Quaternion Fusion Packets

Description

Let p be a prime and S a finite p-group. A p-fusion system on S is a category whose objects are the subgroups of S and whose morphisms are certain injective group homomorphisms. Fusion systems are of interest in modular representation theory, algebraic topology, and local finite group theory. The book provides a characterization of the 2-fusion systems of the groups of Lie type and odd characteristic, a result analogous to the Classical Involution Theorem for groups. The theorem is the most difficult step in a two-part program. The first part of the program aims to determine a large subclass of the class of simple 2-fusion systems, while part two seeks to use the result on fusion systems to simplify the proof of the theorem classifying the finite simple groups. Cover Title page Contents Background and overview Chapter 0. Introduction Chapter 1. The major theorems and some background 1.1. Theorems 1 through 8 1.2. Background 1.3. An outline of the proof Basics and examples Chapter 2. Some basic results 2.1. Preliminary lemmas 2.2. Solvable components 2.3. Intrinsic 𝑆𝐿₂[𝑚]-components 2.4. A sufficient condition for quaternion fusion packets 2.5. Basic results on fusion packets 2.6. The case 𝑧∈𝑍(ℱ). 2.7. ℱ=𝒮𝒪_{𝜏} 2.8. Modules for groups with a strongly embedded subgroup Chapter 3. Results on 𝜏 3.1. Δ(𝜏), 𝜂(𝜏), and 𝜇(𝜏) 3.2. The graph 𝒜 3.3. More basic lemmas 3.4. Generating ℱ Chapter 4. 𝑊(𝜏) and 𝑀(𝜏) 4.1. 3-transposition groups 4.2. The groups in 𝑀(𝜏) 4.3. The groups 𝜔̄(Φ,𝑚) Chapter 5. Some examples 5.1. 𝐴𝐸_{𝑛} 5.2. The 2-share of the order of some groups 5.3. Orthogonal groups and packets 5.4. Linear, unitary, and symplectic groups and packets 5.5. Exceptional groups and packets 5.6. ℱ_{𝒮}(𝒢) is simple 5.7. 𝐿_{𝑑}^{𝜋}[𝑚] and 𝜔̄(𝐴_{𝑑-1},𝑚) 5.8. 𝜔̄(𝐷_{𝑛},𝑚) 5.9. 𝜔̄(𝐶_{𝑛},𝑚) and 2𝜔̄(𝐶_{𝑛},𝑚) 5.10. Some constrained examples 5.11. Summary of basics Theorems 2 through 5 Chapter 6. Theorems 2 and 4 6.1. 𝒟(𝜏)^{𝒸} 6.2. Beginning the case 𝑧∈𝑂₂(ℱ) 6.3. The case 𝐸≰𝑁_{𝐺}(𝐾) 6.4. Subnormal closure 6.5. 𝐹*(ℱ) 6.6. 𝑧 not in 𝑂₂(ℱ) 6.7. The proof of Theorem 2 Chapter 7. Theorems 3 and 5 7.1. Packets of width 1 7.2. 𝒜(𝓏)̸=∅ Coconnectedness Chapter 8. 𝜏^{∘} not coconnected 8.1. 𝒟^{𝒸} disconnected Theorem 6 Chapter 9. Ω=Ω(𝑧) of order 2 9.1. |Ω(𝑧)|=2 9.2. Generation when |Ω(𝑧)|=2 9.3. |Ω(𝑧)|=2 and 𝒵∩𝒪(𝓏)̸={𝓏} 9.4. |Ω(𝑧)|=2 and 𝒟*(𝓏)=𝒟(𝓏) 9.5. |Ω(𝑧)|=2 and 𝜇 isomorphic to 𝑆₄ Chapter 10. |Ω(𝑧)|>2 10.1. |Ω(𝑧)|=4 and 𝜇 isomorphic to \roman{𝑊𝑒𝑦𝑙}(𝐷₄) 10.2. |Ω(𝑧)| large Chapter 11. Some results on generation 11.1. |Ω(𝑧)|=2, 𝜇 isomorphic to \roman{𝑊𝑒𝑦𝑙}(𝐷_{𝑛}), 𝑛≥4 11.2. Generation 11.3. More generation 11.4. Essentials and normal subsystems 11.5. Generating Ω_{𝑑}^{𝜀}[𝑚] 11.6. Generating 𝐴𝐸_{𝑘} Chapter 12. |Ω(𝑧)|=2 and the proof of Theorem 6 12.1. |Ω(𝑧)|=2, 𝜇 isomorphic to \roman{𝑊𝑒𝑦𝑙}(𝐷₄) 12.2. More |Ω(𝑧)|=2 12.3. Completing |Ω(𝑧)|=2 12.4. The proof of Theorem 6 Theorems 7 and 8 Chapter 13. |Ω(𝑧)|=1 and 𝜇 abelian 13.1. Systems with 𝜇 abelian 13.2. Generic systems with 𝜇 abelian 13.3. Symplectic groups and systems 13.4. Linear and unitary groups and systems 13.5. Generating symplectic and linear systems 13.6. Finishing 𝜇 abelian Chapter 14. More generation 14.1. A generation lemma 14.2. A generation lemma for 𝐸₈ Chapter 15. |Ω(𝑧)|=1 and 𝜇 nonabelian 15.1. |Ω(𝑧)|=1 15.2. The case 𝑟>1 15.3. Φ=𝐷_{𝑛} 15.4. Φ=𝐷₄ 15.5. Φ=𝐴_{𝑛} 15.6. Generating linear systems 15.7. Wrapping up Φ=𝐴_{𝑛} 15.8. Φ=𝐸_{𝑛} Theorem 1 and the Main Theorem Chapter 16. Proofs of four theorems 16.1. The proof of Theorem 1 16.2. Proofs of the Main Theorem and Theorems 6, 7, and 8 16.3. Lie fusion packets References and Index Bibliography Index Back Cover