Factorizations of Almost Simple Groups With a Solvable Factor, and Cayley Graphs of Solvable Groups

✍ Scribed by Cai-Heng Li, Binzhou Xia

• Publisher: American Mathematical Society
• Year: 2022
• Tongue: English
• Leaves: 112
• Series: Memoirs of the American Mathematical Society, 279
• Category: Library

✦ Synopsis

Characterizing factorizations of almost simple groups with a solvable factor, Li and Xia conclude that there are only several infinite families of these non-trivial factorizations, and an almost simple group with such a factorization cannot have socle exceptional Lie type or orthogonal of minus type. They apply the characterization to study s-arc-transitive Cayley graphs of solvable groups, leading to a striking corollary that, except for cycles, a non-bipartite connected 3-arc transitive Cayley graph of a finite solvable group is necessarily a normal cover the Petersen graph or the Hoffman-Singleton graph. Annotation ©2022 Ringgold, Inc., Portland, OR (protoview.com)

✦ Table of Contents

Cover
Title page
Chapter 1. Introduction
1.1. Factorizations of almost simple groups
1.2. 𝑠-Arc-transitive Cayley graphs
1.3. Discussions and some open problems
Chapter 2. Preliminaries
2.1. Notation
2.2. Results on finite simple groups
2.3. Elementary facts concerning factorizations
2.4. Maximal factorizations of almost simple groups
Chapter 3. The factorizations of linear and unitary groups of prime dimension
3.1. Singer cycles
3.2. Linear groups of prime dimension
3.3. Unitary groups of prime dimension
Chapter 4. Non-classical groups
4.1. The case that both factors are solvable
4.2. Exceptional groups of Lie type
4.3. Alternating group socles
4.4. Sporadic group socles
Chapter 5. Examples in classical groups
5.1. Examples in unitary groups
5.2. Examples in symplectic groups
5.3. Examples in orthogonal groups of odd dimension
5.4. Examples in orthogonal groups of plus type
Chapter 6. Reduction for classical groups
6.1. Inductive hypothesis
6.2. The case that 𝐴 has at least two non-solvable composition factors
Chapter 7. Proof of Theorem 1.1
7.1. Linear groups
7.2. Symplectic Groups
7.3. Unitary Groups
7.4. Orthogonal groups of odd dimension
7.5. Orthogonal groups of even dimension
7.6. Completion of the proof
Chapter 8. 𝑠-Arc-transitive Cayley graphs of solvable groups
8.1. Preliminaries
8.2. A property of finite simple groups
8.3. Reduction to affine and almost simple groups
8.4. Proof of Theorem 1.3 and Corollary 1.5
Appendix A. Tables for nontrivial maximal factorizations of almost simple classical groups
Bibliography
Back Cover