说明成果03代数代数学finite group theory i martin isaacs ams

1、Finite Group TheoryI. Martin IsaacsFinite Group TheoryI. Martin IsaacsGraduate Studies in MathematicsVolume 92American MathematicalSocietyProvidence, Rhode IslandEditorial BoardDavid Cox (Chair) Steven G. Krantz Rafe MazzeoMartin Scharlemann2000 M a t h e m a t i c s Subjec t Classification . Primar

2、 y 20B15 , 20B20 , 20D06 , 20D10 , 20D20 , 20D25 , 20D35 , 20D45 , 20E22 , 20E36 .20D15 ,Fo r additiona l informatio n an d updates o n this book, visi tLibrary of Congress Cataloging-in-Publication Data Isaacs, I. Martin, 1940-Finite group theory / I. Martin Isaacs.p. cm. (Graduate studies in mathe

3、matics ; v. 92) Includes index.ISBN 978-0-8218-4344-4 (alk. paper)1. Finite groups. 2. Group theory. I. Title.QA177.I835 2008 512'.23dc222008011388Copying and reprinting. Individualers of this publication, and nonprofit librariesacting for them, are permitted to make fair use of the material, su

4、ch as to copy a chapter for usein 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 perm

5、itted 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 to .by© 20

6、08 by the American Mathematical Society.The American Mathematical Society retains.sexcept those granted to the United States Government.Printed in the United States of America.© The paper used in this book is acid-and falls within the guidelinesestablished to ensure permanence and durability.Vi

7、sit the AMS home page at10 9 8 7 6 5 4 3 2 113 12 11 10 09 08T oD e b o r a hContentsPrefaceixChapter1.Sylow Theory1Chapter2.Subnormality45Chapter3.Split Extensions65Chapter4.Commutators113Chapter5.Transfer147Chapter6.Frobeniu ions177Chapter7.The Thompson Subgroup201Chapter8.Permutation Groups223Cha

8、pter9.More on Subnormality271Chapter10.More TransferTheory295The Basics325Appendix:Index345PrefaceThis book is a somewhat expanded version of a graduate course in finite group theory that I often teach at the University of Wisconsin. I offer this course in order to share what I consider to be a beau

9、tiful subject wit h asmany people as possible, and also to provide the solid background in puregroup theory that my doctorali n representation theory.s need to carry out their thesis workThe focus of group theory research has changed profoundly in recent decades. Starting near the beginning of the 2

10、0th century with the work ofW . Burnside, the major problem was to find and classify the finite simplegroups, and, many of the most significant results i n pure group theoryand i n representation theory were directly, or at least peripherally, related to this goal. The simple-group classification no

11、w appears to be complete, and current research has shifted to other aspects of finite group theory including permutation groups, p-groups and especially, representation theory.It is certainly no less essential in this post-classification period that group-theory researchers, whatever their subspecia

12、lty, should have a mas- tery of the classical techniques and results, and so without attempting to be encyclopedic, I have included much of that material here. Bu t my choice of topics was largely determined by my primary goal i n writing this book,which was to convey tofinite group theory.ers my fe

13、eling for the beauty and elegance ofGive n its origin, this book should certainly be suitable as a text for agraduate course like mine. Bu t I have tried to write it so thaters wouldalso be comfortable using it for independent study, and for that reason, Ihave tried to preserve some of the informal

14、flavor of my classroom. I have tried to keep the proofs as short and clean as possible, but without omittingixPrefac eXs, and, i n some of the more difficult material, my arguments aresimpler than can be found in print elsewhere. Finally, since I firmly believe that one cannot learn mathematics with

15、out doing it, I have included a largenumber of problems, many of which are far from routine.Some of the material heas rarely, if ever, appeared previously i nbooks. Just in the first few chapters, for example, we offer Zenkov's mar- velous theorem about intersections of abelian subgroups, Wielan

16、dt' s "zipper lemma" i n subnormality theory and a proof of Horosevskii's theorem that the order of a group automorphism can never exceed the order of the group. Later chapters include many movanced topics that a ard or impos-sible to find elsewhere.Most of thegraduates who attend

17、my group-theory course aresecond-year s, and anpeople hads, wit h a subst al minority of first-yearoccasionalpreviouslywell-prepared undergraduate.Almost all of thesebeen exposed to a standard first-year graduate abstract algebracourse covering the basics of groups, rings and fields. I expect that m

18、ost ers of this book wil l have a similar background, and so I have decidednot to begin at the beginning.Most of myers (like mys) wil l have previously seen basicgroup theory, so I wanted to avoid repeating that material and to start withsomething more exciting: Sylow theory. Bu t I recognize that m

19、y audienceis not homogeneous, and someers wil l have gaps in their preparation,so I have included an appendix that contains most of the assumed materiali n a fairly condensed form. O n the other hand, I expect that many i n myaudience wil l aly know the Sylow theorems, but I am confident that eventh

20、ese well-prepared ers wil l find material that is new to them within the first few sections.M y semester-long graduate course at Wisconsin covers most of the first seven chapters of this book, starting with the Sylow theorems and cul- minating wit h a purely group-theoretic proof of Burnside' s

21、famous p a q b - theorem. Some of the topics along the way are subnormality theory, the Schur-Zassenhaus theorem, transfer theory, coprime group actions, Frobe- nius groups, and the normal p-complement theorems of Frobenius and of Thompson. The last three chapters cover material for which I never ha

22、ve time i n class. Chapter 8 includes a proof of the simplicity of the groups P S L ( n , q) , and also some graph-theoretic techniques for studying subdegrees of primitive and nonprimitive permutation groups. Subnormality theory is revisited i n Chapter 9, which includes Wielandt' s beautiful a

23、utomorphism tower theorem and the Thompson-Wielandt theorem related to the SimsPrefac ex iconjecture. fer theory,theorem".Finally, Chapter 10 presents some advanced topics i n trans-including Yoshida' s theorem and the so-called "principal idealFinally, I thank my manys and colleagues

24、who have contributedideas, suggestions and corrections while this book was being written. In particular, I mention that the comments of Yakov Berkovich and GabrielNavarro were invaluable and very much appreciated.Chapte r1Sylow Theory1 AIt seems appropriate to begin this book wit h a topic that unde

25、rlies virtuallyal l of finite group theory: the Sylow theorems. In this chapter, we state and prove these theorems, and we present some applications and related results.Althoug h much of this material should be very familiar, wethatmostnew toers wil l find that at least some of the content of this c

26、hapter isthem.Althoug h the theorem that proves Sylow subgroups always exist dates back to 1872, the existence proof that we have decided to present is that of H . Wielandt, published i n 1959. Wielandt' s proof is slick and short, but it does have some drawbacks. It is based on a trick that see

27、ms to have no other application, and the proof is not really constructive; it gives no guidance about how, i n practice, one might actually find a Sylow subgroup. Bu t Wielandt' s proof is beautiful, and that is the principal motivation for presenting it here.Also, Wielandt' s proof gives us

28、 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 about actions, althoughwe omit somes from the proofs

29、.Le t G be a group, and let ft the elements of ft as "points".)new element of ft, denoted a-g,be a nonempty set. (We wil l often refer to Suppose we have a rule that determines awhenever we are given a point a e ft andan element g e G. We say that this rule defines an action of G on ft if

30、thefollowing two conditions hold.121 . Sylo wTheor y(1) a - l = a for all a G ft and(2) a-g)- h = a-fo/i) for all a G ft and all group elementsg , h e G .Suppose that G acts on ft. It is easy tosee that if g G G- a-$ has anis arbitrary,inverse: thethen the function a: ft -> ft defined bygfunction

31、. Therefore, a g is a permutationof the set ft, which means thatC T 5 _ Iogis both injective and surjective, and thus a g lies in the symmetric groupSym(ft) consisting of all permutations of ft. In fact, the map g -> aisgeasily seen to be a homomorphism from G into Sym(ft). ( A homomorphism like

32、this, which arises from an action of a group G on some set, is called apermutation representation of G.) The kernel of this homomorphism is,of course, a normal subgroup of G, which is referred to action. The kernel is exactly the set of elements g e G ft, which means that a- g = a for all points a e

33、 ft.Generally, we consider a theorem or a technique to find a normal subgroup of G to be "good", andas the kernel of thethatact trivially onthathas the powerpermutationrepresentations can be good in this sense. (See the problems at the end of this section.) Bu t our goal i n introducing gr

34、oup actions here is not to find normal 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. We can let G act on G by right multiplication, so that x- g = x g for x , g G G. This is the reg

35、ular action of G, and it should be clear that it is faithful, which means that its kernel is trivial. It follows that the corresponding permutation representation of G is an isomorphism of G into Sym(G) , and this proves Cayley' s theorem: every group is isomorphic to a group of permutations on

36、some set.We continue to take ft = G, but this time, we define x- g = g l x g . (Thestandard notation for g x x g is x».) It is trivial to check that x = x and thatl x 9 ) h =9 h for all x , g , h G G, and thus we truly have an action, which isXcalled the conjugation action of G on itself. Note

37、that x= x if and only if9x g = gx, and thus the kernel of the conjugation action is the set of elements g e G that commute wit h all elements x e G. The kernel, therefore, is the center Z (G) .Agai n let G be arbitrary. In each of the previous examples, we took ft = G , but we also get interesting a

38、ctions if instead we take ft to be the setof all subsets of G. In the conjugation action of G on ft we let X - g = X=9 x 9 x e X and i n the right-multiplication 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 al l subset

39、s of G. For 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 all subgroups of G. Also, both right multiplication1 A3and conjugation preserve cardinality, and so each of these actions makes sense i f we take ft to be

40、the collection of all subsets of G of some fixed size. In fact, as we shall see, the trick in Wielandt' s proof of the Sylow existence theorem is to use the right multiplication action of G on its set of subsets wit h a certain fixed cardinality.We mention one other example, which is a special c

41、ase of the right- multiplication action on subsets that we discussed i n the previous paragraph.LetH C G be a subgroup, and let ft = H xof H i n G. If X is any right coset of H , i t x G G , the set of right cosets is easy to see that X g is also aX g = H x g ) . )The n G acts onright coset of H . (

42、, if X = H x , thenthe set ft by right multiplication.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 that Gais a subgroup of G ; it is called the stabilizer of the point a . For example, in the regular action of G on itself, the stabili

43、zer of every point (element of G ) is the trivial subgroup. In the conjugation action of G on G, the stabilizer of x G G is the centralizer C G ( x ) and in the conjugation action of G on subsets, the stabilizer of a subset X is the normalizer N G ( X ) . A useful general fact about point stabilizer

44、s is the following, which is easy to prove. In any action, if a- g = /?, then the stabilizers Gaand G p are conjugate i n G, and in fact, G a ) 3 = Gp.Now consider the action (by right multiplication) of G on the right cosetsof H , wheC G is a subgroup.The stabilizer of the coset H x is theset of al

45、l group elements g such that H x g = H x . It is easy to see that g satisfies this condition if and only if x g G H x . (This is because two cosets H u and H v are identical if and only if u G H v . ) It follows that g stabilizes H x if and only if g G x l H x . Since x l H x = H x , we see that the

46、 stabilizerofx . Gthe point (coset) H x is exactly the subgroup H x , conjugate to H viaIt follows that the kernel of the action of G on the right cosets of H i nis exactly f | H. This subgroup is called the core of H i n G, denotedxx e Gc o r e G ( H ) .The core of H is normal in G because i t is t

47、he kernel of an action, and, clearly, it is contained i n H . In fact, if AT < G is any normal subgroupthat happens to be contained in H , then N = NC Hfor all x G G , andxxthus N C core G (tf) . In other words, the core of H i n G is the unique largest normal subgroup of G contained in H . (It i

48、s "largest" i n the strong sense that it contains all others.)Wehave digressed from our goal, which is to show how to use groupactions to count things. Bu t having come this far, we may as well state theresults that our discussion has essentially proved. theorem and its corollaries can be

49、used to provesubgroups, and so they might be considered to beNote that the following the existence of normal"good" results.41 . Sylo wT h e o r y1.1.o f HTheorem. Le t H C G be a subgroup , an d le t ft be th e set of righ tcosetsi n G. The n G / c o v e G H )i s isomorphi c t o a subgrou

50、p o/Sym(ft) . I nparticular , if th e inde x G : H = n , the n G / c o r e G ( H )i s i s o m o r p h i c t o asubgrou p of Sn,th e s y m m e t r i c grou p o n n symbols .Proof. The action of G on the set ft by right multiplication defines ahomomorphism 6 (the permutation representation) from G int

51、o Sym(ft).Since ker(0) G / c o r e G H ) follows sincethus Sym(ft)= core G (tf) , it follows by the homomorphism theorem that 0 ( G ) , which is a subgroup of Sym(G) . The last statement if G : H = n , then (by definition of the index) |ft| = n , and= Sn.U1.2. Corollary. Le t G be a group , an d sup

52、posetha t H C G i s a subgrou pw i t h G : H = n . The n H contain s a n o r m a l subgrou p N of G such G : N divide s n .tha tProof. Take N = c o r e G ( H ) .The n G / Nis isomorphic to a subgroup ofthe symmetric group Sn,and so by Lagrange's theorem, G / N divides S n = n .1.3. Corollary. Le

53、 t G be simpl e an dThe n G divide s nl .contai n a subgrou p of inde x n > 1.Proof. The normal subgroup N of the and hence it is proper i n G because nthus G = G / N divides nl . Uprevious corollary is contained i n H ,> 1. Since G is simple, N = 1 , andIn order to pursue our main goal, which is counting, we