vi CONTENT S
18. Theorem 4: The An Case 110
19. Theorem 4: The Wide Lie Type Case 112
20. Completion of the Proof for p 2: Theorem 5 113
21. Theorem 6: The Sporadic Cases 118
22. The An Case, n = 9, 10, or 11 121
23. Theorem 7, Case 1 125
24. Theorem 9 128
25. Theorem 7, Case 2: 5/2-Balance 131
26. Theorem 7, Case 2: 9'5/2{G;B) Is Nontrivial 136
27. Theorem 7, Case 2: Bootstrapping 140
28. Theorem 7: Case 3 142
29. Theorem 8: The Extended 2-Source Case 148
30. Fusion and Pumpups 156
31. Failure of 3/2-Balance 160
Chapter 4. Theorem e?: Stage 2 165
1. Introduction 165
2. The Subsidiary Theorems 167
3. Properties of X-Groups and p-Sources 171
4. Proposition 1 174
5. Theorem 1: K Lies in M 175
6. The Embedding of K in M: The Diagonal Case, p 2 179
7. The Embedding of K in M: The General Case 185
8. Theorem 2 188
9. Corollary 2 203
10. Controlling Layers of Centralizers of Elements of Order p 204
11. Theorem 3: p-Component Preuniqueness Subgroups 213
12. H = K: The Non-Strongly-Closed {S)L±(q) Case 216
13. ~H_ = ~K: The Non-Strongly-Closed Case 219
14. H = K: The Strongly Closed Case 225
15. The Embedding of TAil(G) 230
16. Connecting 235
17. Theorem 4 237
18. Theorem 5 244
19. Appendix I: The Semisimple-Neighbor Strongly-Closed Case 249
20. Appendix II: p-Terminal Pairs with Wide Centralizers 255
Chapter 5. Theorem C^: Stage 3a 271
1. Introduction 271
2. The Subsidiary Theorems 274
3. Strong p-Uniqueness Subgroups 275
4. Theorem 1 and Corollary 1 281
5. Theorem 2: Off-Characteristic Action of Subcomponents 285
6. The Isomorphism Type of K When p Is Odd 288
7. The Exceptional Alternating Cases 291
8. Theorem 3: The Centralizing Case 296
9. Bridges 302
Chapter 6. Properties of X-Groups
309
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