 fixes a , and so it lies in Ga = H . It follows tha t y G H x , and

thus H y = H x . Th is proves tha t 6 is injective, as required .

It is easy to check tha t the bi jection 9 of the previous theorem actual ly

defines a "permutat ion isomorphism" between the act ion of G on A and the

action of G on the orbi t O. Formal ly , this means tha t 0 ( X - g ) = 9 ( X ) - g

for a l l "points" X in A and group elements g G G. More informally, this

says tha t the actions of G on A and on O are "essentially the same". Since

every act ion can be thought of as composed of the actions on the ind iv idua l

orbits, and each of these actions is permutat ion isomorphic to the right-

mul t ip l icat ion action of G on the right cosets of some subgroup, we see tha t

these actions on cosets are t ru ly fundamental : every group act ion can be

viewed as being composed of actions on r ight cosets of various subgroups.

We close this section w i th two famil iar and useful appl ications of the

fundamenta l counting pr inciple .

6 1 . S y l o w T h e o r y

1.5. Corol lary . L e t x G, w h e r e G i s a finite g r o u p , a n d l e t K be t h e

c o n j u g a c y class of G c o n t a i n i n g x . T h e n \ K \ = \ G : C G ( x ) \ .

Proof . The class of x is the orbi t of x under the conjugation action of G on

itself, and the stabil izer of x in this act ion is the centralizer C G { x ) . Thus

\ K \ = \ G : C G { x ) \ , as required.

1.6. Corol lary . L e t H C G be a s u b g r o u p , w h e r e G i s finite. T h e n t h e t o t a l

n u m b e r of d i s t i n c t c o n j u g a t e s of H i n G, c o u n t i n g H i t s e l f , i s \ G : N G { H ) \ .

Proof . The conjugates of H form an orbi t under the conjugation act ion of

G on the set of subsets of G. The normalizer N G { H ) is the stabil izer of H

in this act ion , and thus the orbi t size is \ G : N G ( # ) | , as wanted.

P r o b l e m s 1 A

1A.1 . Le t H be a subgroup of pr ime index p in the finite group G, and

suppose that no prime smaller than p divides \ G \ . Prove that H < G.

1A.2 . G iven subgroups H , K C G and an element g G G, the set H g K =

{ h g k \ h e H , k G K } is called an ( H , i ^ -double coset. In the case where

H and K are finite, show that \ H g K \ = \ H \ \ K \ / \ K n H 9 \ .

Hin t . Observe tha t H g K is a union of right cosets of H , and tha t these

cosets form an orbi t under the action of K .

Note . If we take g = 1 in this problem , the result is the famil iar formula

\ H K \ = \ H \ \ K \ / \ H n K \ .

1 A . 3 . Suppose that G is finite and tha t H , K C G are subgroups.

(a) Show that \ H : H n K \ < \ G : K \ , w i th equality i f and only i f

H K = G.

(b) If \ G : H \ and \ G : K \ are coprime, show tha t H K = G.

Note . Proofs of these useful facts appear in the appendix , bu t we suggest

tha t readers t ry to f ind their own arguments. A lso , recal l tha t the produc t

H K of subgroups H and K is not always a subgroup. In fact, H K is a

subgroup i f and only i f H K = K H . (This too is proved in the appendix.)

If H K = K H , we say that H and K are permutable .

1A.4 . Suppose tha t G = H K , where H and K are subgroups. Show that

also G = H x K y for a l l elements x,y G G. Deduce tha t i f G = H H X for a

subgroup H and an element x G G, then H = G.

P r o b l e m s 1 A 7

1A.5 . A n act ion of a group G on a set ft is transitive i f ft consists of a

single orbit . Equivalent ly , G is transit ive on ft i f for every choice of points

a,/? G ft, there exists an element g G G such tha t a - g = 0. Now assume

tha t a group G acts t ransit ively on each of two sets ft and A . Prove tha t

the natura l induced act ion of G on the cartesian produc t ft x A is transit ive

i f and only i f G a G p = G for some choice of a G ft and 0 G A .

Hin t . Show that i f G a G p = G for some a G ft and 0 G A , then in fact, this

holds for a l l a G ft and 0 G A .

1A.6 . Le t G act on ft, where both G and ft are finite. Fo r each element

g G G, wri te x ( g ) = \ { a e n \ a - g = a } \ . The nonnegative-integer-valued

function X is cal led the permutat ion character associated w i th the act ion .

Show that

J]x(0) = J2 \ G a \ = n \ G \ ,

where n is the number of orbits of G on ft.

Note . Thus the number of orbits is

1 1 g e G

which is the average value of x over the group. A l though this orbit-counting

formula is often at t r ibuted to W . Burnside , it should (according to P . Neu

mann) more properly be credited to Cauchy and Frobenius .

1A.7 . Le t G be a finite group, and suppose that H < G is a proper sub

group. Show that the number of elements of G tha t do not lie in any

conjugate of H is at least \ H \ .

Hint . Le t X be the permutat ion character associated w i th the r ight-mul t ip l i

cation act ion of G on the right cosets of H . Then x ( # ) = \ G \ , where the

sum runs over g e G . Show that x W > 2| t f | , where here, the sum

runs over h e H . Use this information to get an estimate on the number of

elements of G where x vanishes.

1A.8 . Le t G be a finite group, let n > 0 be an integer, and let C be

the addit ive group of the integers modulo n . Le t ft be the set of n-tuples

[ x l , x 2 , . . . , x n ) of elements of G such tha t x x x 2 x n = l .

(a) Show tha t C acts on ft according to the formula

( x i , x 2 , , x n ) - k = ( x i + k , x 2 + k , , x n + k ) ,

where k G C and the subscripts are interpreted modulo n .

8 1 . S y l o w T h e o r y

(b) Now suppose tha t n = p is a prime number tha t divides \ G \ . Show

tha t V divides the number of C-orbi ts of size 1 on ft, and deduce

tha t the number of elements of order p in G is congruent to - 1

mod p .

Note . In part icular , i f a pr ime p divides | G | , then G has at least one element

of order p . Th is is a theorem of Cauchy , and the proo f in this problem is

due to J . H . M c K a y . Cauchys theorem can also be derived as a corol lary

of Sylows theorem. Al ternat ively , a proo f of Sylows theorem different from

Wielandt s can be based on Cauchys theorem. (See P rob lem 1B.4.)

1A.9 . Suppose \ G \ = p m , where p > m and p is prime . Show tha t G has a

unique subgroup of order p .

1A.10 . Le t H C G .

(a) Show tha t \ N G ( H ) : H \ is equa l to the number of right cosets of H

in G tha t are invariant under right mul t ip l icat ion by H .

(b) Suppose tha t \ H \ is a power of the prime p and tha t \ G : H \ is

divisible by p . Show tha t \ N G { H ) : H \ is divisible by p .

I B

F i x a prime number p . A finite group whose order is a power of p is called a p-

group . It is often convenient, however, to use this nomenclature somewhat

carelessly, and to refer to a group as a "p-group" even i f there is no part icular

prime p under consideration. For example, in proving some theorem, one

migh t say: it suffices to check that the result holds for p-groups. W h a t is

meant here, of course, is that it suffices to show tha t the theorem holds for

al l p-groups for a l l primes p .

We mention that, al though in this book a p-group is required to be finite,

it is also possible to define infinite p-groups. The more genera l definit ion is

tha t a (not necessarily finite) group G is a p-group i f every element of G has

finite p-power order. O f course, i f G is finite, then by Lagranges theorem,

every element of G has order d iv id ing \ G \ , and so i f \ G \ is a power of p , it

follows tha t the order of every element is a power of p , and hence G is a

p-group according to the more general definition. Conversely, i f G is finite

and has the property tha t the order of every element is a power of p, then

clearly, G can have no element of order q for any prime q different from p.

It follows by Cauchys theorem (Problem 1A.8) tha t no prime q ^ p can

divide | G | , and thus \ G \ must be a power of p, and this shows that the two

definitions of "p-group" are equivalent for finite groups.

Aga in , fix a prime p. A subgroup S o f a finite group G is said to be

a Sylow p-subgroup of G i f \S\ is a power of p and the index \ G : S\ is

I B 9

not divisible by p. A n alternative formulat ion of this definit ion relies on the

observation tha t every positive integer can be (uniquely) factored as a power

of the given pr ime p times some integer not d iv is ible by p . In part icular ,

i f we wri te \ G \ = p a m , where a > 0 and p does not divide m > 1, then a

subgroup S of G is a Sylow p-subgroup of G precisely when \S\ = p a . In

other words, a Sylow p-subgroup of G is a p-subgroup S whose order is as

large as is permit ted by Lagranges theorem , which requires that \S\ must

divide | G | . We mention two t r i v ia l cases: i f | G | is not divisible by p, then

the ident i ty subgroup is a Sylow p-subgroup of G, and i f G is a p-group,

then G is a Sylow p-subgroup of itself. The Sylow existence theorem asserts

tha t Sylow subgroups a l w a y s exist.

1.7. Theorem (Sylow E ) . L e t G be a finite g r o u p , a n d l e t p be a p r i m e .

T h e n G has a S y l o w p - s u b g r o u p .

The Sylow E-theorem can be viewed as a par t ia l converse of Lagranges

theorem. Lagrange asserts that i f i f is a subgroup of G and \ H \ = k, then

k divides \ G \ . The converse, which in genera l is false, would say that i f k

is a posit ive integer tha t divides then G has a subgroup of order k.

(The smallest example of the failure of this assertion is to take G to be the

alternating group A A of order 12; this group has no subgroup of order 6.)

Bu t i f A; is a power of a prime , we shal l see tha t G actual ly does have a

subgroup of order k. If k is the largest power of p tha t divides \ G \ , the

desired subgroup of order k is a Sylow p-subgroup; for smaller powers of p,

we w i l l prove tha t a Sylow p-subgroup of G necessarily has a subgroup of

order k.

We are ready now to begin work toward the proo f of the Sylow E -

theorem. We start w i th a purely ar i thmetic fact abou t b inomia l coefficients.

1.8. Lemma . L e t p be a p r i m e n u m b e r , a n d l e t a > 0 a n d m > 1 be i n t e g e r s .

Proof . Consider the po lynomia l (1 + X ) p . Since p is pr ime , it is easy to

see tha t the b inomia l coefficients (p) are divisible by p for 1 < i < p - 1,

and thus we can write (1 + X ) p = 1 + X p mod p. (The assertion tha t

these polynomials are congruent modulo p means tha t the coefficients of

corresponding powers of X are congruent modulo p.) A p p l y i n g this fact a

second t ime , we see tha t { 1 + X ) p 2 = ( 1 + X P ) P = 1 + X p 2 mod p. Cont inu ing

l ike this, we deduce tha t (1 + X ) p a = 1 + X p a mod p , and thus

T h e n

= m mod p

( l + Xfam = ( 1 + X p a ) m mod p .

10 1 . S y l o w T h e o r y

Since these polynomials are congruent, the coefficients of corresponding

terms are congruent modulo p , and the result follows by considering the

coefficient of X ? a on each side.

Proo f of the Sylow E- theorem (Wielandt) . Wr i te \ G \ = p a m , where

a > 0 and p does not divide m. Le t f t be the set of a l l subsets of G having

cardinal i ty p a , and observe tha t G acts by right mul t ip l icat ion on f t . Because

of this act ion , f t is part i t ioned into orbits, and consequently, \fl\ is the sum

of the orbi t sizes. B u t

and so | i2| is not divisible by p , and it follows that there is some orbi t O

such tha t \ 0 \ is not divisible by p .

Now let X G O, and let H = Gx be the stabil izer of X in G. B y

the fundamenta l counting pr inciple , \ 0 \ = \ G \ / \ H \ , and since p does not

divide \ 0 \ and p a divides \ G \ , we conclude that p a must divide \ H \ , and in

part icular p a < \ H \ .

Since H stabilizes X under right mul t ip l icat ion , we see tha t i f x G X ,

then x H C X , and thus \ H \ = \ x H \ < \X\ = p a , where the fina l equality

holds since X G f t . We now have \ H \ = p a , and since H is a subgroup, it is

a Sylow subgroup of G, as wanted.

In P rob lem 1A.8 , we sketched a proo f of Cauchys theorem. We can now

give another proof, using the Sylow E-theorem .

1.9. Coro l lary (Cauchy) . L e t G be a finite g r o u p , a n d suppose t h a t p i s a

p r i m e d i v i s o r of \ G \ . T h e n G has a n element of o r d e r p .

Proof . Le t S be a Sylow p-subgroup of G, and note that since \S\ is the

max imum power of p that divides | G | , we have \S\ > 1. Choose a non-

identi ty element x of S, and observe tha t the order o { x ) divides \S\ by

Lagranges theorem, and thus 1 < o ( x ) is a power of p . In part icular , we

can wri te o { x ) = pm for some integer m > 1, and we see tha t o { x m ) = p , as

wanted.

We introduce the notat ion S y l p ( G ) to denote the set of a l l Sylow p-

subgroups of G. The assertion of the Sylow E-theorem , therefore, is tha t

the set Sylp(G) is nonempty for a l l finite groups G and a l l primes p . The

intersection n S y l p ( G ) of a l l Sylow p-subgroups of a group G is denoted

O p ( G ) , and as we shal l see, this is a subgroup that plays an importan t role

in finite group theory.

I B 11

Perhaps this is a good place to digress to review some basic facts abou t

characteristic subgroups. (Some of this mater ia l also appears in the appen

dix.) F i rs t , we recal l the definition: a subgroup K C G is characteristic

in G i f every automorphism of G maps K onto itself.

It is often difficult to find a l l automorphisms of a given group, and so the

definit ion of "characteristic" can be hard to apply directly, bu t nevertheless,

in many cases, it easy to establish tha t certain subgroups are characteristic.

Fo r example , the center Z ( G ) , the derived (or commutator) subgroup G,

and the intersection of al l Sylow p-subgroups Op ( G ) are characteristic in

G . More generally, any subgroup that can be described unambiguously as

" t h e something" is characteristic. It is essential tha t the description using

the definite article be unambiguous, however. G iven a subgroup H Q G , for

example, we cannot conclude tha t the normalizer N G ( H ) or the center Z ( H )

is characteristic in G . A l though these subgroups are described using "the",

the descriptions are not unambiguous because they depend on the choice

of H . We can say, however, that Z ( G ) is characteristic in G because it is

t h e center of t h e derived subgroup; it does not depend on any unspecified

subgroups.

A good way to see why "the something" subgroups must be characteris

t ic is to imagine two groups G i and G2 , w i th an isomorphism 9 : G i -> G2 .

Since isomorphisms preserve "group theoretic" properties, it should be clear

that 9 maps the center Z ( G i ) onto Z ( G 2 ) , and indeed 9 maps each un

ambiguously defined subgroup of G i onto the corresponding subgroup of

G 2 . Now specialize to the case where G i and G2 happen to be the same

group G , so 9 is an automorphism of G . Since in the genera l case, we

know that 6(Z(GJ) = Z ( G 2 ) , we see tha t when G i = G = G2 , we have

0(Z(G)) = Z ( G ) , and similarly , i f we consider any "the something" sub

group in place of the center.

O f course, characteristic subgroups are automatical ly normal . Th is

is because the definit ion of normal i ty requires only tha t the subgroup be

mapped onto itself by i n n e r automorphisms while characteristic subgroups

are mapped onto themselves by a l l automorphisms . We have seen that some

characteristic subgroups are easily recognized, and it follows that these sub

groups are obviously and automatical ly normal . Fo r example, the subgroup

O p ( G ) is norma l in G for al l primes p .

The fact that characteristic subgroups are norma l remains true in an

even more genera l context. The fol lowing, which we presume is already

known to most readers of this book, is extremely useful. (This result also

appears in the appendix.)

1.10. Lemma . L e t K C N C G, w h e r e G i s a g r o u p , N i s a n o r m a l

s u b g r o u p of G a n d K i s a c h a r a c t e r i s t i c s u b g r o u p of N . T h e n K < G .

12 I . Sylow T h e o r y

Proof . Le t g G. Then conjugation by g maps N onto itself, and it follows

tha t the restr ict ion of this conjugation map to N is an automorphism of N .

(Bu t note tha t it is not necessarily an inner automorphism of N . ) Since K

is characteristic in N , it is mapped onto itself by this automorphism of N ,

and thus K 9 = K , and it follows tha t K < G. U

P r o b l e m s I B

1B.1 . Le t S e S y l p ( G ) , where G is a finite group.

(a) Le t P C G be a p-subgroup. Show that PS is a subgroup i f and

only i f P C S.

(b) If S < G, show that S y l p ( G ) = { S } , and deduce tha t S is charac

teristic in G.

Note . O f course, it would be "cheating" to do problems in this section using

theory tha t we have not yet developed. In part icular , you should avoid using

the Sylow C-theorem , which asserts that every two Sylow p-subgroups of G

are conjugate in G.

1B.2 . Show tha t O p ( G ) is the unique largest norma l p-subgroup of G. (This

means tha t it is a norma l p-subgroup of G that contains every other norma l

p-subgroup of G.)

1B.3 . Le t S Sy lp(G) , and write N = N G ( 5 ) . Show tha t N = N G ( N ) .

1B.4 . Le t P C G be a p-subgroup such that \ G : P \ is divisible by p. Us ing

Cauchys theorem , bu t wi thou t appealing to Sylows theorem, show that

there exists a subgroup Q of G containing P , and such tha t \ Q : P \ = p .

Deduce that a max ima l p-subgroup of G (which obviously must exist) must

be a Sylow p-subgroup of G.

Hin t . Use P rob lem 1A.10 and consider the group N G ( P ) / P .

Note . Once we know Cauchys theorem, this problem yields an alternative

proo f of the Sylow E-theorem . O f course, to avoid circularity , we appea l to

Prob lem 1A.8 for Cauchys theorem, and not to Coro l lary 1.9.

1B.5 . Le t 7T be any set of pr ime numbers. We say that a finite group H is

a vr-group i f every pr ime divisor of \ H \ lies in TT . A lso , a vr-subgroup H C G

is a Ha l l 7r-subgroup of G i f no pr ime d iv id ing the index \ G : H \ lies in TT.

(So i f TT = {p}, a Ha l l vr-subgroup is exactly a Sylow p-subgroup.)

Now let 9 : G - K be a surjective homomorphism of finite groups.

(a) If H is a H a l l vr-subgroup of G, prove that 9 ( H ) is a Ha l l vr-subgroup

of K .

P r o b l e m s I B 13

(b) Show that every Sylow p-subgroup of K has the form 9 ( H ) , where

H is some Sylow p-subgroup of G.

(c) Show that | S y lp ( G ) | > | S y lp ( X ) l for every pr ime p.

Note . If the set vr contains more than one pr ime number , then a H a l l vr-

subgroup can fai l to exist. B u t a theorem of P . H a l l , after whom these

subgroups are named , asserts that in the case where G is solvable, H a l l TT-

subgroups always do exist. (See Chapte r 3, Section C.) We mention also

that Par t (b) of this problem would not remain true i f "Sylow p-subgroup"

were replaced by "Ha l l 7r-subgroup".

1B.6 . Le t G be a finite group, and let K C G be a subgroup. Suppose that

H C G is a Ha l l 7r-subgroup, where TT is some set of primes . Show tha t i f

H K is a subgroup, then H n K is a H a l l 7r-subgroup of K .

Note . In part icular , K has a Ha l l 7r-subgroup i f either H or K is norma l in

G since in that case, H K is guaranteed to be a subgroup.

1B.7 . Le t G be a finite group, and let vr be any set of primes.

(a) Show that G has a (necessarily unique) norma l 7r-subgroup N such

tha t N D M whenever M < G is a 7r-subgroup.

(b) Show that the subgroup N of Par t (a) is contained in every Ha l l

7r-subgroup of G.

(c) Assuming tha t G has a H a l l 7r-subgroup, show tha t N is exactly

the intersection of a l l of the Ha l l vr-subgroups of G.

Note . The subgroup N of this problem is denoted O n ( G ) . Because of the

uniqueness in (b), it follows tha t this subgroup is characteristic in G . F ina l ly ,

we note tha t i f p is a prime number, then, of course, 0{ p } ( G ) - O p ( G ) .

1B.8 . Le t G be a finite group, and let vr be any set of primes.

(a) Show tha t G has a (necessarily unique) norma l subgroup N such

tha t G / N is a vr-group and M D N whenever M < G and G / M is

a 7r-group.

(b) Show that the subgroup N of Par t (a) is generated by the set of al l

elements of G that have order not divisible by any pr ime in TT.

Note . The characteristic subgroup N of this problem is denoted O f f ( G ) .

Also , we recal l tha t the subgroup generated by a subset of G is the (unique)

smallest subgroup that contains tha t set.

14 I . S y l o w T h e o r y

1 C

We are now ready to study in greater detai l the nonempty set S y l p ( G ) of

Sylow p-subgroups of a finite group G .

1.11. Theorem . L e t P be a n a r b i t r a r y p - s u b g r o u p of a finite g r o u p G, a n d

suppose t h a t S S y l p ( G ) . T h e n PCS9 f o r some element g e G .

Proof . Le t ft = {Sx \ x e G } , the set of right cosets of S in G , and note

that |ft| = \G:S\ is not divisible by p since -S is a Sylow p-subgroup of G .

We know tha t G acts by right mul t ip l icat ion on ft, and thus P acts too , and

ft is part i t ioned into P-orb i ts . A lso , since |ft| is not divisible by p, there

must exist some P-orb i t O such that \ 0 \ is not divisible by p .

B y the fundamenta l counting principle , \ 0 \ is the index in P of some

subgroup. It follows tha t \ 0 \ divides | P | , which is a power of p. Then \ 0 \ is

both a power of p and not divisible by p, and so the only possibi l i ty is tha t

\ 0 \ = 1. Recal l ing that a l l members of ft are right cosets of S in G , we can

suppose tha t the unique member of O is the coset Sg.

Since Sg is alone in a P-orb i t , it follows that it is fixed under the act ion

of P , and thus Sgu = Sg for a l l elements u e P . Then g u Sg, and hence

u g-^Sg = S9. Thus PCS9, as required.

If S is a Sylow p-subgroup of G , and g e G is arbitrary , then the conju

gate S9 is a subgroup having the same order as S. Since the only requirement

on a subgroup that is needed to qualify it for membership in the set S y l p ( G )

is that it have the correct order, and since S S y l p ( G ) and \S9\ = \S\, it

follows tha t S9 also lies in S y l p ( G ) . In fact every member of S y l p ( G ) arises

this way: as a conjugate of S. Th is is the essential content of the Sylow

conjugacy theorem. Pu t t i ng it another way: the conjugation act ion of G on

S y l p ( G ) is transit ive .

1.12. Theorem (Sylow C ) . If S a n d T S y l o w p - s u b g r o u p s of a finite g r o u p

G, t h e n T = S9 f o r some element g e G .

Proof . A p p l y i n g Theorem 1.11 w i th T in place of P , we conclude that

T C S9 for some element g e G . Bu t since both S and T are Sylow p-

subgroups, we have \T\ = \S\ = \S9\, and so the containment of the previous

sentence must actual ly be an equality.

The Sylow C-theorem yields an alternative proo f of Prob lem I B . 1(b),

which asserts that i f a group G has a norma l Sylow p-subgroup S, then

S is the only Sylow p-subgroup of G . Indeed, by the Sylow C-theorem , i f

T e S y l p ( G ) , then we can wri te T = S9 = S, where the second equality is a

consequence of the normal i ty of S.

1 C 15

A frequently used appl icat ion of the Sylow C-theorem is the so-called

"Frat t in i argument", which we are abou t to present. Perhaps the reason

that this result is generally referred to as an "argument" rather than as a

" lemma" or "theorem" is that variations on its proo f are used nearly as often

as its statement.

1.13. L e m m a (Frat t in i Argument ) . L e t N < G w h e r e N i s finite, a n d s u p

pose t h a t P G Sylp( iV) . T h e n G = N G { P ) N .

Proof . Le t g G G, and note that P 9 C N 9 = N , and thus P9 is a subgroup

of N having the same order as the Sylow p-subgroup P . It follows that

P 9 G Sylp( iV) , and so by the Sylow C-theorem appl ied in N , we deduce that

( p g y = p , for some element n J V . Since P 9 n = P , we have g n G NG ( P ) ,

and so g G NG ( P ) n "1 C N G ( P ) N . Bu t g G was arbitrary , and we deduce

that G = N G ( P ) A T , as required.

B y definit ion, a Sylow p-subgroup of a finite group G is a p-subgroup

that has the largest possible order consistent w i th Lagranges theorem. B y

the Sylow E-theorem , we can make a stronger statement: a subgroup whose

order is max ima l among the orders of a l l p-subgroups of G is a Sylow p-

subgroup. A n even stronger assertion of this type is tha t every max ima l p-

subgroup of G is a Sylow p-subgroup. Here, "maximal " is to be interpreted

in the sense of containment: a subgroup H of G is max ima l w i th some

property i f there is no subgroup K > H tha t has the property. The t ru th of

this assertion is the essential content of the Sylow "development" theorem.

1.14. Theorem (Sylow D ) . L e t P be a p - s u b g r o u p of a finite g r o u p G. T h e n

P i s c o n t a i n e d i n some S y l o w p - s u b g r o u p of G.

Proof . Le t S G S y l p ( G ) . Then by Theorem 1.11, we know that PCS9 for

some element g G G. A lso , since \S9\ = \S\, we know tha t S9 is a Sylow

p-subgroup of G. U

Given a finite group G, we consider next the question of how many Sylow

p-subgroups G has. To facil itate this discussion, we introduce the (not quite

standard) notat ion n p { G ) = | S y lp ( G ) | . (Occasionally , when the group we

are considering is clear from the context, we w i l l s imply write n p instead of

n p { G ) . )

Fi rs t , by the Sylow C-theorem , we know tha t S y l p ( G ) is a single orbi t

under the conjugation action of G. The fol lowing is then an immediate

consequence.

1.15. Corol lary . L e t S G S y l p ( G ) , w h e r e G i s a finite g r o u p . T h e n np ( G ) -

| G : N G ( 5 ) | .

16 1 . S y l o w T h e o r y

Proof . Since n p ( G ) = | S y lp ( G ) | is the tota l number of conjugates of S in

G, the result follows by Coro l lary 1.6.

In part icular , it follows that n p ( G ) divides \ G \ , bu t we can say a bi t

more. If 5 S y l p ( G ) , then of course, S C NG ( S ) since S is a subgroup, and

thus | G : S\ = \ G : N G ( 5 ) | |NG ( 5 ) : S\. A lso , n p { G ) = \ G : N G (5 ) | , and

hence n p ( G ) divides \ G : S\. In other words, i f we wri te \ G \ = pa m , where

p does not divide m , we see that n p { G ) divides m. (We mention that the

integer m is often referred to as the p-part of \ G \ . )

The information that n p { G ) divides the p-part of \ G \ becomes even more

useful when it is combined w i th the fact (probably known to most readers)

tha t n p ( G ) = 1 mod p for a l l groups G. In fact, there is a useful stronger

congruence constraint, which may not be quite so wel l known . Before we

present our theorem, we mention that i f S,T S y l p ( G ) , then |5| = |T | , and

thus \S : SnT\ = \S\/\SnT\ = |T | / | 5n r | = \ T : S n T \ . The statement of

the fol lowing result, therefore, is not really as asymmetric as it may appear.

1.16. Theorem . Suppose t h a t G i s a finite g r o u p such t h a t np { G ) > 1 , a n d

choose d i s t i n c t S y l o w p - s u b g r o u p s S a n d T of G such t h a t t h e o r d e r \S D T\

i s as l a r g e as p o s s i b l e . T h e n np { G ) = 1 m o d \S : S n T | .

1.17. Corol lary . If G i s a finite g r o u p a n d p i s a p r i m e , t h e n np ( G ) = 1

m o d p .

Proof . If n p ( G ) = 1, there is nothing to prove. Otherwise , Theorem 1.16

applies, and there exist dist inc t members S,T e S y l p ( G ) such that n p ( G ) =

1 mod \S :SnT\, and thus it suffices to show tha t \S :SnT\ is divisible by

p. B u t \S : S n T\ = \ T : S n T\ is certainly a power of p, and it exceeds 1

since otherwise S = S n T = T , which is not the case because S and T are

dist inct .

In order to see how Theorem 1.16 can be used, consider a group G of

order 21,952 = 26 -7 3 . We know that n 7 must divide 26 = 64, and it must

be congruent to 1 modulo 7. We see, therefore, that n 7 must be one of 1,

8 or 64. Suppose that G does not have a norma l Sylow 7-subgroup, so that

n 7 > 1. Since neither 8 nor 64 is congruent to 1 modulo 72 = 49, we see

by Theorem 1.16 that there exist dist inc t Sylow 7-subgroups S and T of G

such that \S : SnT\ = 7.

Let s pursue this a bi t further. Wr i te D = S ( I T i n the above si tuation ,

and note tha t since |5 : D \ = 7 is the smallest pr ime divisor of \S\ = 73 ,

it follows by Prob lem 1A .1 , tha t D < S. Simi la r reasoning shows tha t also

D < T , and hence S and T are both contained in N = NG (L>) . Now S

and T are dist inc t Sylow 7-subgroups of N , and it follows tha t n 7 ( N ) > 1,

and hence n 7 ( N ) > 8 by Corol lary 1.17. Since n 7 ( N ) is a power of 2 that

1 C 17

divides |AT | , we deduce tha t 2 3 divides \ N \ . Since also 73 divides \ N \ , we

have \ G : N \ < 8.

We can use wha t we have established to show that a group G of order

21,952 cannot be simple. Indeed, i f n7 ( G ) = 1, then G has a norma l sub

group of order 73 , and so is not simple . Otherwise , our subgroup N has

index at most 8, and we see tha t | G | does not d iv ide \ G : N \ \ . B y the n \ -

theorem (Corol lary 1.3), therefore, G cannot be simple i f N < G. F ina l ly ,

\{ N = G then D < G and G is not simple in this case too.

In the last case, where D < G, we see tha t D is contained in al l Sylow

7-subgroups of G, and thus D is the intersection of every two dist inc t Sylow

7-subgroups of G. In most situations, however, Theorem 1.16 can be used

to prove only the existence of some pair of dist inc t Sylow subgroups w i th

a "large" intersection; it does not usually follow tha t every such pair has a

large intersection.

To prove Theorem 1.16, we need the fol lowing .

1.18. L e m m a . L e t P S y l p ( G ) , w h e r e G i s a finite g r o u p , a n d suppose

t h a t Q i s a p - s u b g r o u p o / N G ( P ) . T h e n Q C P .

Proof . We apply Sylow theory in the group N = NG ( P ) . Clearly , P is

a Sylow p-subgroup of N , and since P < N , we deduce that P is the only

Sylow p-subgroup of N . B y the Sylow D-theorem , however, the p-subgroup

Q of N must be contained in some Sylow p-subgroup. The only possibi l i ty

is Q C P , as required.

A n alternative method of proo f for L e m m a 1.18 is to observe that since

Q C ^ N G ( P ) , it follows that Q P = P Q . Then Q P is a subgroup, and it is

easy to see that it is a p-subgroup tha t contains the Sylow p-subgroup P . It

follows tha t P = Q P 5 Q, as wanted.

Proo f of Theorem 1.16. Le t S act on the set S y l p ( G ) by conjugation.

One orbi t is the set { S } , of size 1, and so i f we can show tha t a l l other orbits

have size divisible by \S : S D T\, it w i l l follow tha t n p ( G ) = | S y lp ( G ) | = 1

mod \S : S D T\, as wanted. Le t O be an arbi trary S-orbi t in S y l p ( G ) other

than { S } and let P G O, so that P ^ S. B y the fundamenta l counting pr in

ciple, \ 0 \ = \S : Q\, where Q is the stabil izer of P in S under conjugation.

Then Q C NG ( P ) , and so Q C P by L e m m a 1.18. B u t also Q C 5 , and thus

| Q | < | 5 n P | < | 5 n T | ; where the latter inequali ty is a consequence of the

fact that |5 D T\ is as large as possible among intersections of two dist inc t

Sylow p-subgroups of G . It follows that \ 0 \ = \S : Q\ > \S : S D T\. Bu t

since the integers \ 0 \ and \S : S n T\ are powers of p and \ 0 \ > \S : S n T\,

we conclude that \ 0 \ is a mul t ip le of \S : S n T\. Th is completes the

proof.

18 1 . S y l o w T h e o r y

P r o b l e m s I C

1C.1 . Le t P e S y l p ( G ) , and suppose that N G ( P ) C H C G , where H is a

subgroup. Prove that H = N G { H ) .

Note . Th is generalizes Prob lem 1B.3 .

1C.2 . Le t H C G , where G is a finite group.

(a) If P Sy lp (P" ) , prove that P = H n 5 for some member 5 G

S y l p ( G ) .

(b) Show that np (P f ) < n p ( G ) for a l l primes p.

1C.3 . Le t G be a finite group, and let X be the subset of G consisting of

al l elements whose order is a power of p , where p is some fixed prime .

(a) Show that X = ( J S y lp ( G ) .

1C.4 . Le t \ G \ = 120 = 23-3-5 . Show that G has a subgroup of index 3 or a

subgroup of index 5 (or both) .

Hin t . Ana lyze separately the four possibil it ies for n2 ( G ) .

1C.5 . Le t P e Sy lp(G) , where G = A p + U the alternating group on p + 1

symbols . Show that | N G ( P ) | = p(p - l ) / 2 .

Hin t . Coun t the elements of order p in G .

1C.6 . Le t G = PTif , where P" and K are subgroups, and fix a prime p.

(a) Show that there exists P S y l p ( G ) such that P n H S y l p ( P )

and P H f i S y l p ( X ) .

(b) If P is as in (a), show that P - ( P n H ) ( P n AT).

Hin t . Fo r (a), first choose Q S y l p ( G ) and g G such that Q n H e

S y l p ( P ) and Q 9 n f i e S y l p ( X ) . Wr i te 0 = Ziifc, w i th h e H and k

I D 19

1C.8 . Le t P be a Sylow p-subgroup of G. Show tha t for every nonnegative

integer a, the numbers of subgroups of order p a in P and in G are congruent

modulo p .

Note . If p a = | P | , then the number of subgroups of order p a in P is clearly

1, and it follows that the number of such subgroups in G is congruent to 1

modulo p . Th is provides a somewhat different proo f that n p ( G ) = 1 mod p .

It is true in genera l that i f p a < \P\, then the number of subgroups of order

p a in P is congruent to 1 modulo p , and thus it follows tha t i f p a divides the

order of an arbi trary finite group G, then the number of subgroups of order

p a in G is congruent to 1 mod p .

I D

We now digress from our study of Sylow theory in order to review some

basic facts abou t p-groups and ni lpoten t groups. A lso , we discuss the F i t t i ng

subgroup, and in the problems at the end of the section, we present some

results about the Fra t t in i subgroup.

Al though p-groups are not at a l l typ ica l of finite groups in general, they

play a prominen t role in group theory, and they are ubiquitous in the study

of finite groups. Th is ubiqui ty is, of course, a consequence of the Sylow

theorems, and perhaps tha t justifies our digression.

We should mention tha t although their structure is atypica l when com

pared w i th finite groups in general, p-groups are, nevertheless, extremely

abundan t in comparison w i th non-p-groups. There are, for example, 2,328

isomorphism types of groups of order 128 = 27 ; the number of types of order

256 = 28 is 56,092; for 512 = 29 the number is 10,494,213; and there are

exactly 49,487,365,422 isomorphism types of groups of order 1,024 = 21 0 .

(These numbers were computed by a remarkable a lgor i thm for counting p-

groups tha t was developed by E . OBr ien. )

There is an extensive theory of finite p-groups (and also of their infinite

cousins, pro-p-groups), and there are severa l books entirely devoted to them .

Ou r brief presentation here w i l l be quite superficial; later, we study p-groups

a bi t more deeply, but s t i l l , we shal l see only a t iny part of wha t is known .

Perhaps the most fundamenta l fact abou t p-groups is that nontr iv ia l

finite p-groups have nontr iv ia l centers. (By our definit ion , "p-group" means

"finite p-group" , bu t we included the redundant adjective in the previous

sentence and in wha t follows in order to stress the fact that finiteness is

essential here. Infinite p-groups can have t r i v ia l centers, and in fact, they

can be simple groups.)

In fact, a stronger statement is true.

20 1 . S y l o w T h e o r y

1.19. Theorem . L e t P be a finite p - g r o u p a n d l e t N be a n o n i d e n t i t y n o r m a l

s u b g r o u p of P . T h e n N n Z ( P ) > 1. I n p a r t i c u l a r , if P i s n o n t n v i a l , t h e n

Z { P ) > 1.

Proof . Since N < P , we can let P act on N by conjugation, and we observe

that N f l Z ( P ) is exactly the set of elements of N that lie in orbits of size

1. B y the fundamenta l counting principle , every orbi t has p-power size, and

so each nontr iv ia l orbi t ( i . e . , orbi t of size exceeding 1) has size divisible by

p . Since the set N - ( N n Z ( P ) ) is a union of such orbits, we see tha t

\ N \ \ N n Z ( P ) | is divisible by p , and thus \ N n Z ( P ) | = \ N \ = 0 mod

p , where the second congruence follows because TV is a nontr iv ia l subgroup.

Now N n Z ( P ) contains the identity element, and so \ N n Z ( P ) | > 0. It

follows tha t \ N n Z ( P ) | > p > 1, and hence N n Z ( P ) is nontr iv ia l , as

required. The fina l assertion follows by tak ing N = P . U

It is now easy to show that (finite, of course) p-groups are ni lpotent ,

and thus we can obtain addit iona l information about p-groups by studying

genera l ni lpoten t groups. Bu t first, we review some definitions.

A finite collection of norma l subgroups Nt of a (not necessarily finite)

group G is a norma l series for G provided that

1 = iVo C N i C C N r = G.

This norma l series is a centra l series i f in addi t ion , we have N i / N i - i C

Z ( G / J V i _ i ) for 1 < i < r . F ina l ly , a group G is nilpotent i f it has a centra l

series. It is worth not ing tha t subgroups and factor groups of ni lpoten t

groups are themselves ni lpotent , al though we omi t the easy proofs of these

facts.

Given any group G, we can attemp t to construct a centra l series as fol

lows. (Bu t of course, this attemp t is doomed to failure unless G is ni lpotent.)

We start by defining Z0 = 1 and Zx = Z ( G ) . The second center Z 2 is

defined to be the unique subgroup such tha t Z 2 / Z 1 = Z { G / Z 1 ) . (Note that

Z 2 exists and is norma l in G by the correspondence theorem.) We continue

l ike this, induct ively defining Zn for n > 0 so that Z n / Z n - i = Z ( G / Z _ i ) .

The chain of norma l subgroups

1 = Z 0 C Z i C Z 2 C

constructed this way is called the upper centra l series of G. We hasten

to poin t out, however, tha t in general, the upper centra l series may not

actual ly be a centra l series for G because it may happen that Z { < G for a l l

i. In other words, the upper centra l series may never reach the whole group

G. Bu t i f Z r = G for some integer r , then { Z x \ 0 < i < r } is a true centra l

series, and G is ni lpotent .

I D 21

Conversely, i f G is ni lpotent , the upper centra l series of G really is a

centra l series. Fo r finite groups G, this is especially easy to prove.

1.20. L e m m a . L e t G be finite. T h e n t h e f o l l o w i n g a r e e q u i v a l e n t .

(1) G i s n i l p o t e n t .

(2) E v e r y n o n t r i v i a l h o m o m o r p h i c i m a g e of G has a n o n t r i v i a l c e n t e r .

(3) G a p p e a r s as a member of i t s u p p e r c e n t r a l s e r i e s .

Proof . We have already remarked tha t homomorphic images of ni lpoten t

groups are ni lpotent . A lso , since the first nontr iv ia l term of a centra l series

for a ni lpoten t group is contained in the center of the group, it follows tha t

nontr iv ia l ni lpoten t groups have nontr iv ia l centers. Th is shows tha t (1)

implies (2).

Assuming (2) now, i t follows tha t i f Z { < G, where is a term in the

upper centra l series for G, then Z l + 1 / Z i = Z { G / Z i ) is nontr iv ia l , and thus

Z i < Z i + 1 . Since G is finite and the proper terms of the upper centra l series

are str ict ly increasing, we see tha t not every term can be proper, and this

establishes (3).

Fina l ly , (3) guarantees tha t the upper centra l series for G is actual ly a

centra l series, and thus G is ni lpotent , proving (1).

If P is a finite p-group, then of course, every homomorphic image of P is

also a finite p-group, and thus every nontr iv ia l homomorphic image of P has

a nontr iv ia l center. It follows by L e m m a 1.20, therefore, tha t finite p-groups

are ni lpotent . In fact, we shal l see in Theorem 1.26 tha t much more is true:

a finite group G is ni lpoten t i f and only i f every Sylow subgroup of G is

normal .

Next , we show tha t the terms of the upper centra l series of a ni lpoten t

group contain the corresponding terms of an arbi t rary centra l series, and

this explains why the upper centra l series is called "upper". It also provides

an alternative proo f of the impl icat ion (1) => (3) of L e m m a 1.20, wi thou t

the assumption tha t G is finite.

1.21. Theorem . L e t G be a ( n o t n e c e s s a r i l y finite) n i l p o t e n t g r o u p w i t h

c e n t r a l series

l = N Q C N 1 C . . - C N r = G,

a n d as u s u a l , l e t

1 = Z 0 C Z x C Z 2 C

be t h e u p p e r c e n t r a l series f o r G. T h e n N t C Z { f o r 0 < % < r , a n d i n

p a r t i c u l a r , Z r = G.

22 1 . S y l o w T h e o r y

Proof . We prove that N i C Z i by induct ion on i. Since Z0 = 1 = A^o, we

can suppose that i > 0, and by the inductive hypothesis, we can assume

tha t N i - i C Z i - i . Fo r notat iona l simplici ty , write N = and Z = Z % - X ,

and observe tha t since N C Z , there is a natura l surjective homomorphism

0 : G / N G / Z , defined by 0 (W5 ) = Z

I D 23

some subgroup H Q G , and al though there_are usual ly many subgroups of

G whose image in G is the given subgroup H , exact ly oneo f them contains

N . If H Q G is arbitrary, we see tha t H N = H N = H since overbar is

a homomorphism and N is its kernel. _It follows tha t H N is the unique

subgroup containing N whose image in G is H . In part icular , since indices

of corresponding subgroups are equal, we have \ G : H \ = \ G : A^P"! for al l

subgroups H Q G .

The correspondence theorem also yields information_abou t normality .

If AT C H Q K C G, then < K i f and only _if H < K . In part icular ,

i f N C H , then since 7f C N G ( / f ) , we see tha t # < N G ( P ) and we have

N G ( H ) C N q ( H ) . In fact, equality holds here. To see this , observe tha t

since Ng ( f f ) is a subgroup of G, it can be wr i t ten in_the form U for some

(unique) subgroup U w i th N Q U Q G. Then H < U, and so H < U and

U C N G ( f f ) . Th is yields N ^ T ? ) = t7 C N G ( 7 f ) , as c laimed .

We w i l l use the bar convention in the fol lowing proof.

Proo f o f Theorem 1.22. Since G is ni lpotent , it has (by definition) a cen

t ra l series { N i | 0 < i < r } , and we have NQ = 1 C H and Nr = G H . It

follows tha t there is some subscript k w i th 0 < k < r such tha t N k C H bu t

N k + 1 % H . We w i l l show tha t in fact, A ^ f c + 1 C N G ( f f ) , and it w i l l follow

tha t N G ( f f ) > H , as required.

Wri te G = G/Nk and use the bar convention. Since the subgroups N i

form a centra l series, we have

N k V i C Z ( G ) C N ^ H ) = N G W ) ,

where the equality holds because Nk C H . Now because Nk C N G ( f f ) ,

we can remove the overbars to obtain A T f c + i C N G ( H ) . The proo f is now

complete.

We return now to p-groups, w i th another appl icat ion of Theorem 1.19.

1.23. Lemma . L e t P be a finite p - g r o u p a n d suppose t h a t N < M a r e

n o r m a l s u b g r o u p s of P . Then t h e r e exists a s u b g r o u p L < P such t h a t N C

L C M a n d \ L : JV| = p .

Proof . W r i t e J P - P / N and note that M is nontr iv ia l and norma l in P .

Now Z ( P ) i l l is nontr iv ia l by Theorem 1.19, and so this subgroup contains

an element of order p . (Choose any nonidenti ty element and take an appro

priate power.) Because our element is centra l and of order p , it generates a

norma l subgroup of order p , and we can write I to denote this subgroup,

where N C L . Now I C M , and thus AT C L C M , as wanted. A lso , as

L < P , we see tha t L < P . F ina l ly , \ L : N \ = | Z | = p , as required.

24 1 . S y l o w T h e o r y

1.24. Corol lary . L e t P be a p - g r o u p of o r d e r p a . T h e n f o r every i n t e g e r b

w i t h 0 < b < a , t h e r e i s a s u b g r o u p L < P such t h a t \ L \ = p b .

Proof . The assertion is t r i v ia l i f b = 0, and so we can assume tha t b > 0

and we work by induct ion on b. B y the inductive hypothesis, there exists a

subgroup N < P such tha t |7V| = p b ~ l and we can apply Theorem 1.22 (wi th

M = P ) to produce a subgroup L < P w i th \ L : N \ = p . Then | L | = p b and

the proo f is complete.

Recal l now that the Sylow E-theorem can be viewed as a part ia l converse

to Lagranges theorem. It asserts tha t for certain divisors k of an integer n ,

every group of order n has a subgroup of order k. (The divisors to which

we refer, of course, are prime powers k such tha t n / k is not divisible by the

relevant prime.)

We can now enlarge the set of divisors for which we know tha t the

converse of Lagranges theorem holds.

1.25. Corol lary . L e t G be a f i n i t e g r o u p , a n d suppose t h a t p b d i v i d e s \ G \ ,

w h e r e p i s p r i m e a n d b > 0 i s a n i n t e g e r . T h e n G has a s u b g r o u p of o r d e r

p b .

Proof . Le t P be a Sylow p-subgroup of G and write \ P \ = p a . Since p b

divides \ G \ , we see tha t b < a , and the result follows by Coro l lary 1.23.

In fact, in the s i tuat ion of Coro l lary 1.25, the number of subgroups of G

having order p b is congruent to 1 modulo p . (By P rob lem 1C.8 and the note

following i t, it suffices to prove this in the case where G is a p-group, and

while this is not especially difficult, we have decided not to present a proo f

here.) We mention also tha t it does not seem to be known whether or not

there are any integers n other than powers of primes such tha t every group

of order divisible by n has a subgroup of order n .

Sylow theory is also related to the theory of ni lpoten t groups in another

way: a finite group is ni lpoten t i f and only i f a l l of its Sylow subgroups are

normal . In fact, we can say more.

1.26. Theorem . L e t G be a finite g r o u p . T h e n t h e f o l l o w i n g a r e e q u i v a l e n t .

(1) G i s n i l p o t e n t .

(2) N G ( H ) > H f o r every p r o p e r s u b g r o u p H < G.

(3) E v e r y m a x i m a l s u b g r o u p of G i s n o r m a l .

(4) E v e r y S y l o w s u b g r o u p of G i s n o r m a l .

(5) G i s t h e ( i n t e r n a l ) d i r e c t p r o d u c t of i t s n o n t r i v i a l S y l o w s u b g r o u p s .

I D 25

Note tha t in statement (3), a "max ima l subgroup" is max ima l among

p r o p e r subgroups. Bu t in most other situations , the word "maximal " does

not imp ly proper. If a group G happens to be ni lpotent , for example, then

the whole group is a max ima l ni lpoten t subgroup of G.

To help w i th the proo f that (4) implies (5), we establish the following.

1.27. Lemma . L e t X be a c o l l e c t i o n of finite n o r m a l s u b g r o u p s of a g r o u p G,

a n d assume t h a t t h e o r d e r s of t h e members of X a r e p a i r w i s e c o p r i m e . T h e n

t h e p r o d u c t H = \ \ X of t h e members of X i s d i r e c t . A l s o , \ H \ = T\ \X\. xex

Proof . Cer ta in ly \ H \ < U\x\- A lso , by Lagranges theorem , \X\ divides

\ H \ , for every member X of X , and since the orders of the members of X

are pairwise coprime, i t follows tha t f j \X\ divides \ H \ . We conclude tha t

\ H \ = n i ^ l , as wanted.

Now to see tha t T \ X direct, it suffices to show tha t

X n Y [ { Y e x | Y + X } = l

for every member X EX. Th is follows since by the previous paragraph , the

order of n Y for Y ^ X is equa l to n \Y\, and this is coprime to | X | .

Proo f of Theorem 1.26. We saw tha t (1) implies (2) in Theorem 1.22.

Tha t (2) implies (3) is clear, since i f M < G is a max ima l subgroup, then

N G ( M ) > M , and so we must have NG ( M ) = G.

Now assume (3), and let P E S y \ p ( G ) for some pr ime p . If N G ( P )

is proper in G, it is contained in some max ima l subgroup M , and we have

M < G. Since P S y l p ( M ) , it follows by L e m m a 1.13, the F ra t t in i argument,

tha t G = N G ( P ) M C M , and this is a contradict ion . Thus P < G, and this

proves (4).

Tha t (4) implies (5) is immediate from L e m m a 1.27. Now assume (5). It

is clear tha t (4) holds, and it follows tha t (4) also holds for every homomor-

phic image of G. (See Prob lem 1B.5.) Since we know tha t (4) implies (5),

we see tha t every homomorphic image of G is a direct produc t of p-groups

for various primes p . The center of a direct product , however, is the d i

rect produc t of the centers of the factors, and since nontr iv ia l p-groups have

nontr iv ia l centers, i t follows tha t every nonidenti ty homomorphic image of

G has a nontr iv ia l center. We conclude by L e m m a 1.20 tha t G is ni lpotent ,

thereby establishing (1).

Recal l tha t O p ( G ) is the unique largest norma l p-subgroup of G, by

which we mean tha t it contains every norma l p-subgroup. We define the

Fi t t ing subgroup of G, denoted F ( G ) , to be the produc t of the subgroups

26 1 . S y l o w T h e o r y

O p { G ) as p runs over the prime divisors of G. O f course, F ( G ) is character

ist ic in G, and in part icular , i t is normal .

1.28. Corol lary . L e t G be a finite g r o u p . T h e n F ( G ) i s a n o r m a l n i l p o t e n t

s u b g r o u p of G. I t c o n t a i n s every n o r m a l n i l p o t e n t s u b g r o u p of G, a n d so i t

i s t h e u n i q u e l a r g e s t such s u b g r o u p .

Proof . B y L e m m a 1.27, we know that | F ( G ) | is the produc t of the orders of

the subgroups Op ( G ) as p runs over the prime divisors of G . Then Op ( G )

S y l p ( F ( G ) ) , and thus F ( G ) has a norma l Sylow subgroup for each pr ime . It

follows by Theorem 1.26 tha t F ( G ) is ni lpotent .

Now let N < G be ni lpotent . If P e S y l p ( i V ) , then P < N by Theorem 1.26,

and thus P is characteristic in N and hence is norma l in G . It follows tha t

P C O p ( G ) C F ( G ) . Since N is the produc t of its Sylow subgroups and

each of these is contained in F ( G ) , it follows tha t N C F ( G ) , and the proo f

is complete.

1.29. Corol lary . L e t K a n d L be n i l p o t e n t n o r m a l s u b g r o u p s of a finite

g r o u p G. T h e n K L i s n i l p o t e n t .

Proof . We have K C F ( G ) and L C F ( G ) , and thus K L C F ( G ) . Since

F ( G ) is ni lpotent , it follows tha t K L is ni lpotent .

P r o b l e m s I D

1D.1 . Le t P 6 S y l p ( P ) , where H C G , and suppose tha t N G ( P ) C H .

Show tha t p does not divide | G : H \ .

Note . In these problems, we are, as usual, dealing w i th finite groups.

1D.2 . F i x a pr ime p , and suppose tha t a subgroup H C G has the property

tha t C G ( x ) C H for every element x E H having order p . Show that p

cannot divide both | P | and | G : P | .

1D.3 . Le t H C G have the property that H n P 9 = 1 for al l elements

P r o b l e m s I D 27

W h a t is much more difficult is the theorem of Frobenius , which asserts tha t

i f H is a Frobenius complement in G in the sense denned here, then it is,

in fact, a Frobenius complement in the sense of Chapte r 6. Unfortunately ,

al l known proofs of Frobenius theorem require character theory, and so we

cannot present a proo f in this book.

1D.4 . Le t G = N H , where I < N < G and N n H 1 . Show tha t H is a

Frobenius complement in G (as denned above) i f and only i f C N ( h ) = 1 for

al l nonidenti ty elements h H .

Hin t . If x and x n lie in H , where n N , observe tha t x ~ x x n N .

1D.5 . Le t H < G, and suppose tha t N G ( P ) C H for a l l p-subgroups P C

H , for al l primes p . Show that H is a Frobenius complemen t in G .

Hin t . Observe tha t the hypothesis is satisfied by H n H 9 for 5 6 G. If this

intersection is nontr iv ia l , consider a nontr iv ia l Sylow subgroup Q o i H n H 9

and show tha t Q and Q9 " 1 a r e conjugate in H .

1D.6 . Show tha t a subgroup of a ni lpoten t group is max ima l i f and only i f

it has pr ime index .

1D.7 . For any finite group G, the Frat t in i subgroup $ ( G ) is the inter

section of al l max ima l subgroups. Show tha t $ ( G ) is exactly the set of

"useless" elements of G , by which we mean the elements g e G such tha t i f

( X U { g } } = G for some subset X of G , then ( X ) = G.

1D.8 . A finite p-group is elementary abel ian i f it is abelian and every

nonidenti ty element has order p . If G is ni lpotent , show tha t G / $ ( G ) is

abelian, and tha t i f G is a p-group, then G / $ ( G ) is elementary abelian .

Note . It is not hard to see tha t i f P is a p-group, then $(P) is the unique

norma l subgroup of P m in ima l w i th the property tha t the factor group is

elementary abelian .

1D.9 . If P is a noncycl ic p-group, show tha t \ P : $(P) | > p 2 , and deduce

that a group of order p 2 must be either cycl ic or elementary abelian .

ID.10 . Le t A be max ima l among the abel ian norma l subgroups of a p-group

P . Show tha t A = C P { A ) , and deduce tha t \ P : A \ divides ( \ A \ - 1)!.

Hin t . Le t C = C P { A ) . If C > A , apply L e m m a 1.23.

28 1 . S y l o w T h e o r y

1D.11 . Le t n be the max imum of the orders of the abelian subgroups of a

finite group G . Show tha t | G | divides n \ .

Hin t . Show tha t for each prime p , the order of a Sylow p-subgroup of G

divides n \ .

Note . There exist infinite groups in which the abelian subgroups have

bounded order, so finiteness is essential here.

ID.12 . Le t p be a pr ime d iv id ing the order of a group G. Show tha t the

number of elements of order p in G is congruent to - 1 modulo p .

ID.13 . If Z C Z ( G ) and G / Z is ni lpotent , show tha t G is ni lpotent .

ID.14 . Show tha t the F ra t t in i subgroup $ ( G ) of a finite group G is ni lpo

tent.

Hin t . A p p l y the F ra t t in i argument. The proo f here is somewhat s imi lar to

the proo f tha t (3) implies (4) in Theorem 1.26.

Note . Th is problem shows that $ ( G ) C F ( G ) .

ID.15 . Suppose tha t $ ( G ) C N < G and that N / $ ( G ) is ni lpotent . Show

tha t N is ni lpotent . In part icular , i f G / $ ( G ) is ni lpotent , then G is ni lpo

tent.

Note . Th is generalizes the previous problem , which follows by setting N =

$ ( G ) . Note tha t this problem proves tha t F ( G / $ ( G ) ) = F ( G ) / $ ( G ) .

ID.16 . Le t N < G, where G is finite. Show tha t $(JV) C $ ( G ) .

Hin t . If some max ima l subgroup M of G fails to contain $ ( jV ) , then

$ ( W ) A f = G , and it follows that N = < P ( N ) ( N n M ) .

1D.17 . Le t N < G, where AT is ni lpoten t and G/AT is ni lpotent . Prove tha t

G is ni lpotent .

Hin t . The derived subgroup N is contained in $(JV) by P rob lem 1D.8 .

Note . If we weakened the assumption that G/N is ni lpoten t and assumed

instead that G / N is ni lpotent , i t would not follow tha t G is necessarily

nilpotent .

1D.18. Show that F ( G / Z ( G ) ) = F ( G ) / Z ( G ) for a l l finite groups G .

ID.19 . Le t F = F ( G ) , where G is an arbi trary finite group, and let C =

C G { F ) . Show tha t G / ( G D F ) has no nontr iv ia l abel ian norma l subgroup.

Hin t . Observe tha t F ( G ) < G .

I E 29

I E

From the earliest days of group theory, researchers have been intr igued by

the question: wha t are the finite simple groups? O f course, the abelian

simple groups are exactly the groups of pr ime order, and (up to isomorphism)

there is jus t one group of order p for each pr ime p : the cycl ic group of

tha t order. Bu t n o n a b e l i a n simple groups are comparat ively rare. There

are, for example, only five numbers less than 1,000 tha t occur as orders of

nonabel ian simple groups, and up to isomorphism , there is jus t one simple

group of each of these orders. (These numbers are 60, 168, 360, 504 and 660.)

There do exist numbers, however, such tha t there are two nonisomorphic

simple groups of tha t order. (The smallest such number is 8!/2 = 20,160.)

B u t no number is the order of three nonisomorphic simple groups.

Perhaps their rar i ty is one reason tha t nonabel ian finite simple groups

have inspired such intense interest over the years. It seems quite natura l to

collect rare objects and to attemp t to acquire a complete col lection . B u t a

more "pract ical" explanat ion is tha t a knowledge of al l finite simple groups

and their properties would be a major step in understanding a l l finite groups.

The reason for this is that, in some sense, al l finite groups are bui l t from

simple groups.

To be more precise, suppose tha t G is any nontr iv ia l finite group. B y

finiteness, G has at least one max ima l norma l subgroup N . (We mean,

of course, a max ima l p r o p e r norma l subgroup, bu t as is customary in this

context, we have not made.the word "proper" expl ici t .) Then by the corre

spondence theorem, the group G / N is simple . (This is because the norma l

subgroups of G / N are in natura l correspondence w i th the norma l subgroups

of G that contain N , and there are jus t two of these: N and G.) Now i f N

is nontr iv ia l , we can repeat the process by choosing a max ima l norma l sub

group M of N . (In general, of course, M w i l l not be norma l in G.) Because

G is finite, we see tha t i f we continue like this , repeatedly choosing a max

ima l norma l subgroup of the previously selected group , we must eventually

reach the identi ty subgroup. If we number our subgroups from the bo t tom

up, we see that we have constructed (or more accurately "chosen") a chain

of subgroups N such that

1 = jV 0 < N x < < N r = G

such tha t each of the factors A ^ - i is simple for 1 < i < r . In this sit

uation , the subgroups iV? are said to form a composit ion series for G,

and the simple groups N / N - x are the corresponding composit ion fac

tors. The Jordan-Holder theorem asserts tha t despite the arbitrariness of

the construct ion of the composit ion series, the set of composi t ion factors

( including mult ipl ic i t ies) is uniquely determined up to isomorphism . The

30 1 . S y l o w T h e o r y

composit ion factors of G are the simple groups from which we migh t say

tha t G is constructed . (We mention tha t the finite groups for which al l

composit ion factors are cycl ic of prime order are exactly the "solvable" f i

nite groups, which we study in more detai l in Chapter 3.)

The problem of f inding al l nonabehan finite simple groups can be ap

proached from two directions: construct as many simple groups as you can,

and prove tha t every finite nonabelian simple group appears in your l ist.

In part icular , as part of the second of these programs, it is useful to prove

"nonsimpl ic i ty" theorems that show tha t groups that look different from the

known simple groups cannot, in fact, be simple. A major result of this type

from early in the 20th century is due to W . Burnside , who showed tha t the

order of a nonabel ian simple group must have at least three different prime

divisors. (This is Burnsides classic pY-theorem.)

Burnside also observed tha t al l of the then known nonabel ian simple

groups had even order. He conjectured that this holds in general: tha t al l

odd-order simple groups are cycl ic of prime order. Burnsides conjecture,

which can be paraphrased as the assertion tha t every group of odd order

is solvable, was eventually proved by W . Feit and J . G . Thompson in the

early 1960s. (The celebrated Fei t -Thompson paper, at about 250 pages,

may have been the longest publ ished proo f of a single theorem at the t ime.)

Since around 1960, there has been dramatic progress w i th bo th aspects of

the simple-group-classification problem , and it appears tha t now, in the

early years of the 21st century, the classification of finite simple groups is

complete.

So wha t are the nonabehan finite simple groups? A h ighly abbreviated

descript ion is this . Every finite nonabelian group is either:

(1) One of the al ternating groups An f o r n > 5,

(2) A member of one of a number of infinite families parameterized by

prime-powers q and (usually) by integers n > 2, or

(3) One of 26 other "sporadic" simple groups tha t do not fit into types

(1) or (2).

O f the simple groups in parameterized families, the easiest to describe

are the projective specia l linear groups P S L ( n , q ) , where q is a prime-

power and n > 2. These are constructed as follows. Le t F be the field of

order q and construct the genera l linear group G L ( n , q ) consisting of al l

invertible n x n matrices over F . The specia l linear group S L ( n , q ) is the

norma l subgroup of G L { n , q ) consisting of those matrices w i th determinant

1. It is not hard to see that the center Z = Z ( S L ( n , q ) ) consists exactly

of the scalar matrices of determinant 1, and by definit ion, P S L ( n , q ) is

the factor group S L ( n , q ) / Z . It turns out that P S L ( n , q ) is simple except

I E 31

when n = 2 and q is 2 or 3. (We say more abou t the groups P S L ( n , q ) in

Chapte r 7, and we prove their s impl ic i ty in Chapte r 8, where we also prove

tha t the al ternating groups An are simple for n > 5.)

In fact, a l l of the simple groups w i th order less than 1,000 are of the

form P S L ( 2 , q ) , where q is one of 5, 7, 8, 9 or 11, and the corresponding

group orders are 60, 168, 504, 360 and 660. The unique (up to isomor

phism) simple group P S L { 2 , 5) of order 60 has two other realizations: it is

isomorphic to P S L ( 2 , 4 ) and also to the al ternat ing group A 5 . The simple

groups P S L ( 2 , 7) of order 168 and P S L ( 2 , 9) of order 360 also have mult ip le

realizations: the first of these is isomorphic to P S L ( 3 , 2) and the second is

isomorphic to A 6 .

The smallest of the 26 sporadic simple groups is the smal l Ma th ieu group ,

denoted M n , of order 7,920; the largest is the Fischer-Griess "monster" of

order

24 6-32 0-59-76- l l 2-17-19-23-29-31-41.47-59-71 =

808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000.

Fo r the remainder of this section, we discuss nonsimpl ic i ty theorems of

the form: " i f the order