The Classification of the Finite Simple Groups, Number 9, Part 9: Part V, Chapters 1-8: Theorem C5 and Theorem C6, Stage 1

Cover
Title page
Preface
Chapter 1. Introduction to Theorem ?₅
1. Statement of Theorem ?₅
2. The Four Stages of the Proof of Theorem ?₅
Chapter 2. General Group-Theoretic Lemmas, and Recognition Theorems
1. Signalizer Functor Theory
2. Signalizers
3. Fusion
4. ?-Groups
5. Uniqueness Subgroups
6. Miscellaneous
7. Components
8. Rigidity, Semirigidity, and Terminality
9. Centralizers of Components
10. Recognition Theorems
Chapter 3. Theorem ?₅: Stage 1
1. Introduction
2. The Strong Balance Lemma
3. Theorem 1: Balance
4. ?₂(?^{?}) Field Triples
5. Corollaries to Theorem 1
6. Theorem 2: Signalizers in ?
7. The Centralizer of a Sylow 2-Subgroup of ?_{?’}(?)
8. Sufficient Conditions for Faithful Action on ?
9. ?₂(?^{?}) Field Triples Do Not Exist
10. A Covering 2-Local Result
11. Theorem 3: Γ_{?,2}^{?}(?)\le?
12. Theorem 4: Γ_{?,2}(?)\le?
13. Theorem 5
Chapter 4. Theorem ?₅: Stage 2
1. Introduction
2. The Principal Subsidiary Theorems
3. Some Generalities
4. Uniqueness Subgroups from ?-Terminal ?-Components
5. Some Sporadic ?-Components
6. Theorem 1: Generalities
7. Theorem 1: ?-Terminal ?-Components
8. Theorem 1: The ?-Terminality of ?₆
9. Theorem 1: Standard 2-Components
10. Theorem 2
11. Corollaries: \BtK^{?}(?), and Components in ?_{?}-??ℯ?(?)
12. Theorem 3: Regular Triples and Mates
13. Theorem 3: ? Must Be 3
14. The Nondegenerate Case: Theorem 4
15. Theorem 5: The Degenerate Case ?>?
16. Theorem 5: The Degenerate Case ?=?
17. Theorem 5: The Degenerate Case \overline?<\overline?
18. Theorem 5: The Degenerate Case \overline?=\overline?
Chapter 5. Theorem ?₅: Stage 3
1. Introduction
2. The Principal Subsidiary Theorems
3. Theorem 1
4. Maximal Symplectic Triples
5. Theorem 2: ?=3 and Constrained Neighborhoods
6. Theorem 2: Nonconstrained Neighborhoods of Type ??₂₄’
7. Theorem 2: Nondegenerate Constrained Neighborhoods
8. Theorem 2: Constrained Neighborhoods of Sporadic Type
Chapter 6. Theorem ?₅: Stage 4
1. Introduction
2. The Case ?*=Ω₇(3) or Ω₈^{±}(3)
3. The Case ?*=??₂₂
4. The Case ?*=??₂₃
5. The Case ?*=??₂₄’ or ??₂₄
6. The Case ?*=?₂
7. The Case ?*=??₁
8. The Case ?*=?₁
9. The Case ?*=²?₅(2), ²?₆(2), or ?₇(2)
Chapter 7. Theorem ?₆*: Stage 1
1. Theorems ?₆ and ?₆*
2. The Proof of Theorem ?₇
3. Theorems 1 and 1*
4. Theorem 2
5. Theorem 2 Completed: The ?₃(8) Case
Chapter 8. Preliminary Properties of ?-Groups
1. Small Groups
2. Outer Automorphisms and Covering Groups
3. Pumpups and Subcomponents
3.1. Key Generalities
3.2. Pumpups of Specific Groups
3.3. Pumpups and ?-Rank
3.4. Pumpups and Schur Multipliers
3.5. Pumpups of ?₆
3.6. ?-Subcomponents and ?-Subcomponents, ?≠?
3.7. Action of ?_{???(?)}(?) on ?(?_{?}(?))
3.8. Pumpups and the Sets ?_{?}, ? Prime
3.9. Solvable Components
3.10. Other
4. Computations in Groups of Lie Type
5. Representations
6. Factorizations
7. Signalizers and Balance
7.1. Signalizers
7.2. Balance and Cores
8. Generation
8.1. Generation with respect to Non-elementary ?-Groups, ?=2
8.2. Generation with respect to Elementary Abelian ?-Groups, ?=2
8.3. Generation with respect to Elementary Abelian ?-Groups, ?>2
8.4. Generation with respect to Non-elementary ?-Groups, ?>2
8.5. Other
9. ?-Structure, ? odd
9.1. Self-centralizing ?_{?²}-Subgroups, ?≥5
9.2. Sylow ?-Subgroups ?, and ?(?) and ?(?)
9.3. Other ?-Subgroups
9.4. Other
10. 2-Structure
10.1. Orthogonal Groups over ?₃
10.2. Sylow 2-Subgroups and Their Overgroups in Quasisimple ?-Groups
10.3. Other Involution Centralizers
10.4. Other
11. ?₂-Groups and ?_{?}-Groups, ? odd
12. {2,?}-Structure
12.1. Wide 2-Components
12.2. Subgroups of Order 2?
12.3. Components for Permutable Subgroups of Orders 2 and ?
12.4. 2-Local ?-Rank
12.5. Mates
12.6. ?-Components in ?₂-Groups
12.7. Other
13. {2,3}-Neighborhoods
14. Semirigidity
14.1. Conditions for Semirigidity
14.2. ?₄-Semirigidity
15. Miscellaneous
16. Preliminary Lemmas for Theorem ?₆*: Stage 1
16.1. ?-Ranks, ?_{?}ⁱ-Groups, and Flat ??_{?}-Groups
16.2. Pumpups and Subcomponents
16.3. 3-Structure
16.4. Other
Bibliography
Index
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