The Classification of the Finite Simple Groups, Number 10 10

This book is the tenth in a series of volumes whose aim is to provide a complete proof of the classification theorem for the finite simple groups based on a fairly short and clearly enumerated set of background results. Specifically, this book completes our identification of the simple groups of bicharacteristic type begun in the ninth volume of the series (see SURV/40.9). This is a fascinating set of simple groups which have properties in common with matrix groups (or, more generally, groups of Lie type) defined both over fields of characteristic 2 and over fields of characteristic 3. This set includes 11 of the celebrated 26 sporadic simple groups along with several of their large simple subgroups. Together with SURV/40.9, this volume provides the first unified treatment of this class of simple groups. Cover Title page Contents Preface Chapter 9. General Group-Theoretic Lemmas 1. Fusion 2. Groups X with e(X)≤3 3. Signalizer Functors 4. Balance 5. p-Groups 6. p-Components 7. Standard Components of Involution Centralizers 8. Coprime Action 9. Generation 10. Subgroups of GL4(3) 11. Strong p-Embedding 12. Characteristic-p Representations 13. Miscellaneous Chapter 10. Theorems C6 and C6* 1. Introduction 2. T_{p}-Groups and TG_{p}-Groups in L_{p}^{o}(G) 3. Centralizers with T_{p}-Components 4. Balance and Generation 5. Theorem 3: 3/2-Balance 6. Theorem 3: 5/2-Balance and 3-Balance 7. Nonquasisimple Pumpups of K 8. Nontrivial Signalizer Functors 9. The Subgroup M 10. K Lies in M 11. The Pumpup of K in M 12. The L-Preuniqueness of M: the Nonsimple Case 13. The L-Preuniqueness of M: the Simple Rank 2 Case 14. The Rank 1 Case: 3/2-Balance 15. The Rank 1 Case: Controlling Cores 16. The L-Preuniqueness of M: the Rank 1 Case 17. Theorem 4: Almost Strong p-Embedding, the General Case 18. The Residual Cases 19. The Case O_{p’}(M)=1 Chapter 11. Theorems C4 and C4*: Introduction 1. Theorem C4* and Its Cases 2. The Stages of Theorem C4* (Case A) 3. Some Definitions Chapter 12. Theorem C4*: Stage A1. First Steps 1. Introduction 2. Initial Remarks 3. An Exceptional Configuration Involving L3(4) 4. An Exceptional Configuration Involving L2(p^{p}) 5. Balance with Respect to P0 6. The Subgroup M, and the Second Case of Balance 7. Balance in the Remaining Cases 8. The Residual Case 9. B-Signalizers and I_{p}^{o}(G): The Centralizing Case 10. The Non-Centralizing Case: A Dichotomy 11. The Case B_{T}∩I_{p}^{o}(G)=∅ 12. The M12 Case 13. The L3(4^{a}) Case 14. Completion of the Proof Chapter 13. Theorem C4*: Stage A2. Nonconstrained p-Rank 3 Centralizers 1. Introduction 2. Components and p-Components 3. Theorem 1: Introduction 4. Some Nonsimple Cases 5. 2F4(√2), and Preliminaries for 2F4(√32), Sp4(8), and 3D4(2) 6. The L2(8) and 2B2(√32) Cases 7. Some Cases of p-Rank 2 8. Configurations Involving M12 and HJ 9. L2(q), q∈{5,7,17} 10. The Remaining Cases of Theorem 1 11. Theorem 2: Introduction 12. Components with Outer p-Automorphisms 13. K Is Not J3 14. From U4(2) to 2Sp6(2) 15. K≅2Sp6(2) Implies G≅Co3 16. The Z2×Sp6(2) Case Chapter 14. Theorem C4*: Stage A3. KM-Singularities 1. Introduction 2. Centralizers of Noncyclic p-Groups 3. KM-Singularities of Rank 2 and Type A6: The Case H^{inv}≠∅ 4. The Case H^{inv}=∅: {2,3}-Local Subgroups 5. p=3; L2(3n) 3-Components 6. 2-Subgroups of G Normalized by E_{33}-Subgroups 7. Subgroups H∈H and Involution Centralizers 8. The Residual Cases Chapter 15. Theorem C4*: Stage A4. Setups for Recognizing G 1. Introduction 2. Pumpups of a Rank 2 KM-Singularity of Type A6 3. Centralizers of 2-Central Involutions Chapter 16. Theorem C4*: Stage A5. Recognition 1. Introduction 2. The Extraspecial Setup 3. The 32U4(3) and G2(3) Cases: G≅Suz or F3 4. The U4(2) Setup 5. 3-Local Structure 6. G≅U5(2) 7. G≅U6(2) 8. G≅D4(2) 9. G≅Co2 10. The A9–Sp6(2) Setup 11. 3-Local Structure 12. The Mixed Case 13. The A9 Case 14. G≅A12 15. G≅F5 16. The Sp6(2) Case 17. G≅Sp8(2) 18. G≅F4(2) Chapter 17. Properties of K-Groups 1. Simple Sections of G, e(G)=3 2. Flat TG_{p}-Groups 3. Groups of Low p-Rank, p Odd 4. Balance and Signalizers 5. Generation 6. The Balance-Generation Dichotomy 7. C_{p}-Groups, p odd 8. T_{p}-Groups and TG_{p}-Groups 9. C2-Groups 10. Some Familiar C2-Groups and Their Subgroups 10.1. O8+(2) 10.2. Sp6(2) 10.3. U6(2) 10.4. L6(2) 10.5. 2F4(q)’ 10.6. L3(4) and L3(4n) 10.7. 2B2(2^{n/2}), n≥3 10.8. Sp4(2)≅Σ6 10.9. U4(2)≅PSp4(3) 10.10. L4^{±}(3) 10.11. M12, HJ 10.12. M22 10.13. J3 10.14. Suz 11. Some Other Familiar Groups 11.1. L2(q), q odd 11.2. L2(27) 11.3. (S)L3^{ε}(q) 11.4. L3(4) and Fi22 11.5. A9 11.6. A10 12. Sylow Subgroups 13. Pumpups and Subcomponents 14. Cross-Characteristic “Pumpups” 15. Connecting p-Groups 16. Small Representations 17. Elements of Order 2p 18. p-Ranks 19. p-Components 20. Miscellaneous Bibliography Index Back Cover Presents the analysis of a minimal counterexample $G$ to the theorem. This book examines 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.