The Classification of the Finite Simple Groups, Number 7

Cover......Page 1
Title page......Page 4
Contents......Page 8
Preface......Page 10
1. Introduction......Page 12
2. The Ranking Function \FF......Page 17
3. The Set \J_{}*(), and the Four Cases......Page 18
4. The Sets \J_{}¹() and \J_{}²(), and How to Modify Stages 2 and 3a......Page 19
5. Component Pairs......Page 22
6. The Remaining Stages of Theorem \C₇*......Page 23
1. -Groups......Page 30
2. Pumpups and -Terminality......Page 31
3. Balance and Signalizers......Page 38
4. Components and Coverings......Page 39
5. Representations and Cohomology......Page 40
6. Fusion and Transfer......Page 42
7. Miscellaneous......Page 46
1. Introduction......Page 50
2. Theorem 2: The Setup and Special Cases......Page 52
3. The Cases ≅_{} or ≅, with =2......Page 56
4. The ₅ Case......Page 65
5. -Thin Configurations,  Odd......Page 67
6. The General Lie Type Case, Begun......Page 74
7. The General Lie Type Case, Concluded......Page 86
8. The =2, ₂() case......Page 99
9. Theorem 3......Page 106
10. Theorem 4: The Prime ......Page 113
11. The -Uniqueness Case......Page 122
12. Theorem 5: The General Case......Page 126
13. Theorem 5: Residual Cases, =2......Page 132
14. Theorem 5: Residual Cases, >2......Page 137
15. Theorem 6......Page 154
1. Introduction......Page 156
2. Theorems 1 and 2: Most Sporadic Components......Page 158
3. Theorems 1 and 2: ₁ and Ree Components......Page 160
4. The Alternating Case: Theorem 3–Centralizers of Rank 2......Page 163
5. Theorem 4–Centralizers of Rank 1: Preliminaries......Page 168
6. Theorem 4–All Alternating Group Neighbors......Page 171
7. Theorem 4–Non-Alternating Group Neighbors......Page 178
8. Theorem 4–The Final Case: 2 Neighbors......Page 186
1. Everyday Tools......Page 192
2. Automorphisms and Coverings......Page 200
3. -Rank and -Sylow Structure of Quasisimple Groups......Page 202
4. Computations in Alternating Groups......Page 207
5. Computations in Sporadic Groups......Page 209
6. Computations in Groups of Lie Type......Page 214
7. Small Groups......Page 228
8. Embeddings......Page 233
10. Balance......Page 234
11. Subcomponents......Page 235
12. Pumpups and the Ranking Function \FF......Page 237
13. Pumpups......Page 242
14. Acceptable Subterminal Pairs, I......Page 273
15. Acceptable Subterminal Pairs, II......Page 282
16. Neighborhoods......Page 288
17. Generation......Page 299
18. Splitting Primes......Page 303
19. Double Pumpups......Page 309
20. The Double Pumpup Propositions......Page 314
21. Monotonicity of \FF, and \J_{}²()......Page 331
22. Miscellaneous......Page 348
Bibliography......Page 352
Index......Page 354
Back Cover......Page 362