From the Preface: "From the 1950s until 1968, the theory of finite groups underwent an intense period of growth, including the first major classification theorem concerning simple groups as well as the construction of the first new sporadic simple group in a hundred years. In writing this book, my aim was to describe that development in sufficient detail for the interested reader to reach the frontiers of the subject and thereby participate in the excitement that then surrounded the study of simple groups ... In the intervening ten years [since the first edition of this work], an equally dramatic change has occurred ... a complete classification of the finite simple groups has now almost approached a reality ... Literally thousands of journal pages have been devoted to their study and powerful new techniques have been developed ... In most instances, these developments constitute a continuation rather than a replacement of the material in this book."
Author(s): Daniel Gorenstein
Series: AMS Chelsea Publishing
Edition: 2 Sub
Publisher: Chelsea Pub Co
Year: 1980
Language: English
Pages: 514
Finite Groups......Page 1
Daniel Gorenstein......Page 3
Second Edition, 1980......Page 4
Contents......Page 7
Preface to the Second Edition......Page 11
Preface to the First Edition......Page 13
Part I. Methods......Page 19
1. Preliminaries......Page 21
1.1. Notation and Terminology......Page 22
1.2. Assumed Results......Page 23
1.3. Related Elementary Results......Page 27
Exercises......Page 31
2.1. Characteristic Subgroups......Page 33
2.2. Elementary Properties of Commutators......Page 36
2.3. Nilpotent Groups......Page 39
2.4. Solvable Groups......Page 41
2.5. Semidirect and Central Products......Page 43
2.6. Automorphisms as Linear Transformations......Page 47
2.7. Transitive and Doubly Transitive Permutation Groups......Page 51
2.8. The Two-Dimensional Linear and Projective Groups......Page 57
Exercises......Page 73
3.1. Basic Concepts......Page 76
3.2. Representations of Abelian Groups......Page 82
3.3. Complete Reducibility......Page 84
3.4. Clifford's Theorem......Page 88
3.5. G-Homomorphisms......Page 93
3.6. Irreducible Representations and Group Algebras......Page 100
3.7. Representations of Direct and Central Products......Page 117
3.8. p-Stable Representations......Page 120
Exercises......Page 127
4.1. Basic Properties......Page 130
4.2. The Orthogonality Relations......Page 137
4.3. Some Applications......Page 148
4.4. Induced Characters and Trivial Intersection Sets......Page 152
4.5. Frobenius Groups......Page 158
4.6. Coherence......Page 166
4.7. Brauer's Characterization of Characters......Page 178
Exercises......Page 188
5. Groups of Prime Power Order......Page 190
5.1. The Frattini Subgroup......Page 191
5.2. p'-Automorphisms of Abelian p-Groups......Page 193
5.3. p'-Automorphisms of p-Groups......Page 196
5.4. p-Groups of Small Depth......Page 206
5.5. Extra-special p-Groups......Page 221
5.6. The Associated Lie Ring......Page 226
Exercises......Page 232
6.1. The Fitting and Frattini Subgroups......Page 235
6.2. The Schur-Zassenhaus Theorem......Page 238
6.3. π-Separable and π-Solvable Groups......Page 244
6.4. Solvable Groups......Page 249
6.5. p-Stability in p-Solvable Groups......Page 252
Exercises......Page 254
7.1. Local Fusion......Page 256
7.2. Alperin's Theorem......Page 258
7.3. Transfer and the Focal Subgroup......Page 263
7.4. Theorems of Burnside, Frobenius, and Grün......Page 269
7.5. Weak Closure and p-Normality......Page 273
7.6. Elementary Applications......Page 275
7.7. Groups with Dihedral Sylow 2-Subgroups......Page 278
Exercises......Page 282
8. p-Constrained and p-Stable Groups......Page 285
8.1. p-Constraint and p-Stability......Page 286
8.2. Glauberman's Theorem......Page 288
8.3. The Glauberman-Thompson Normal p-Complement Theorem......Page 298
8.4. Groups with Subgroups of Glauberman Type......Page 299
8.5. The Thompson Transitivity Theorem......Page 306
8.6. The Maximal Subgroup Theorem......Page 312
Exercises......Page 316
9. Groups of Even Order......Page 318
9.1. Elementary Properties of Involutions......Page 319
9.2. The Feit-Suzuki-Thompson Theorems......Page 324
9.3. Two Applications......Page 328
9.4. Group Order Formulas......Page 333
Exercises......Page 346
Part II. Applications......Page 349
10.1. Elementary Properties......Page 351
10.2. Fixed-Point-Free Automorphisms of Prime Order......Page 355
10.3. Frobenius Groups and Groups with Nilpotent Maximal Subgroups......Page 357
10.4. Fixed-Point-Free Automorphisms of Order 4......Page 358
10.5. Fixed-Point-Free Four-Groups of Automorphisms......Page 363
Exercises......Page 374
11. The Hall-Higman Theorem......Page 376
11.1. Statement and Initial Reductions......Page 377
11.2. The Extra-Special Case......Page 381
Exercises......Page 389
12. Groups with Generalized Quaternion Sylow 2-Subgroups......Page 391
13. Zassenhaus Groups......Page 396
13.1. Elementary Properties......Page 397
13.2. Feit's Theorem......Page 401
13.3. Classification of Certain Zassenhaus Groups......Page 406
14. Groups in which Centralizers are Nilpotent......Page 417
14.1. Basic Properties of CN-Groups......Page 418
14.2. CN-Groups of Odd Order......Page 423
14.3. Solvability of CN-Groups of Odd Order......Page 427
14.4. CN-Groups with Abelian Sylow 2-Subgroups......Page 433
15. Groups with Self-Centralizing Sylow 2-Subgroups of Order 4......Page 435
15.1. Some Properties of L_2(q)......Page 436
15.2. Statement of the Theorem and Initial Reduction......Page 438
15.3. The Structure of the Centralizer of an Involution......Page 441
15.4. The Brauer-Suzuki-Wall Theorem......Page 451
Exercises......Page 456
Part III. General Classification Problems......Page 459
16. Simple Groups of Low Rank......Page 461
16.1. General Methods and Objectives......Page 463
16.2. Groups of Odd Order......Page 468
16.3. Groups with Dihedral Sylow 2-Subgroups......Page 480
16.4. C-Groups......Page 483
16.5. N-Groups......Page 491
16.6. Groups with Abelian Sylow 2-Subgroups......Page 498
16.7. Other Classification Theorems......Page 504
Addendum to Page 479......Page 506
17. The Known Simple Groups......Page 507
Bibliography......Page 513
List of Symbols......Page 527
Index......Page 531
