# The Local Structure of Finite Groups of Characteristic 2 Type

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*Daniel Gorenstein; Richard Lyons*

#### Table of Contents

# Table of Contents

## The Local Structure of Finite Groups of Characteristic 2 Type

- Table of Contents iii4 free
- Acknowledgements vii8 free
- General Introduction 110 free
- Part I: Properties of K-groups and Preliminary Lemmas 1524 free
- Chapter 1. Introduction 1726
- Chapter 2. Decorations of the known simple groups 7180
- Chapter 3. Local subgroups of the known simple groups 91100
- §8. The local structure of classical groups, as seen on their natural modules 91100
- §9. Outer automorphisms of Chevalley groups 102111
- §10. Sylow structure, p-rank, and 2-local p-rank of the known simple groups 110119
- §11. The permutation action of large elementary abelian p-groups on p-components of K-groups 143152
- §12. p-fusion in the known simple groups 158167
- §13. Parabolic subgroups of Chevalley groups 167176
- §14. Subcomponents and pumpups of the known simple groups 172181
- §15. Action of commuting p-elements on subcomponents of known simple groups 204213
- §16. Splitting primes for groups in Chev (2) 211220
- §17. Standard subcomponents of groups in Chev (2) 222231

- Chapter 4. Balance and signalizers 233242
- Chapter 5. Generational properties of K-groups 283292
- Chapter 6. Factorizations 387396
- Chapter 7. Miscellaneous general results and lemmas about K-groups 397406
- Appendix 425434

- Part II: The Trichotomy Theorem 433442
- Chapter 1. Odd standard form 435444
- §1. Introduction 435444
- §2. A preliminary lemma 440449
- §3. The p-constrained case 441450
- §4. Maximal p-components 446455
- §5. Partial standard form 460469
- §6. The characteristic of L 466475
- §7. The alternating case 471480
- §8. The sporadic case 476485
- §9. Quasisimplicity 484493
- §10. The embedding of B: field automorphisms 490499
- §11. The embedding of B: splitting and half-splitting primes 503512
- §12. Neighbors 513522
- §13. Coda 520529

- Chapter 2. Signalizer functors and weak proper 2-generated p- cores 523532
- §1. Introduction 523532
- §2. Balance 528537
- §3. The p- components of 1[sub(b)] 537546
- §4. Signalizer functors and pre-uniqueness subgroups 539548
- §5. Case (IV): existence of extremal elements of B[sup(1)][sub(max)] (G;3) 544553
- §6. Case (IV): some properties of extremal elements 555564
- §7. Case (IV): the existence of a pre-uniqueness subgroup 562571
- §8. Pre-uniqeness 2-local subgroups 568577
- §9. Uniqueness 2-local subgroups for B 571580
- §10. Connected elementary abelian p-subgroups 584593
- §11. Moving around: controlling cores and layers of centralizers of elements of order p 588597
- §12. Moving around: controlling cores and layers of 2-locals 610619
- §13. Completion of the proof of Theorem 2* 628637

- Chapter 3. Almost strongly p-embedded maximal 2-local subgroups 645654

- References 723732