x CONTENTS
Chapter 4. Local Subgroups of Groups of Lie Type, II 119
4.1. Semisimple Automorphisms of Algebraic Groups 119
4.2. Semisimple Automorphisms of Finite Groups of Lie Type 134
4.3. Centralizers of Involutions in Auto (if): Statements 143
4.4. Centralizers of Involutions in Auto(K): Proofs 153
4.5. Centralizers of Involutions in Groups in £jie(r), r Odd: Statements 170
4.6. Centralizers of Involutions in Groups in £ie(r), r Odd: Proofs 186
4.7. Inner-Diagonal and Graph Automorphisms of Exceptional Odd
Prime Orders 204
4.8. Some Other p-Local Subgroups 219
4.9. The Remaining Types of Non-Inner Automorphisms 227
4.10. Cross-Characteristic Sylow Structure 236
Chapter 5. The Alternating Groups and the Twenty-Six Sporadic
Groups 249
5.1. Schur Multipliers and Coverings of Simple Groups 249
5.2. The Alternating Groups 252
5.3. Structure of the Twenty-Six Sporadic Groups 259
5.4. Justification of the Tables: The Simple Case 288
5.5. Justification of the Tables: The Non-Simple Case 294
5.6. Ranks of Sporadic Groups 302
Chapter 6. Coverings and Embeddings of Quasisimple DC-Groups 311
6.1. The Schur Multipliers and Covering Groups of the Simple
X-Groups 311
6.2. Some Embeddings of Quasisimple Groups 313
6.3. The Action of Out (if) on M(K) 317
6.4. Splitting and Stable Involutions 324
6.5. Subgroups of Small Simple Groups 328
Chapter 7. General Properties of X-Groups 335
7.1. Verification of the Fundamental X-Group Properties 335
7.2. Generation of Semisimple Groups: Generalities 343
7.3. Generation for Groups in Qhev: Statements of Main Results 346
7.4. Generation for Groups in Qhev: Proofs 349
7.5. Generation for Groups in Alt U Spor 380
7.6. Strongly p-Embedded Subgroups and Proper fc-Generated Cores 383
7.7. Balance and Signalizers 385
7.8. Fusion 399
Background References 403
Expository References 404
Errata for Numbers 1 and 2 409
Glossary 410
Index 414
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