Mathematical Surveys and Monographs
Volume: 40;
2002;
467 pp;
Hardcover
MSC: Primary 20;
Print ISBN: 978-0-8218-2776-5
Product Code: SURV/40.5
List Price: $110.00
AMS Member Price: $88.00
MAA Member Price: $99.00
Electronic ISBN: 978-1-4704-1270-8
Product Code: SURV/40.5.E
List Price: $110.00
AMS Member Price: $88.00
MAA Member Price: $99.00
Supplemental Materials
The Classification of the Finite Simple Groups, Number 5
Share this pageDaniel Gorenstein; Richard Lyons; Ronald Solomon
The classification of finite simple groups is a landmark
result of modern mathematics. The original proof is spread over scores of
articles by dozens of researchers. In this multivolume book, the authors are
assembling the proof with explanations and references. It is a monumental
task. The book, along with background from sections of the previous volumes,
presents critical aspects of the classification.
In four prior volumes
(Surveys of Mathematical Monographs, Volumes 40.1, 40.2, 40.3, and 40.4), the
authors began the proof of the classification theorem by establishing certain
uniqueness and preuniqueness results. In this volume, they now begin the proof
of a major theorem from the classification grid, namely Theorem \(\mathcal{C}_7\).
The book is suitable for graduate students and researchers interested in group
theory.
Readership
Graduate students and research mathematicians interested in group theory.
Reviews & Endorsements
This series of volumes, and this one in particular, is a model for all mathematicians of the standards and clarity that should be achieved.
-- Mathematical Reviews
Table of Contents
Table of Contents
The Classification of the Finite Simple Groups, Number 5
- Contents v6 free
- Preface ix10 free
- Part III, Chapters 1–6: The Generic Case, Stages 1–3a 114 free
- Chapter 1. Theorem C[sub(7)]: General Introduction 114
- Chapter 2. General Group-Theoretic Results 922
- Chapter 3. Theorem C*[sub(7)]: Stage 1 4356
- 1. Introduction 4356
- 2. p-Sources 5063
- 3. Failure of Local Balance 6073
- 4. The Subsidiary Theorems, General Case 6174
- 5. The Preferred Set J[sup(o)][sub(p)](G) 6376
- 6. The Remaining Subsidiary Theorems 6780
- 7. Theorem 1: The 2A[sub(n)] Case 6780
- 8. The Tight Sp[sub(4)](q) Case 7083
- 9. Theorem 2: The Nonsolvable Case 7184
- 10. Theorem 2: The Solvable p-Component Case 7285
- 11. Corollary 2 and Extended p-Sources 7689
- 12. Theorem 3: The Wide Lie Type Case, p = 2 8497
- 13. Theorem 3: The A[sub(n)] Case, n ≥ 12, and the F[sub(5)] Case, p = 2 8598
- 14. The Extended (x, K) p-Source Strategy 88101
- 15. Bootstrapping from an Easier Level of Balance 89102
- 16. The L[sub(3)](4) Case of the Reduction 95108
- 17. Applications to Extended p-Sources 101114
- 18. Theorem 4: The A[sub(n)] Case 110123
- 19. Theorem 4: The Wide Lie Type Case 112125
- 20. Completion of the Proof for p > 2: Theorem 5 113126
- 21. Theorem 6: The Sporadic Cases 118131
- 22. The A[sub(n)] Case, n = 9, 10, or 11 121134
- 23. Theorem 7, Case 1 125138
- 24. Theorem 9 128141
- 25. Theorem 7, Case 2: 5/2-Balance 131144
- 26. Theorem 7, Case 2: Θ'[sub(5/2)](G;B) Is Nontrivial 136149
- 27. Theorem 7, Case 2: Bootstrapping 140153
- 28. Theorem 7: Case 3 142155
- 29. Theorem 8: The Extended 2-Source Case 148161
- 30. Fusion and Pumpups 156169
- 31. Failure of 3/2-Balance 160173
- Chapter 4. Theorem C*[sub(7)]: Stage 2 165178
- 1. Introduction 165178
- 2. The Subsidiary Theorems 167180
- 3. Properties of k-Groups and p-Sources 171184
- 4. Proposition 1 174187
- 5. Theorem 1: K Lies in M 175188
- 6. The Embedding of K in M: The Diagonal Case, p > 2 179192
- 7. The Embedding of K in M: The General Case 185198
- 8. Theorem 2 188201
- 9. Corollary 2 203216
- 10. Controlling Layers of Centralizers of Elements of Order p 204217
- 11. Theorem 3: p-Component Preuniqueness Subgroups 213226
- 12. H = K: The Non-Strongly-Closed (S)L[sup(+)][sub(p)](q) Case 216229
- 13. H = K: The Non-Strongly-Closed Case 219232
- 14. H = K: The Strongly Closed Case 225238
- 15. The Embedding of Γ[sub(A,1)](G) 230243
- 16. Connecting 235248
- 17. Theorem 4 237250
- 18. Theorem 5 244257
- 19. Appendix I: The Semisimple- Neighbor Strongly-Closed Case 249262
- 20. Appendix II: p-Terminal Pairs with Wide Centralizers 255268
- Chapter 5. Theorem C*[sub(7)]: Stage 3a 271284
- 1. Introduction 271284
- 2. The Subsidiary Theorems 274287
- 3. Strong p- Uniqueness Subgroups 275288
- 4. Theorem 1 and Corollary 1 281294
- 5. Theorem 2: Off-Characteristic Action of Subcomponents 285298
- 6. The Isomorphism Type of K When p Is Odd 288301
- 7. The Exceptional Alternating Cases 291304
- 8. Theorem 3: The Centralizing Case 296309
- 9. Bridges 302315
- Chapter 6. Properties of k-Groups 309322
- 1. The g[sub(p)]-Groups 309322
- 2. Specific Groups 310323
- 3. Balance and Signalizers 323336
- 4. Small Groups 329342
- 5. Pumpups and Subcomponents 336349
- 6. Sylow Structure 358371
- 7. Generation 361374
- 8. Miscellaneous 385398
- 9. p-Sources 389402
- 10. Subterminal Pairs 425438
- 11. Bridges 443456
- 12. The Weak Signalizer Property 451464
- 13. Fusion 458471
- Background References 463476
- Expository References 464477
- Glossary of Symbols and Terms 465478