Contents

Preface xi

Background and overview 1

Chapter 0. Introduction 3

0.1. The Classification Theorem 3

0.2. Principle I: Recognition via local subgroups 4

0.3. Principle II: Restricted structure of local subgroups 7

0.4. The finite simple groups 16

0.5. The Classification grid 19

Chapter 1. Overview: The classification of groups of Gorenstein-Walter type 25

The Main Theorem for groups of Gorenstein-Walter type 25

1.1. A strategy based on components in centralizers 26

1.2. The Odd Order Theorem 28

1.3. (Level 1) The Strongly Embedded Theorem

and the Dichotomy Theorem 29

1.4. The 2-Rank 2 Theorem 33

1.5. (Level 1) The Sectional 2-Rank 4 Theorem

and the 2-Generated Core Theorem 35

1.6. The B-Conjecture and the Standard Component Theorem 41

1.7. The Unbalanced Group Theorem, the 2An-Theorem,

and the Classical Involution Theorem 44

1.8. Finishing the Unbalanced Group Theorem and the B-Theorem 48

1.9. The Odd Standard Component Theorem

and the Aschbacher-Seitz reduction 53

1.10. The Even Standard Component Theorem 55

Summary: Statements of the major subtheorems 59

Chapter 2. Overview: The classification of groups of characteristic 2 type 63

The Main Theorem for groups of characteristic 2 type 63

2.1. The Quasithin Theorem covering e(G) ≤ 2 65

2.2. The trichotomy approach to treating e(G) ≥ 3 66

2.3. The Trichotomy Theorem for e(G) ≥ 4 69

2.4. The e(G) = 3 Theorem (including trichotomy) 75

2.5. The Standard Type Theorem 77

2.6. The GF (2) Type Theorem 77

2.7. The Uniqueness Case Theorem 78

Conclusion: The proof of the Characteristic 2 Type Theorem 80

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