# The Theory of Finite Groups: An Introduction

###### Hans Kurzweil and Bernd Stellmacher
Publisher:
Springer Verlag
Publication Date:
2004
Number of Pages:
387
Format:
Hardcover
Price:
69.95
ISBN:
0-387-40510-0.
Category:
Textbook
[Reviewed by
Michael Berg
, on
07/1/2004
]

I once had a friend whose all-consuming passion was the theory of finite groups. When Gorenstein's dream of the classification of the finite simple groups was achieved, my friend half-joked that a lot of mathematicians, himself included, had just been robbed of their raison d'être. He made some noise about John Horton Conway's monstrous moonshine, which was music to my number-theoretic ears, of course, but, as far as I know, he didn't go anywhere with it. I lost track over the years, both of my friend and of finite group theory. Regarding the latter, however, I found out recently that cross-fertilization from physics has evidently opened up some very beautiful and dramatic lines, having to do with vertex algebras, originating in physics. The original moonshine phenomenon, noted first by McKay and then expounded by Conway and Norton, is now part of a brightly lit landscape (with Borcherds' work figuring very prominantly), but apparently some related material is still hidden in dark shadows and a lot of exciting things may be expected. (See Terry Gannon's fascinating recent arXiv article, "Monstrous Moonshine: the first twenty-five years," forthcoming in Bull. London Math. Soc.)

So, indeed, the theory of finite groups is alive and kicking. Moreover, it would be disingenuous to suggest that the field has been taken over by methodologies coming from physics and number theory. The book under review is incontrovertible proof that the theory of finite groups per se is alive and well, too. Indeed, while serving to introduce a relative novice to the subject, The Theory of Finite Groups: An Introduction is also a marvelous treatment of a large chunk of what's going on today. It is presented as "the first book which shows us the amalgam method and moreover shows us how it works," this according to G. Stroth's Zentralblatt review, cited on the back cover. The authors' discussion of this important method can be found on p. 281 ff. They note that "[the amalgam] method was introduced by Goldschmidt at the end of the 1970's and since then has become an integral part of the local structure theory of finite groups. The name amalgam method refers to the fact that this method does not require a finite group but can be carried out already in the amalgamated product of ... finite groups." This is, in and of itself, very interesting material and obviously of interest to any aspiring hard-core group theorist.

And, as already hinted, there's a lot more to this book. It starts with baby group theory, so to speak, using the notion of a group action both systematically and often, very often: it's a central pedagogical and expositional device, which works well! This introductory material is presented at a fast pace: Sylow appears on p. 62. But it works, even if the reader really ought to be mathematically mature beyond the level of a garden-variety junior mathematics major.

The book goes on to finite group theory properly so-called, as the philosophers say, while omitting representation-theoretic aspects as well as any real treatment of the classification problem. (The only space devoted to the classification as such is a statement of the theorem on pp. 370-371: they authors are busy about other things.) There are a lot of nice exercises, the scholarship is phenomenally thorough, and many very interesting (and some exotic) things are covered. The entire presentation is quite elegant.

I think Kurzweil and Stellmacher's book would serve beautifully as a source for a year-long seminar (or longer?) on group theory at the level of advanced undergraduate students or beginning graduate students. If I, personally, should wish to learn a lot of serious finite group theory I'd go with this book, perhaps coupled with Rose's well-known Second Course in Group Theory.

Michael Berg (mberg@lmu.edu) is professor of mathematics at Loyola Marymount University.

Preface v
List of Symbols xi
1 Basic Concepts 1
1.1 Groups and Subgroups . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Homomorphisms and Normal Subgroups . . . . . . . . . . . . 10
1.3 Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.5 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.6 Products of Groups . . . . . . . . . . . . . . . . . . . . . . . . 27
1.7 Minimal Normal Subgroups . . . . . . . . . . . . . . . . . . . 36
1.8 Composition Series . . . . . . . . . . . . . . . . . . . . . . . . 39
2 Abelian Groups 43
2.1 The Structure of Abelian Groups . . . . . . . . . . . . . . . . 43
2.2 Automorphisms of Cyclic Groups . . . . . . . . . . . . . . . . 49
3 Action and Conjugation 55
3.1 Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Sylow's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 Complements of Normal Subgroups . . . . . . . . . . . . . . . 71
vii
viii Contents
4 Permutation Groups 77
4.1 Transitive Groups and Frobenius Groups . . . . . . . . . . . . 77
4.2 Primitive Action . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3 The Symmetric Group . . . . . . . . . . . . . . . . . . . . . . 87
4.4 Imprimitive Groups and Wreath Products . . . . . . . . . . . 91
5 p-Groups and Nilpotent Groups 99
5.1 Nilpotent Groups . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 Nilpotent Normal Subgroups . . . . . . . . . . . . . . . . . . 104
5.3 p-Groups with Cyclic Maximal Subgroups . . . . . . . . . . . 108
6 Normal and Subnormal Structure 121
6.1 Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 The Theorem of Schur-Zassenhaus . . . . . . . . . . . . . . . 125
6.3 Radical and Residue . . . . . . . . . . . . . . . . . . . . . . . 128
6.4 p-Separable Groups . . . . . . . . . . . . . . . . . . . . . . . 133
6.5 Components and the Generalized
Fitting Subgroup . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.6 Primitive Maximal Subgroups . . . . . . . . . . . . . . . . . . 145
6.7 Subnormal Subgroups . . . . . . . . . . . . . . . . . . . . . . 156
7 Transfer and p-Factor Groups 163
7.1 The Transfer Homomorphism . . . . . . . . . . . . . . . . . . 163
7.2 Normal p-Complements . . . . . . . . . . . . . . . . . . . . . 169
8 Groups Acting on Groups 175
8.1 Action on Groups . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.2 Coprime Action . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.3 Action on Abelian Groups . . . . . . . . . . . . . . . . . . . . 190
8.4 The Decomposition of an Action . . . . . . . . . . . . . . . . 197
8.5 Minimal Nontrivial Action . . . . . . . . . . . . . . . . . . . . 203
8.6 Linear Action and the Two-Dimensional
Linear Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Contents ix
9.1 Quadratic Action . . . . . . . . . . . . . . . . . . . . . . . . . 225
9.2 The Thompson Subgroup . . . . . . . . . . . . . . . . . . . . 230
9.3 Quadratic Action in p-Separable Groups . . . . . . . . . . . . 239
9.4 A Characteristic Subgroup . . . . . . . . . . . . . . . . . . . . 249
9.5 Fixed-Point-Free Action . . . . . . . . . . . . . . . . . . . . . 257
10 The Embedding of p-Local Subgroups 261
10.1 Primitive Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . 262
10.2 The paqb-Theorem . . . . . . . . . . . . . . . . . . . . . . . . 276
10.3 The Amalgam Method . . . . . . . . . . . . . . . . . . . . . . 281
11 Signalizer Functors 303
11.1 Definitions and Elementary Properties . . . . . . . . . . . . . 304
11.2 Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
11.3 The Completeness Theorem of Glauberman . . . . . . . . . . 325
12 N-Groups 335
12.1 An Application of the Completeness
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
12.2 J(T)-Components . . . . . . . . . . . . . . . . . . . . . . . . 347
12.3 N-Groups of Local Characteristic 2 . . . . . . . . . . . . . . . 356
Appendix 369
Bibliography 373
Books and Monographs . . . . . . . . . . . . . . . . . . . . . . . . 373
Journal Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
Index 381