Finite Group Theory-Martin Isaac

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 2 0 B 1 5 , 2 0 B 2 0 , 2 0 D 0 6 , 2 0 D 1 0 , 2 0 D 1 5 , 20D20, 20D25, 20D35, 20D45, 20E22, 20E36.

For

additional information a n d updates o n this book, visit www.ams.org/bookpages/gsm-92

Library of Congress Cataloging-in-Publication Data Isaacs, I. Martin, 1940Finite 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'.23—dc22

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

To

Deborah

Contents

Preface

ix

C h a p t e r 1.

Sylow T h e o r y

1

C h a p t e r 2.

Subnormality

45

C h a p t e r 3.

Split Extensions

65

C h a p t e r 4.

Commutators

113

C h a p t e r 5.

Transfer

147

C h a p t e r 6.

Frobenius A c t i o n s

177

C h a p t e r 7.

T h e T h o m p s o n Subgroup

201

C h a p t e r 8.

Permutation Groups

223

C h a p t e r 9.

M o r e on S u b n o r m a l i t y

271

C h a p t e r 10. M o r e Transfer T h e o r y Appendix: Index

T h e Basics

295 325 345

Preface

T h i s b o o k is a somewhat expanded version of a graduate course i n finite group theory that I often teach at the U n i v e r s i t y of W i s c o n s i n . I offer this course i n order to share what I consider to be a beautiful subject w i t h as m a n y people as possible, and also to provide the solid b a c k g r o u n d i n pure group theory t h a t m y d o c t o r a l students need to carry out their thesis work i n representation theory. T h e focus of group theory research has changed profoundly i n recent decades. S t a r t i n g near the beginning of the 20th century w i t h the work of W . B u r n s i d e , the major p r o b l e m was to find a n d classify the finite simple groups, and indeed, m a n y of the most significant results i n pure group theory and i n representation theory were directly, or at least peripherally, related to this goal. T h e simple-group classification now appears to be complete, and current research has shifted to other aspects of finite group theory i n c l u d i n g p e r m u t a t i o n groups, p-groups and especially, representation theory. It is certainly no less essential i n this post-classification p e r i o d that group-theory researchers, whatever their subspecialty, s h o u l d have a mastery of the classical techniques and results, and so w i t h o u t a t t e m p t i n g to be encyclopedic, I have included m u c h of t h a t m a t e r i a l here. B u t m y choice of topics was largely determined by m y p r i m a r y goal i n w r i t i n g this book, w h i c h was to convey to readers m y feeling for the beauty and elegance of finite group theory. G i v e n its origin, this b o o k should certainly be suitable as a text for a graduate course like mine. B u t I have t r i e d to write it so that readers w o u l d also be comfortable using it for independent study, a n d for t h a t reason, I have t r i e d to preserve some of the informal flavor of m y classroom. I have tried to keep the proofs as short and clean as possible, b u t w i t h o u t o m i t t i n g

ix

X

Preface

details, and indeed, i n some of the more difficult m a t e r i a l , m y arguments are simpler t h a n can be found i n p r i n t elsewhere. F i n a l l y , since I firmly believe t h a t one cannot learn mathematics w i t h o u t doing it, I have included a large number of problems, m a n y of w h i c h are far from routine. Some of the m a t e r i a l here has rarely, if ever, appeared previously i n books. Just i n the first few chapters, for example, we offer Zenkov's marvelous theorem about intersections of abelian subgroups, W i e l a n d t ' s "zipper l e m m a " i n s u b n o r m a l i t y theory and a proof of Horosevskii's theorem that the order of a group a u t o m o r p h i s m can never exceed the order of the group. L a t e r chapters include m a n y more advanced topics that are h a r d or impossible to find elsewhere. M o s t of the students who attend m y group-theory course are second-year graduate students, w i t h a substantial m i n o r i t y of first-year students, and an occasional well-prepared undergraduate. A l m o s t all 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 b o o k w i l l have a similar background, and so I have decided not to begin at the beginning. M o s t of m y readers (like m y students) w i l l have previously seen basic group theory, so I wanted to avoid repeating that m a t e r i a l and to start w i t h something more exciting: Sylow theory. B u t I recognize that m y audience is not homogeneous, and some readers w i l l have gaps i n their preparation, so I have included an a p p e n d i x that contains most of the assumed m a t e r i a l in a fairly condensed form. O n the other hand, I expect that m a n y i n m y audience w i l l already know the Sylow theorems, but I a m confident that even these well-prepared readers w i l l find m a t e r i a l that is new to t h e m w i t h i n the first few sections. M y semester-long graduate course at W i s c o n s i n covers most of the first seven chapters of this book, starting w i t h the Sylow theorems and culm i n a t i n g w i t h a purely group-theoretic proof of B u r n s i d e ' s famous p q theorem. Some of the topics along the way are s u b n o r m a l i t y theory, the Schur-Zassenhaus theorem, transfer theory, coprime group actions, Frobenius groups, and the n o r m a l p-complement theorems of Frobenius and of T h o m p s o n . T h e last three chapters cover material for w h i c h I never have time i n class. C h a p t e r 8 includes a proof of the s i m p l i c i t y of the groups P S L ( n , q ) , and also some graph-theoretic techniques for s t u d y i n g subdegrees of p r i m i t i v e and n o n p r i m i t i v e p e r m u t a t i o n groups. S u b n o r m a l i t y theory is revisited i n C h a p t e r 9, w h i c h includes W i e l a n d t ' s beautiful a u t o m o r p h i s m tower theorem and the T h o m p s o n - W i e l a n d t theorem related to the Sims a

b

Preface

xi

conjecture. F i n a l l y , C h a p t e r 10 presents some advanced topics i n transfer theory, i n c l u d i n g Y o s h i d a ' s theorem a n d the so-called "principal ideal theorem". F i n a l l y , I t h a n k m y m a n y students and colleagues who have contributed ideas, suggestions and corrections while this b o o k was being w r i t t e n . I n particular, I m e n t i o n that the comments of Y a k o v B e r k o v i c h and G a b r i e l N a v a r r o were invaluable and very m u c h appreciated.

Chapter 1

Sylow Theory

1A It seems appropriate to begin this b o o k w i t h a t o p i c t h a t underlies v i r t u a l l y all of finite group theory: the Sylow theorems. I n this chapter, we state and prove these theorems, a n d we present some applications a n d related results. A l t h o u g h m u c h of this m a t e r i a l should be very familiar, we suspect t h a t most readers w i l l find t h a t at least some of the content of this chapter is new to t h e m . A l t h o u g h the theorem t h a t proves Sylow subgroups always exist dates back to 1872, the existence proof t h a t we have decided to present is t h a t of H . W i e l a n d t , published i n 1959. W i e l a n d t ' s proof is slick a n d short, but it does have some drawbacks. It is based o n a t r i c k t h a t seems to have no other a p p l i c a t i o n , a n d the proof is not really constructive; it gives no guidance about how, i n practice, one might a c t u a l l y find a Sylow subgroup. B u t W i e l a n d t ' s proof is beautiful, a n d t h a t is the p r i n c i p a l m o t i v a t i o n for presenting it here. A l s o , W i e l a n d t ' s proof gives us an excuse to present a quick review of the theory of group actions, w h i c h are nearly as u b i q u i t o u s i n the study of finite groups as are the Sylow theorems themselves. W e devote the rest of this section to the relevant definitions a n d basic facts about actions, a l t h o u g h we o m i t some details from the proofs. Let G be a group, and let ft be a n o n e m p t y set. (We w i l l often refer to the elements of ft as "points".) Suppose we have a rule t h a t determines a new element of ft, denoted a-g, whenever we are given a point a e ft and a n element g e G. W e say t h a t this rule defines a n a c t i o n of G on ft i f the following two conditions hold.

1

2

1. S y l o w

Theory

(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 t h a t G acts on ft. It is easy to see t h a t i f g G G is arbitrary, then the function a : ft -> ft defined by - a-$ has an inverse: the function C T _ I . Therefore, a is a p e r m u t a t i o n of the set ft, w h i c h means t h a t o is b o t h injective and surjective, and thus a lies i n the s y m m e t r i c group Sym(ft) consisting of a l l permutations of ft. In fact, the m a p g •-> a is easily seen to be a h o m o m o r p h i s m from G into Sym(ft). ( A h o m o m o r p h i s m like this, w h i c h arises from an action of a group G on some set, is called a p e r m u t a t i o n representation of G.) T h e kernel of this h o m o m o r p h i s m is, of course, a n o r m a l subgroup of G, w h i c h is referred to as the kernel of the action. T h e kernel is exactly the set of elements g e G t h a t act t r i v i a l l y on ft, w h i c h means t h a t a - g = a for a l l points a e ft. g

5

g

g

g

g

Generally, we consider a theorem or a technique t h a t has the power to find a n o r m a l subgroup of G to be "good", and indeed p e r m u t a t i o n representations can be good i n this sense. (See the problems at the end of this section.) B u t our goal i n i n t r o d u c i n g group actions here is not to find n o r m a l subgroups; it is to count things. Before we proceed i n that direction, however, it seems appropriate to mention a few examples. Let G be arbitrary, and take ft = G. W e can let G act on G by right m u l t i p l i c a t i o n , so t h a t x - g = x g for x , g G G. T h i s is the regular action of G, a n d it should be clear t h a t it is faithful, w h i c h means t h a t its kernel is t r i v i a l . It follows t h a t the corresponding p e r m u t a t i o n representation of G is an i s o m o r p h i s m of G into S y m ( G ) , and this proves C a y l e y ' s theorem: every group is isomorphic to a group of permutations on some set. l

We continue to take ft = G, but this time, we define x - g = g ~ x g . (The standard n o t a t i o n for g ~ x g is x».) It is t r i v i a l to check t h a t x = x and that { x 9 ) = 9 for a l l x , g , h G G, and thus we t r u l y have a n action, w h i c h is called the conjugation action of G on itself. N o t e t h a t x = x i f and o n l y i f x g = gx, and thus the kernel of the conjugation action is the set of elements g e G t h a t commute w i t h a l l elements x e G. T h e kernel, therefore, is the center Z ( G ) . x

h

l

h

X

9

A g a i n let G be arbitrary. I n each of the previous examples, we took ft = G , but we also get interesting actions i f instead we take ft to be the set of a l l subsets of G. I n the conjugation action of G on ft we let X - g = X = { x \ x e X } and i n the r i g h t - m u l t i p l i c a t i o n action we define X - g = X g = { x g | x 6 X } . O f course, i n order to make these examples work, we do not really need ft to be a l l subsets of G. F o r example, since a conjugate of a subgroup is always a subgroup, the conjugation action is well defined if we take ft to be the set of a l l subgroups of G. A l s o , b o t h right m u l t i p l i c a t i o n 9

9

1A

3

a n d conjugation preserve cardinality, and so each of these actions makes sense i f we take ft to be the collection of a l l subsets of G of some fixed size. In fact, as we shall see, the t r i c k i n W i e l a n d t ' s proof of the Sylow existence theorem is to use the right m u l t i p l i c a t i o n action of G on its set of subsets w i t h a certain fixed cardinality. We mention one other example, w h i c h is a special case of the rightm u l t i p l i c a t i o n action on subsets that we discussed i n the previous paragraph. Let H C G be a subgroup, and let ft = { H x \ x G G } , the set of right cosets of H i n G. If X is any right coset of H , it is easy to see that X g is also a right coset of H . (Indeed, if X = H x , then X g = H { x g ) . ) T h e n G acts on the set ft by right m u l t i p l i c a t i o n . In general, i f a group G acts on some set ft a n d a G ft, we write G = { g G G | a - g = a } . It is easy to check t h a t G is a subgroup of G ; it is called the stabilizer of the point a . For example, i n the regular action of G on itself, the stabilizer of every point (element of G ) is the t r i v i a l subgroup. In the conjugation action of G on G, the stabilizer of x G G is the centralizer C ( x ) and i n the conjugation a c t i o n of G on subsets, the stabilizer of a subset X is the normalizer N ( X ) . A useful general fact about point stabilizers is the following, w h i c h is easy to prove. In any action, if a - g = /?, then the stabilizers G a n d G p are conjugate i n G, and i n fact, { G ) 3 = G. a

a

G

G

a

a

p

Now consider the action (by right m u l t i p l i c a t i o n ) of G on the right cosets of H , where H C G is a subgroup. T h e stabilizer 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 t h a t g satisfies this c o n d i t i o n if and only if x g G H x . ( T h i s is because two cosets H u and H v are identical if and only if u G H v . ) It follows t h a t g stabilizes H x if and only if g G x ~ H x . Since x ~ H x = H , we see that the stabilizer of the point (coset) H x is exactly the subgroup H , conjugate to H v i a x . It follows t h a t the kernel of the action of G on the right cosets of H i n l

l

x

x

G is exactly

f| H

x

. T h i s subgroup is called the core of H i n G, denoted

xeG

c o r e ( H ) . T h e core of H is n o r m a l i n G because i t is the kernel of an action, and, clearly, it is contained i n H . In fact, if AT < G is any n o r m a l subgroup that happens to be contained i n H , then N = N C H for a l l x G G , and thus N C c o r e ( t f ) . In other words, the core of H i n G is the unique largest n o r m a l subgroup of G contained i n H . (It is "largest" i n the strong sense t h a t it contains a l l others.) G

x

x

G

We have digressed from our goal, w h i c h is to actions to count things. B u t having come this far, results that our discussion has essentially proved. theorem and its corollaries can be used to prove subgroups, and so they might be considered to be

show how to use group we m a y as well state the N o t e t h a t the following the existence of n o r m a l "good" results.

4

1. S y l o w

1.1. T h e o r e m . L e t H C G be o f H i n G. T h e n G / c o v e { H ) p a r t i c u l a r , if t h e i n d e x \ G : H s u b g r o u p of S , t h e s y m m e t r i c G

n

Theory

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 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(ft). I n \ = n , then G / c o r e ( H ) is isomorphic to a group o nn symbols. G

P r o o f . T h e action of G on the set ft by right m u l t i p l i c a t i o n defines a h o m o m o r p h i s m 6 (the p e r m u t a t i o n representation) from G into Sym(ft). Since ker(0) = c o r e ( t f ) , it follows by the h o m o m o r p h i s m theorem t h a t G / c o r e { H ) ^ 0 ( G ) , w h i c h is a subgroup of S y m ( G ) . T h e last statement follows since if \ G : H \ = n , then (by definition of the index) |ft| = n , and thus Sym(ft) = S . U G

G

n

1.2. C o r o l l a r y . L e t G be a g r o u p , a n d suppose that H C G is a subgroup 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 that \ G : N \ divides n\. P r o o f . Take N = c o r e ( H ) . T h e n G / N is isomorphic to a subgroup of the s y m m e t r i c group S , and so by Lagrange's theorem, \ G / N \ divides \S \=n\. • G

n

n

1.3. C o r o l l a r y . 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\ divides n l . P r o o f . T h e n o r m a l subgroup N of the previous corollary is contained i n H , a n d hence it is proper i n G because n > 1. Since G is simple, N = 1 , a n d thus \ G \ = \ G / N \ divides n l . U In order to pursue our m a i n goal, w h i c h is counting, we need to discuss the "orbits" of an action. Suppose t h a t G acts on ft, a n d let a G ft. T h e set O = { a - g | g G G } is called the orbit of a under the given action. It is routine to check t h a t i f ¡3 G O , then O p = O , and it follows that distinct orbits are a c t u a l l y disjoint. A l s o , since every point is i n at least one orbit, it follows t h a t the orbits of the action of G on ft p a r t i t i o n ft. I n particular, if ft is finite, we see t h a t |ft| = where i n this s u m , O runs over the a

a

a

full set of G - o r b i t s on ft. We mention some examples of orbits a n d orbit decompositions. F i r s t , i f H C G is a subgroup, we can let H act on G by right m u l t i p l i c a t i o n . It is easy to see t h a t the orbits of this action are exactly the left cosets of H i n G . (We leave to the reader the p r o b l e m of realizing the right cosets of H in G as the orbits of an appropriate action 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 action of G on itself, where the orbits are exactly the conjugacy classes of G . T h e fact

5

1A

t h a t for a finite group, the order \ G \ is the s u m of the sizes of the classes is sometimes called the class equation of G. How 1.4.

b i g is an orbit? T h e key result here is the following.

Theorem

(The F u n d a m e n t a l C o u n t i n g P r i n c i p l e ) . 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 = G 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 : G \. a>

a

P r o o f . W e observe first t h a t i f H x = H y , t h e n a-x = a-y. T o see w h y this is so, observe t h a t we c a n w r i t e y = hx for some element h s H . T h e n a-y

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

where the last equality holds because h G H = G , a

a n d so h stabilizes a .

G i v e n a coset H x G A , the point a-x lies i n O, a n d we know t h a t it is determined by the coset H x , a n d not just b y the p a r t i c u l a r element x . It is therefore permissible to define the function 9 : A O by 9 { H x ) = a-x, and it remains to show t h a t 9 is b o t h injective a n d surjective. The surjectivity is easy, a n d we do t h a t first. If 0 G O, then by the definition of an orbit, we have 0 = a-x for some element x G G. T h e n H x G A satisfies 6 { H x ) = a-x = 0, as required. a-x

To prove that 9 is injective, suppose t h a t 9 { H x ) = 9 { H y ) . = a-y, and hence a = a-1 = { a - x ) - x -

1

1

= (a-y)-x" =

W e have

1

a-iyx' ).

1

T h e n yx' fixes a , and so it lies i n G = H . It follows t h a t y G H x , and thus H y = H x . T h i s proves t h a t 6 is injective, as required. • a

It is easy to check t h a t the bijection 9 of the previous theorem a c t u a l l y defines a " p e r m u t a t i o n i s o m o r p h i s m " between the a c t i o n of G on A and the action of G on the orbit O. Formally, this means t h a t 0 ( X - g ) = 9 ( X ) - g for a l l "points" X i n A and group elements g G G. M o r e informally, this says t h a t the actions of G on A and on O are "essentially the same". Since every action can be thought of as composed of the actions on the i n d i v i d u a l orbits, and each of these actions is p e r m u t a t i o n isomorphic to the rightm u l t i p l i c a t i o n action of G on the right cosets of some subgroup, we see t h a t these actions on cosets are t r u l y fundamental: every group action can be viewed as being composed of actions on right cosets of various subgroups. We close this section w i t h two familiar a n d useful applications of the fundamental counting principle.

1. S y l o w

6

Theory

1.5. C o r o l l a r y . 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 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 ( x ) \ .

be t h e

G

P r o o f . T h e class of x is the orbit of x under the conjugation action of G on itself, and the stabilizer of x i n this action is the centralizer C { x ) . T h u s \ K \ = \ G : C { x ) \ , as required. • G

G

1.6. C o r o l l a r y . 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 { H ) \ . G

P r o o f . T h e conjugates of H form a n orbit under the conjugation action of G on the set of subsets of G. T h e normalizer N { H ) is the stabilizer of H in this action, a n d thus the orbit size is \ G : N ( # ) | , as wanted. • G

G

Problems

1A

1 A . 1 . L e t H be a subgroup of p r i m e index p i n the finite group G, and suppose that no prime smaller t h a n p divides \ G \ . P r o v e that H < G. 1 A . 2 . G i v e n 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 ^ - d o u b l e coset. I n the case where H and K are finite, show that \ H g K \ = \ H \ \ K \ / \ K n H \ . 9

H i n t . Observe t h a t H g K is a u n i o n of right cosets of H , and t h a t these cosets form an orbit under the action of K . N o t e . If we take g = 1 i n this problem, the result is the familiar formula \HK\ = \H\\K\/\HnK\. 1 A.3.

Suppose that G is finite a n d t h a t H , K C G are subgroups. (a) Show that \ H : H n K \ < \ G : K \ , w i t h equality if and o n l y if H K

= G.

(b) If \ G : H \ a n d \ G : K \ are coprime, show t h a t H K = G. N o t e . Proofs of these useful facts appear i n the appendix, but we suggest t h a t readers t r y to find their o w n arguments. A l s o , recall t h a t the product H K of subgroups H and K is not always a subgroup. In fact, H K is a subgroup if a n d o n l y i f H K = K H . (This too is proved i n the appendix.) If H K = K H , we say that H and K are p e r m u t a b l e . 1 A . 4 . Suppose t h a t G = H K , where H and K are subgroups. Show that also G = H K for a l l elements x,y G G. Deduce t h a t if G = H H for a subgroup H a n d a n element x G G, then H = G. x

y

X

Problems 1 A

7

1 A . 5 . A n action of a group G on a set ft is transitive i f ft consists of a single orbit. Equivalently, G is transitive on ft i f for every choice of points a,/? G ft, there exists an element g G G such t h a t a - g = 0. N o w assume t h a t a group G acts t r a n s i t i v e l y on each of two sets ft a n d A . Prove t h a t the n a t u r a l induced action of G on the cartesian p r o d u c t ft x A is transitive if and only i f G G p = G for some choice of a G ft a n d 0 G A . a

H i n t . Show that if G G p = G for some a G ft and 0 G A , then i n fact, this holds for a l l a G ft and 0 G A . a

1 A . 6 . L e t G act on ft, where b o t h G a n d ft are finite. F o r each element g G G, w r i t e ( g ) = \ { a e n \ a - g = a } \ . T h e nonnegative-integer-valued function is called the p e r m u t a t i o n character associated w i t h the action. Show that x

X

J]x(0)

=

J2

\Ga\=n\G\,

where n is the number of orbits of G on ft. N o t e . T h u s the number of orbits is

1

1

geG

w h i c h is the average value of x over the group. A l t h o u g h this orbit-counting formula is often a t t r i b u t e d to W . B u r n s i d e , it s h o u l d (according to P . N e u mann) more p r o p e r l y be credited to C a u c h y a n d Frobenius. 1 A . 7 . L e t G be a finite group, and suppose that H < G is a proper subgroup. Show that the number of elements of G t h a t do not lie i n any conjugate of H is at least \ H \ . H i n t . L e t be the p e r m u t a t i o n character associated w i t h the cation action of G on the right cosets of H . T h e n £ x ( # ) = sum runs over g e G . Show that £ x W > 2 | t f | , where runs over h e H . Use this information to get a n estimate o n elements of G where vanishes. X

right-multipli\ G \ , where the here, the s u m the number of

x

1 A . 8 . L e t G be a finite group, let n > 0 be a n integer, and let C be the additive group of the integers m o d u l o n . L e t ft be the set of n-tuples [x ,x ,...,x ) l

2

n

of elements of G such t h a t x x x

2

•••x

= l.

n

(a) Show t h a t C acts on ft according to the formula ( x i , x , •••, x ) - k = ( x i 2

n

+ k

, x

2 +

k , • ••,

x

n

+

k

) ,

where k G C and the subscripts are interpreted m o d u l o n .

1. S y l o w

8

Theory

(b) N o w suppose t h a t n = p is a prime number t h a t divides \ G \ . Show that divides the number of C - o r b i t s of size 1 on ft, and deduce t h a t the number of elements of order p i n G is congruent to - 1 mod p . V

N o t e . I n p a r t i c u l a r , if a p r i m e p divides | G | , then G has at least one element of order p . T h i s is a theorem of Cauchy, and the proof i n this p r o b l e m is due to J . H . M c K a y . C a u c h y ' s theorem can also be derived as a corollary of Sylow's theorem. A l t e r n a t i v e l y , a proof of Sylow's theorem different from W i e l a n d t ' s can be based on C a u c h y ' s theorem. (See P r o b l e m 1B.4.) 1 A . 9 . Suppose \ G \ = p m , where p > m and p is prime. Show t h a t G has a unique subgroup of order p . 1A.10. Let H C G . (a) Show t h a t \ N ( H ) : H \ is equal to the number of right cosets of H in G t h a t are invariant under right m u l t i p l i c a t i o n by H . G

(b) Suppose t h a t \ H \ is a power of the prime p and t h a t \ G : H \ is divisible by p . Show t h a t \ N { H ) : H \ is divisible b y p . G

IB F i x a prime number p . A finite group whose order is a power of p is called a pgroup. It is often convenient, however, to use this nomenclature somewhat carelessly, and to refer to a group as a "p-group" even if there is no p a r t i c u l a r prime p under consideration. F o r example, i n proving some theorem, one might 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 t h a t the theorem holds for all p-groups for a l l primes p . We m e n t i o n that, a l t h o u g h i n this book a p-group is required to be finite, it is also possible to define infinite p-groups. T h e more general definition is t h a t a (not necessarily finite) group G is a p-group if every element of G has finite p-power order. O f course, i f G is finite, then by Lagrange's theorem, every element of G has order d i v i d i n g \ G \ , and so i f \ G \ is a power of p, it follows t h a t the order of every element is a power of p, a n d hence G is a p-group according to the more general definition. Conversely, if G is finite and has the property t h a 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 C a u c h y ' s theorem ( P r o b l e m 1A.8) t h a 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. A g a i n , 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

IB

9

not divisible by p. A n alternative formulation of this definition relies on the observation t h a t every positive integer can be (uniquely) factored as a power of the given p r i m e p times some integer not divisible by p . In p a r t i c u l a r , if we w r i t e \ G \ = p 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 w h e n \S\ = p . I n other words, a Sylow p-subgroup of G is a p-subgroup S whose order is as large as is p e r m i t t e d by Lagrange's theorem, w h i c h requires that \S\ must divide | G | . W e m e n t i o n two t r i v i a l cases: i f | G | is not divisible by p, then the i d e n t i t y subgroup is a Sylow p-subgroup of G, a n d if G is a p-group, then G is a Sylow p-subgroup of itself. T h e Sylow existence theorem asserts t h a t Sylow subgroups a l w a y s exist. a

a

1.7. T h e o r e m (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 - t h e o r e m can be viewed as a p a r t i a l converse of Lagrange's theorem. Lagrange asserts that i f i f is a subgroup of G a n d \ H \ = k, t h e n k divides \ G \ . T h e converse, w h i c h i n general is false, w o u l d say that i f k is a positive integer t h a 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 of order 12; this group has no subgroup of order 6.) But i f A; is a power of a prime, we shall see t h a t G a c t u a l l y does have a subgroup of order k. If k is the largest power of p t h a 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 t h a t a Sylow p-subgroup of G necessarily has a subgroup of order k. A

We are ready now to begin work t o w a r d the proof of the Sylow E theorem. W e start w i t h a purely a r i t h m e t i c fact about b i n o m i a l coefficients. 1.8. L e m m a . 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 . Then = m

mod p

p

P r o o f . C o n s i d e r the p o l y n o m i a l (1 + X ) . Since p is prime, it is easy to see t h a t the b i n o m i a l coefficients ( ) are divisible by p for 1 < i < p - 1, and thus we can write (1 + X ) = 1 + X m o d p. ( T h e assertion t h a t these p o l y n o m i a l s are congruent m o d u l o p means t h a t the coefficients of corresponding powers of X are congruent m o d u l o p.) A p p l y i n g this fact a second t i m e , we see t h a t { 1 + X ) = (1+X ) = 1+X m o d p. C o n t i n u i n g like this, we deduce t h a t (1 + X ) = 1+ X m o d p, and thus p

p

p

p 2

P

p

am

( l + Xf

P

a

= (1 + X

p

p

a

)

m

p 2

a

mod p .

1. S y l o w

10

Theory

Since these p o l y n o m i a l s are congruent, the coefficients of corresponding terms are congruent m o d u l o p, and the result follows b y considering the coefficient of X ? on each side. • a

a

P r o o f of the Sylow E - t h e o r e m ( W i e l a n d t ) . W r i t e \ G \ = p m , where a > 0 a n d p does not divide m. L e t f t be the set of a l l subsets of G having c a r d i n a l i t y p , and observe t h a t G acts by right m u l t i p l i c a t i o n on f t . Because of this action, f t is p a r t i t i o n e d into orbits, and consequently, \fl\ is the s u m of the orbit sizes. B u t a

and so |i2| is not divisible by p , and it follows that there is some orbit O such t h a t \ 0 \ is not divisible by p . Now let X G O, a n d let H = G be the stabilizer of X i n G. B y the fundamental counting principle, \ 0 \ = \ G \ / \ H \ , a n d since p does not divide \ 0 \ and p divides \ G \ , we conclude that p must d i v i d e \ H \ , and i n particular p < \ H \ . x

a

a

a

Since H stabilizes X under right m u l t i p l i c a t i o n , we see t h a t if x G X , t h e n x H C X , and thus \ H \ = \ x H \ < \X\ = p , where the final equality holds since X G f t . W e now have \ H \ = p , and since H is a subgroup, it is a Sylow subgroup of G, as wanted. • a

a

In P r o b l e m 1 A . 8 , we sketched a proof of C a u c h y ' s theorem. W e c a n now give another proof, using the Sylow E-theorem. 1.9. C o r o l l a r y ( C a u c h y ) . L e t G be a

finite

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

g r o u p , a n d suppose

that p is a

of o r d e r p .

P r o o f . L e t S be a Sylow p-subgroup of G, and note that since \S\ is the m a x i m u m power of p that divides | G | , we have \S\ > 1. Choose a nonidentity element x of S, and observe t h a t the order o { x ) divides \S\ by Lagrange's theorem, a n d thus 1 < o ( x ) is a power of p . I n particular, we can w r i t e o { x ) = pm for some integer m > 1, and we see t h a t o { x ) = p , as wanted. • m

We introduce the n o t a t i o n S y l ( G ) to denote the set of a l l Sylow psubgroups of G. T h e assertion of the Sylow E-theorem, therefore, is t h a t the set Sylp(G) is nonempty for a l l finite groups G a n d a l l primes p . T h e intersection n S y l ( G ) of a l l Sylow p-subgroups of a group G is denoted O ( G ) , a n d as we shall see, this is a subgroup that plays an i m p o r t a n t role in finite group theory. p

p

p

IB

11

Perhaps this is a good place to digress to review some basic facts about characteristic subgroups. (Some of this m a t e r i a l also appears i n the appendix.) F i r s t , we recall the definition: a subgroup K C G is characteristic in G i f every a u t o m o r p h i s m of G maps K onto itself. It is often difficult to find a l l automorphisms of a given group, and so the definition of "characteristic" can be h a r d to apply directly, but nevertheless, in m a n y cases, it easy to establish t h a t certain subgroups are characteristic. For example, the center Z ( G ) , the derived (or commutator) subgroup G', and the intersection of all Sylow p-subgroups O ( G ) are characteristic i n G . M o r e generally, any subgroup that can be described unambiguously as " t h e something" is characteristic. It is essential t h a t the description using the definite article be unambiguous, however. G i v e n a subgroup H Q G , for example, we cannot conclude t h a t the normalizer N ( H ) or the center Z ( H ) is characteristic i n G . A l t h o u g h these subgroups are described using "the", the descriptions are not unambiguous because they depend on the choice of H . W e can say, however, that Z ( G ' ) is characteristic i n G because it is t h e center of t h e derived subgroup; it does not depend on any unspecified subgroups. p

G

A good way to see w h y "the something" subgroups must be characteristic is to imagine two groups G i and G , w i t h an i s o m o r p h i s m 9 : G i -> G . Since isomorphisms preserve "group theoretic" properties, it should be clear that 9 maps the center Z ( G i ) onto Z ( G ) , and indeed 9 maps each unambiguously defined subgroup of G i onto the corresponding subgroup of G . N o w specialize to the case where G i and G h a p p e n to be the same group G , so 9 is an a u t o m o r p h i s m of G . Since i n the general case, we know that 6 » ( Z ( G J ) = Z ( G ) , we see t h a t when G i = G = G , we have 0(Z(G)) = Z ( G ) , and similarly, if we consider any "the something" subgroup i n place of the center. 2

2

2

2

2

2

2

Of course, characteristic subgroups are a u t o m a t i c a l l y n o r m a l . T h i s is because the definition of n o r m a l i t y requires o n l y t h a t the subgroup be m a p p e d onto itself by i n n e r automorphisms while characteristic subgroups are m a p p e d onto themselves by a l l automorphisms. W e have seen that some characteristic subgroups are easily recognized, and it follows that these subgroups are obviously and a u t o m a t i c a l l y n o r m a l . F o r example, the subgroup O (G) is n o r m a l i n G for all primes p . p

The fact that characteristic subgroups are n o r m a l remains true i n an even more general context. T h e following, w h i c h we presume is already k n o w n to most readers of this book, is extremely useful. (This result also appears i n the appendix.) 1.10. L e m m a . 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 .

I . Sylow

12

Theory

P r o o f . L e t g € G. T h e n conjugation by g maps N onto itself, and it follows t h a t the restriction of this conjugation m a p to N is an a u t o m o r p h i s m of N . ( B u t note t h a t it is not necessarily an inner a u t o m o r p h i s m of N . ) Since K is characteristic i n N , it is m a p p e d onto itself by this a u t o m o r p h i s m of N , and thus K = K , and it follows t h a t K < G. U 9

Problems

IB

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

(a) L e t P C G be a p-subgroup. only i f P C S.

Show that PS

is a subgroup if and

(b) If S < G, show that S y l ( G ) = { S } , and deduce t h a t S is characteristic i n G. p

N o t e . O f course, it w o u l d be "cheating" to do problems i n this section using theory t h a t we have not yet developed. I n particular, y o u should avoid using the Sylow C-theorem, w h i c h asserts that every two Sylow p-subgroups of G are conjugate i n G. 1B.2. Show t h a t O ( G ) is the unique largest n o r m a l p-subgroup of G. (This means t h a t it is a n o r m a l p-subgroup of G that contains every other n o r m a l p-subgroup of G.) p

1 B . 3 . L e t S £ S y l p ( G ) , and write N = N ( 5 ) . Show t h a t N = G

N (N). G

1B.4. L e t P C G be a p-subgroup such that \ G : P \ is divisible by p. U s i n g C a u c h y ' s theorem, but w i t h o u t appealing to Sylow's theorem, show that there exists a subgroup Q of G containing P , and such t h a t \ Q : P \ = p . Deduce that a m a x i m a l p-subgroup of G (which obviously must exist) must be a Sylow p-subgroup of G. H i n t . Use P r o b l e m 1A.10 a n d consider the group N ( P ) / P . G

N o t e . Once we k n o w C a u c h y ' s theorem, this p r o b l e m yields an alternative proof of the Sylow E - t h e o r e m . O f course, to avoid circularity, we appeal to P r o b l e m 1 A . 8 for C a u c h y ' s theorem, and not to C o r o l l a r y 1.9. 1B.5. L e t 7T be any set of p r i m e numbers. W e say that a finite group H is a vr-group i f every p r i m e divisor of \ H \ lies i n TT. A l s o , a vr-subgroup H C G is a H a l l 7r-subgroup of G i f no p r i m e d i v i d i n g the index \ G : H \ lies i n TT. (So i f TT = {p}, a H a l l vr-subgroup is exactly a Sylow p-subgroup.) N o w let 9 : G -»• K be a surjective h o m o m o r p h i s m of finite groups. (a) If H is a H a l l vr-subgroup of G, prove that 9 ( H ) is a H a l l vr-subgroup of K .

Problems I B

13

(b) Show that every Sylow p-subgroup of K has the form 9 ( H ) , H is some Sylow p-subgroup of G.

where

(c) Show that | S y l ( G ) | > | S y l ( X ) l for every p r i m e p. p

p

N o t e . If the set vr contains more t h a n one p r i m e number, then a H a l l vrsubgroup can fail to exist. B u t a theorem of P . H a l l , after w h o m these subgroups are named, asserts that i n the case where G is solvable, H a l l TTsubgroups always do exist. (See C h a p t e r 3, Section C . ) W e m e n t i o n also that P a r t (b) of this p r o b l e m would not r e m a i n true if "Sylow p-subgroup" were replaced by " H a l l 7r-subgroup". 1B.6. L e t G be a finite group, and let K C G be a subgroup. Suppose that H C G is a H a l l 7r-subgroup, where TT is some set of primes. Show t h a t i f H K is a subgroup, t h e n H n K is a H a l l 7r-subgroup of K . N o t e . In p a r t i c u l a r , K has a H a l l 7r-subgroup if either H or K is n o r m a l i n G since i n that case, H K is guaranteed to be a subgroup. 1 B . 7 . L e t G be a finite group, a n d let vr be any set of primes. (a) Show that G has a (necessarily unique) n o r m a l 7r-subgroup N such t h a t N D M whenever M < G is a 7r-subgroup. (b) Show that the subgroup N of P a r t (a) is contained i n every H a l l 7r-subgroup of G. (c) A s s u m i n g t h a t G has a H a l l 7r-subgroup, show t h a t N is exactly the intersection of a l l of the H a l l vr-subgroups of G. N o t e . T h e subgroup N of this p r o b l e m is denoted O ( G ) . Because of the uniqueness i n (b), it follows t h a t this subgroup is characteristic i n G . F i n a l l y , we note t h a t i f p is a prime number, then, of course, 0 ( G ) - O ( G ) . n

{ p }

p

1 B . 8 . L e t G be a finite group, and let vr be any set of primes. (a) Show t h a t G has a (necessarily unique) n o r m a l subgroup N such t h a t G / N is a vr-group and M D N whenever M < G a n d G / M is a 7r-group. (b) Show that the subgroup N of P a r t (a) is generated by the set of a l l elements of G that have order not divisible by any p r i m e i n TT. f f

N o t e . T h e characteristic subgroup N of this p r o b l e m is denoted O ( G ) . A l s o , we recall t h a t the subgroup generated by a subset of G is the (unique) smallest subgroup that contains t h a t set.

I. Sylow

14

Theory

1C We are now ready to study i n greater detail the nonempty set S y l ( G ) of Sylow p-subgroups of a finite group G . p

1.11. T h e o r e m . 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 suppose t h a t S € S y l ( G ) . T h e n PCS f o r some element 9

p

finite g r o u p G, a n d g e G .

P r o o f . L e t ft = {Sx \ x e G } , the set of right cosets of S i n G , and note that |ft| = \G:S\ is not divisible by p since -S is a Sylow p-subgroup of G . We k n o w t h a t G acts by right m u l t i p l i c a t i o n on ft, and thus P acts too, and ft is p a r t i t i o n e d into P - o r b i t s . A l s o , since |ft| is not divisible by p, there must exist some P - o r b i t O such that \ 0 \ is not divisible by p . B y the fundamental counting principle, \ 0 \ is the index i n P of some subgroup. It follows t h a t \ 0 \ divides | P | , w h i c h is a power of p. T h e n \ 0 \ is b o t h a power of p and not divisible by p, and so the only possibility is t h a t \ 0 \ = 1. R e c a l l i n g that a l l members of ft are right cosets of S i n G , we can suppose t h a t the unique member of O is the coset Sg. Since Sg is alone i n a P - o r b i t , it follows that it is fixed under the a c t i o n of P , and thus Sgu = Sg for a l l elements u e P . T h e n g u € Sg, and hence g-^Sg = S . T h u s PCS , as required. • 9

u

9

€

If S is a Sylow p-subgroup of G , and g e G is arbitrary, then the conjugate S is a subgroup h a v i n g the same order as S. Since the only requirement on a subgroup that is needed to qualify it for membership i n the set S y l ( G ) is that it have the correct order, and since S € S y l ( G ) and \S \ = \S\, it follows t h a t S also lies i n S y l ( G ) . In fact every member of S y l ( G ) arises this way: as a conjugate of S. T h i s is the essential content of the Sylow conjugacy theorem. P u t t i n g it another way: the conjugation a c t i o n of G on S y l ( G ) is transitive. 9

p

9

p

9

p

p

p

1.12. T h e o r e m (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 G,

then T = S

9

f o r some

element

finite

group

g e G .

P r o o f . A p p l y i n g T h e o r e m 1.11 w i t h T i n place of P , we conclude that T C S for some element g e G . B u t since b o t h S and T are Sylow psubgroups, we have \T\ = \S\ = \S \, and so the containment of the previous sentence must a c t u a l l y be an equality. • 9

9

The Sylow C-theorem yields an alternative proof of P r o b l e m I B . 1(b), w h i c h asserts that i f a group G has a n o r m a l Sylow p-subgroup S, then S is the only Sylow p-subgroup of G . Indeed, by the Sylow C-theorem, if T e S y l ( G ) , then we c a n w r i t e T = S = S, where the second equality is a consequence of the n o r m a l i t y of S. 9

p

15

1C

A frequently used a p p l i c a t i o n of the Sylow C - t h e o r e m is the so-called " F r a t t i n i argument", w h i c h we are about to present. Perhaps the reason that this result is generally referred to as a n "argument" rather t h a n as a "lemma" or "theorem" is that variations on its proof are used nearly as often as its statement. 1.13. L e m m a ( F r a t t i n i A r g u m e n t ) . L e t N < G w h e r e N i s pose t h a t P G Sylp(iV). T h e n G = N { P ) N .

finite,

and

sup-

G

9

9

9

P r o o f . L e t g G G, and note that P C N = N , and thus P is a subgroup of N h a v i n g the same order as the Sylow p-subgroup P . It follows that P G Sylp(iV), and so by the Sylow C-theorem a p p l i e d i n N , we deduce that ( p g y = p , for some element n £ J V . Since P = P , we have g n G N ( P ) , and so g G N ( P ) n " C N ( P ) N . B u t g € G was arbitrary, and we deduce that G = N ( P ) A T , as required. • 9

9

n

G

1

G

G

G

B y definition, a Sylow p-subgroup of a finite group G is a p-subgroup that has the largest possible order consistent w i t h Lagrange's theorem. B y the Sylow E-theorem, we can make a stronger statement: a subgroup whose order is m a x i m a l among the orders of a l l p-subgroups of G is a Sylow psubgroup. A n even stronger assertion of this type is t h a t every m a x i m a l psubgroup of G is a Sylow p-subgroup. Here, " m a x i m a l " is to be interpreted in the sense of containment: a subgroup H of G is m a x i m a l w i t h some property i f there is no subgroup K > H t h a t has the property. T h e t r u t h of this assertion is the essential content of the Sylow "development" theorem. 1.14. T h e o r e m (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. P

i s c o n t a i n e d i n some

Then

S y l o w p - s u b g r o u p of G. 9

P r o o f . L e t S G S y l ( G ) . T h e n by T h e o r e m 1.11, we k n o w that PCS for some element g G G. A l s o , since \S \ = \S\, we know t h a t S is a Sylow p-subgroup of G. U p

9

9

G i v e n a finite group G, we consider next the question of how m a n y Sylow p-subgroups G has. T o facilitate this discussion, we introduce the (not quite standard) n o t a t i o n n { G ) = | S y l ( G ) | . (Occasionally, when the group we are considering is clear from the context, we w i l l s i m p l y write n instead of n {G).) p

p

p

p

F i r s t , by the Sylow C-theorem, we k n o w t h a t S y l ( G ) is a single orbit under the conjugation action of G. T h e following is then an immediate consequence. p

1.15. C o r o l l a r y . L e t S G S y l ( G ) , w h e r e G i s a finite g r o u p . p

|G:N (5)|. G

Thenn (G) p

-

16

1. S y l o w

Theory

P r o o f . Since n ( G ) = | S y l ( G ) | is the t o t a l number of conjugates of S i n G, the result follows by C o r o l l a r y 1.6. • p

p

In particular, it follows that n ( G ) divides \ G \ , but we can say a bit more. If 5 € S y l ( G ) , then of course, S C N ( S ) since S is a subgroup, and thus | G : S\ = \ G : N ( 5 ) | | N ( 5 ) : S\. A l s o , n { G ) = \ G : N ( 5 ) | , and hence n ( G ) divides \ G : S\. I n other words, if we write \ G \ = p m , where p does not divide m , we see that n { G ) divides m. (We mention that the integer m is often referred to as the p'-part of \ G \ . ) p

p

G

G

G

p

G

a

p

p

T h e information that n { G ) divides the p'-part of \ G \ becomes even more useful w h e n it is combined w i t h the fact (probably k n o w n to most readers) t h a t n ( G ) = 1 m o d p for a l l groups G. In fact, there is a useful stronger congruence constraint, w h i c h may not be quite so well k n o w n . Before we present our theorem, we m e n t i o n that i f S,T € S y l ( G ) , then |5| = | T | , and thus \S : SnT\ = \S\/\SnT\ = | T | / | 5 n r | = \ T : S n T \ . T h e statement of the following result, therefore, is not really as asymmetric as it may appear. p

p

p

1.16. T h e o r e m . Suppose t h a t G i s a finite g r o u p such t h a t n { 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 n { G ) = 1 m o d \S : S n T | . p

p

1.17. C o r o l l a r y . If G i s a mod p.

finite

group and p is a prime, then n ( G ) = 1 p

P r o o f . If n ( G ) = 1, there is n o t h i n g to prove. Otherwise, T h e o r e m 1.16 applies, a n d there exist distinct members S,T e S y l ( G ) such that n ( G ) = 1 m o d \S :SnT\, and thus it suffices to show t h a 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 , w h i c h is not the case because S and T are distinct. • p

p

p

In order to see how T h e o r e m 1.16 can be used, consider a group G of order 21,952 = 2 - 7 . W e know that n must divide 2 = 64, and it must be congruent to 1 m o d u l o 7. W e see, therefore, that n must be one of 1, 8 or 64. Suppose that G does not have a n o r m a l Sylow 7-subgroup, so that n > 1. Since neither 8 nor 64 is congruent to 1 m o d u l o 7 = 49, we see by T h e o r e m 1.16 that there exist distinct Sylow 7-subgroups S and T of G such that \S : SnT\ = 7. 6

3

6

7

7

2

7

L e t ' s pursue this a b i t further. W r i t e D = S ( I T i n the above situation, and note t h a t since |5 : D \ = 7 is the smallest prime divisor of \S\ = 7 , it follows by P r o b l e m 1 A . 1 , t h a t D < S. S i m i l a r reasoning shows t h a t also D < T , a n d hence S and T are b o t h contained i n N = N ( L > ) . N o w S and T are distinct Sylow 7-subgroups of N , and it follows t h a t n ( N ) > 1, a n d hence n ( N ) > 8 by C o r o l l a r y 1.17. Since n ( N ) is a power of 2 that 3

G

7

7

7

1C

17

divides |AT|, we deduce t h a t 2 have \ G : N \ < 8.

3

divides \ N \ .

Since also 7

3

divides \ N \ , we

We can use what we have established to show that a group G of order 21,952 cannot be simple. Indeed, if n ( G ) = 1, then G has a n o r m a l subgroup of order 7 , and so is not simple. Otherwise, our subgroup N has index at most 8, and we see t h a t | G | does not d i v i d e \ G : N \ \ . B y the n \ theorem ( C o r o l l a r y 1.3), therefore, G cannot be simple i f N < G. F i n a l l y , \{ N = G then D < G and G is not simple i n this case too. 7

3

In the last case, where D < G, we see t h a t D is contained i n a l l Sylow 7-subgroups of G, and thus D is the intersection of every two distinct Sylow 7-subgroups of G. I n most situations, however, T h e o r e m 1.16 can be used to prove only the existence of some pair of distinct Sylow subgroups w i t h a "large" intersection; it does not usually follow t h a t every such pair has a large intersection. To prove T h e o r e m 1.16, we need the following. 1.18. L e m m a . L e t P € S y l ( G ) , w h e r e G i s a that Q is a p-subgroup o/N (P). Then Q C P . p

finite

group, and

suppose

G

P r o o f . W e a p p l y Sylow theory i n the group N = N ( 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 - t h e o r e m , however, the p-subgroup Q of N must be contained i n some Sylow p-subgroup. T h e only possibility is Q C P , as required. • G

A n alternative m e t h o d of proof for L e m m a 1.18 is to observe that since Q C ^ N ( P ) , it follows that Q P = P Q . T h e n Q P is a subgroup, and it is easy to see that it is a p-subgroup t h a t contains the Sylow p-subgroup P . It follows t h a t P = Q P 5 Q, as wanted. G

P r o o f of T h e o r e m 1.16. L e t S act on the set S y l ( G ) by conjugation. One orbit is the set { S } , of size 1, a n d so i f we can show t h a t a l l other orbits have size divisible by \S : S D T\, it w i l l follow t h a t n ( G ) = | S y l ( G ) | = 1 mod \S : S D T\, as wanted. L e t O be an a r b i t r a r y S - o r b i t i n S y l ( G ) other t h a n { S } and let P G O, so that P ^ S. B y the fundamental counting p r i n ciple, \ 0 \ = \S : Q\, where Q is the stabilizer of P i n S under conjugation. T h e n Q C N ( 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 inequality is a consequence of the fact that |5 D T\ is as large as possible a m o n g intersections of two distinct Sylow p-subgroups of G . It follows that \ 0 \ = \S : Q\ > \S : S D T\. B u 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 m u l t i p l e of \S : S n T\. T h i s completes the proof. • p

p

p

p

G

18

1. S y l o w

Problems

Theory

IC

1 C . 1 . L e t P e S y l ( G ) , and suppose that N ( P ) C H C G , where H is a subgroup. P r o v e that H = N { H ) . p

G

G

N o t e . T h i s generalizes P r o b l e m 1 B . 3 . 1 C . 2 . L e t H C G , where G is a finite group. (a) If P € S y l ( P " ) , prove that P = H n 5 for some member 5 G Syl (G). p

p

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

p

1 C . 3 . L e t G be a finite group, and let X be the subset of G consisting of all elements whose order is a power of p , where p is some fixed prime. (a) Show that X = ( J S y l ( G ) . p

3

1 C . 4 . L e t \ G \ = 120 = 2 -3-5. Show that G has a subgroup of index 3 or a subgroup of index 5 (or b o t h ) . H i n t . A n a l y z e separately the four possibilities for n ( G ) . 2

1 C . 5 . L e t P e S y l p ( G ) , where G = A the alternating group on p + 1 symbols. Show that | N ( P ) | = p(p - l ) / 2 . p

+

U

G

H i n t . C o u n t the elements of order p i n G . 1 C . 6 . L e t G = PTif, where P" and K are subgroups, and fix a prime p. (a) Show that there exists P € S y l ( G ) such that P n H € S y l ( P ) and P H f i £ S y l ( X ) . p

p

p

(b) If P is as i n (a), show that P - ( P n H ) ( P n AT). H i n t . F o r (a), first choose Q € S y l ( G ) and g £ G such that Q n H S y l ( P ) and Q n f i e S y l ( X ) . W r i t e 0 = Ziifc, w i t h h e H and k