x CONTENTS
Chapter 3. p-Component Uniqueness Theorems 131
1. Introduction 131
2. Theorem PUi: The Subsidiary Results 137
3. Exceptional Triples and 3C-Group Lemmas 138
4. M-Exceptional Subgroups of G 142
5. Centralizers of Elements of XP(K*) 147
6. The Proofs of Theorems 1 and 2 150
7. Theorem 3: The Nonsimple Case 153
8. Theorem 3: The Simple Case 163
9. Theorem 4: Preliminaries 165
10. Normalizers of p-Subgroups of K 168
11. The Case K £ S£(p) 171
12. The S£{p) case 172
13. Theorem 5: The M-Exceptional Case 179
14. The Residual M-Exceptional Cases 188
15. Completion of the Proof of Theorem PUi: The p-Rank 1 Case 193
16. Reductions to Theorem PUi 194
17. Further Reductions: The Non-Normal Case 201
18. Theorem PU2: The Setup 206
19. The Case r p: A Reduction 208
20. The Case r = 1: Conclusion 211
21. The Case r = p = 2: A. Reduction 214
22. The Case r = p = 2, ~K ^ L2(q) 216
23. Theorem PU3: The Simple Case 220
24. The Residual Simple Cases 223
25. The Nonsimple Case 224
26. Theorem PU4 226
27. Corollaries PU2 and PU4 227
28. Aschbacher's Reciprocity Theorem 228
29. Theorem PU5 232
Chapter 4. Properties of K-Groups 237
1. Automorphisms 237
2. Schur Multipliers and Covering Groups 238
3. Bender Groups 245
4. Groups of Low 2-Rank 253
5. Groups of Low p-Rank, p Odd 256
6. Centralizers of Elements of Prime Order 257
7. Sylow 2-Subgroups of Specified DC-Groups 268
8. Disconnected Groups 271
9. Strongly Closed Abelian Subgroups 276
10. Generation 283
11. Generation and Terminal Components 293
12. Preuniqueness Subgroups and Generation 302
13. 2-Constrained Groups 328
14. Miscellaneous Results 331
Background References 333
Expository References 334
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