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.
Author(s): Michael Aschbacher
Series: Contemporary Mathematics, 765
Publisher: American Mathematical Society
Year: 2022
Language: English
Pages: 455
City: Providence
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 ??₂[?]-components
2.4. A sufficient condition for quaternion fusion packets
2.5. Basic results on fusion packets
2.6. The case ?∈?(ℱ).
2.7. ℱ=??_{?}
2.8. Modules for groups with a strongly embedded subgroup
Chapter 3. Results on ?
3.1. Δ(?), ?(?), and ?(?)
3.2. The graph ?
3.3. More basic lemmas
3.4. Generating ℱ
Chapter 4. ?(?) and ?(?)
4.1. 3-transposition groups
4.2. The groups in ?(?)
4.3. The groups ?̄(Φ,?)
Chapter 5. Some examples
5.1. ??_{?}
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_{?}(?) is simple
5.7. ?_{?}^{?}[?] and ?̄(?_{?-1},?)
5.8. ?̄(?_{?},?)
5.9. ?̄(?_{?},?) and 2?̄(?_{?},?)
5.10. Some constrained examples
5.11. Summary of basics
Theorems 2 through 5
Chapter 6. Theorems 2 and 4
6.1. ?(?)^{?}
6.2. Beginning the case ?∈?₂(ℱ)
6.3. The case ?≰?_{?}(?)
6.4. Subnormal closure
6.5. ?*(ℱ)
6.6. ? not in ?₂(ℱ)
6.7. The proof of Theorem 2
Chapter 7. Theorems 3 and 5
7.1. Packets of width 1
7.2. ?(?)̸=∅
Coconnectedness
Chapter 8. ?^{∘} not coconnected
8.1. ?^{?} disconnected
Theorem 6
Chapter 9. Ω=Ω(?) of order 2
9.1. |Ω(?)|=2
9.2. Generation when |Ω(?)|=2
9.3. |Ω(?)|=2 and ?∩?(?)̸={?}
9.4. |Ω(?)|=2 and ?*(?)=?(?)
9.5. |Ω(?)|=2 and ? isomorphic to ?₄
Chapter 10. |Ω(?)|>2
10.1. |Ω(?)|=4 and ? isomorphic to \roman{????}(?₄)
10.2. |Ω(?)| large
Chapter 11. Some results on generation
11.1. |Ω(?)|=2, ? isomorphic to \roman{????}(?_{?}), ?≥4
11.2. Generation
11.3. More generation
11.4. Essentials and normal subsystems
11.5. Generating Ω_{?}^{?}[?]
11.6. Generating ??_{?}
Chapter 12. |Ω(?)|=2 and the proof of Theorem 6
12.1. |Ω(?)|=2, ? isomorphic to \roman{????}(?₄)
12.2. More |Ω(?)|=2
12.3. Completing |Ω(?)|=2
12.4. The proof of Theorem 6
Theorems 7 and 8
Chapter 13. |Ω(?)|=1 and ? abelian
13.1. Systems with ? abelian
13.2. Generic systems with ? 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 ? abelian
Chapter 14. More generation
14.1. A generation lemma
14.2. A generation lemma for ?₈
Chapter 15. |Ω(?)|=1 and ? nonabelian
15.1. |Ω(?)|=1
15.2. The case ?>1
15.3. Φ=?_{?}
15.4. Φ=?₄
15.5. Φ=?_{?}
15.6. Generating linear systems
15.7. Wrapping up Φ=?_{?}
15.8. Φ=?_{?}
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
