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

• Record Nr.: UNINA9910970915503321
• Autore: Li Cai Heng
• Pubbl/distr/stampa: Providence : , : American Mathematical Society, , 2022; ©2022
• ISBN: 9781470472290; 1470472295
• Edizione: [1st ed.]
• Descrizione fisica: 1 online resource (112 pages)
• Collana: Memoirs of the American Mathematical Society ; ; v.279
• Classificazione: 20D4020D0620D0805E18
• Altri autori (Persone): XiaBinzhou
• Disciplina: 512/.23; 512.23
• Soggetti: Finite groups; Group theory; Group theory and generalizations -- Abstract finite groups -- Products of subgroups; Group theory and generalizations -- Abstract finite groups -- Simple groups: alternating groups and groups of Lie type; Group theory and generalizations -- Abstract finite groups -- Simple groups: sporadic groups; Combinatorics -- Algebraic combinatorics -- Group actions on combinatorial structures
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di contenuto: Cover -- Title page -- Chapter 1. Introduction -- 1.1. Factorizations of almost simple groups -- 1.2. -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 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. -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.
• Sommario/riassunto: "A characterization is given for the factorizations of almost simple groups with a solvable factor. It turns out that there are only several infinite families of these nontrivial factorizations, and an almost simple group with such a factorization cannot have socle exceptional Lie type or orthogonal of minus type. The characterization is then applied 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 of the Petersen graph or the Hoffman-Singleton graph"--