Mathematical Surveys and Monographs
Volume: 40;
1999;
341 pp;
Hardcover
MSC: Primary 20;
Print ISBN: 978-0-8218-1379-9
Product Code: SURV/40.4
List Price: $92.00
AMS Member Price: $73.60
MAA Member Price: $82.80
Electronic ISBN: 978-1-4704-1269-2
Product Code: SURV/40.4.E
List Price: $92.00
AMS Member Price: $73.60
MAA Member Price: $82.80
Supplemental Materials
The Classification of the Finite Simple Groups, Number 4: Part II, Chapters 1–4: Uniqueness Theorems
Share this pageDaniel Gorenstein; Richard Lyons; Ronald Solomon
After three introductory volumes on the classification of the
finite simple groups, (
Two fundamental and powerful theorems in finite group theory are
examined: the Bender-Suzuki theorem on strongly embedded subgroups
(for which the non-character-theoretic part of the proof is provided)
and Aschbacher's Component theorem. Included are new generalizations
of Aschbacher's theorem which treat components of centralizers of
involutions and \(p\)-components of centralizers of elements of
order \(p\) for arbitrary primes \(p\).
This book, with background from sections of the previous volumes,
presents in an approachable manner critical aspects of the
classification of finite simple groups.
Features:
- Treatment of two fundamental and powerful theorems in finite group theory.
- Proofs that are accessible and largely self-contained.
- New results generalizing Aschbacher's Component theorem and related component uniqueness theorems.
Readership
Graduate students and research mathematicians working in group theory and generalizations.
Table of Contents
Table of Contents
The Classification of the Finite Simple Groups, Number 4: Part II, Chapters 1-4: Uniqueness Theorems
- Contents ix10 free
- Preface xiii14 free
- Part II, Chapters 1–4: Uniqueness Theorems 118 free
- Chapter 1. General Lemmas 118
- Chapter 2. Strongly Embedded Subgroups and Related Conditions on Involutions 1936
- 1. Introduction and Statement of Results 1936
- 2. The Subsidiary Theorems 2542
- 3. The Basic Setup and Counting Arguments 2744
- 4. Reduction to the Simple Case: Theorem 1 3350
- 5. p-Subgroups Fixing Two or More Points: Theorem 2 3855
- 6. A Reduction: Corollary 3 4259
- 7. Double Transitivity and X[sub(αββ[sup(z)])]: Theorem 4 4461
- 8. Reductions for Theorem SE 5067
- 9. Good Subgroups of V Exist 5471
- 10. The Structures of V and D 5875
- 11. Proof of Theorem 6 (and Theorem SE) 6279
- 12. The Group L = 〈z,t, [0[sub(2)],(D),t]〈 6582
- 13. The Unitary Case 7289
- 14. The Suzuki Case 7491
- 15. The Linear Case 7996
- 16. The Structure of D and C[sub(x)](u) 8299
- 17. The Proof of Theorem ZD 88105
- 18. The Weak 2-Generated 2-Core 91108
- 19. The J[sub(2)] Case 95112
- 20. Bender Groups as 2-Components 98115
- 21. Theorem SU: The 2-Central Case 101118
- 22. The J[sub(1)] Case 103120
- 23. The 2A[sub(9)] Case 105122
- 24. Theorem SA: Strongly Closed Abelian 2-Subgroups 109126
- 25. Theorem SF: Terminal Bender Components 112129
- 26. Theorem SF: Product Disconnection 119136
- 27. Theorem A[sub(9)] 127144
- Chapter 3. p-Component Uniqueness Theorems 131148
- 1. Introduction 131148
- 2. Theorem PU[sub(1)]: The Subsidiary Results 137154
- 3. Exceptional Triples and K-Group Lemmas 138155
- 4. M-Exceptional Subgroups of G 142159
- 5. Centralizers of Elements of I[sub(P)](K*) 147164
- 6. The Proofs of Theorems 1 and 2 150167
- 7. Theorem 3: The Nonsimple Case 153170
- 8. Theorem 3: The Simple Case 163180
- 9. Theorem 4: Preliminaries 165182
- 10. Normalizers of p-Subgroups of K 168185
- 11. The Case K ∉ S[omitted](p) 171188
- 12. The S[omitted](p) case 172189
- 13. Theorem 5: The M-Exceptional Case 179196
- 14. The Residual M-Exceptional Cases 188205
- 15. Completion of the Proof of Theorem PU[sub(1)]: The p-Rank 1 Case 193210
- 16. Reductions to Theorem PU[sub(1)] 194211
- 17. Further Reductions: The Non-Normal Case 201218
- 18. Theorem PU[sub(2)]: The Setup 206223
- 19. The Case r < p: A Reduction 208225
- 20. The Case r = 1: Conclusion 211228
- 21. The Case r = p = 2: A. Reduction 214231
- 22. The Case r = p = 2, K ≅ L[sub(2)](q) 216233
- 23. Theorem PU[sub(3)]: The Simple Case 220237
- 24. The Residual Simple Cases 223240
- 25. The Nonsimple Case 224241
- 26. Theorem PU[sub(4)] 226243
- 27. Corollaries PU[sub(2)] and PU[sub(4)] 227244
- 28. Aschbacher's Reciprocity Theorem 228245
- 29. Theorem PU[sub(5)] 232249
- Chapter 4. Properties of K-Groups 237254
- 1. Automorphisms 237254
- 2. Schur Multipliers and Covering Groups 238255
- 3. Bender Groups 245262
- 4. Groups of Low 2-Rank 253270
- 5. Groups of Low p-Rank, p Odd 256273
- 6. Centralizers of Elements of Prime Order 257274
- 7. Sylow 2-Subgroups of Specified K-Groups 268285
- 8. Disconnected Groups 271288
- 9. Strongly Closed Abelian Subgroups 276293
- 10. Generation 283300
- 11. Generation and Terminal Components 293310
- 12. Preuniqueness Subgroups and Generation 302319
- 13. 2-Constrained Groups 328345
- 14. Miscellaneous Results 331348
- Background References 333350
- Expository References 334351
- Errata for Number 3, Chapter 1[sub(A)]: Almost Simple K-Groups 335352
- Glossary 338355
- Index of Terminology 340357 free