LEADER 05023nam 22006253 450 001 9910970915503321 005 20251116153855.0 010 $a9781470472290 010 $a1470472295 035 $a(MiAaPQ)EBC29731927 035 $a(Au-PeEL)EBL29731927 035 $a(CKB)24767778700041 035 $a(OCoLC)1343248790 035 $a(PPN)270358773 035 $a(EXLCZ)9924767778700041 100 $a20220904d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aFactorizations of Almost Simple Groups with a Solvable Factor, and Cayley Graphs of Solvable Groups 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2022. 210 4$dİ2022. 215 $a1 online resource (112 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vv.279 311 08$aPrint version: Li, Cai-Heng Factorizations of Almost Simple Groups with a Solvable Factor, and Cayley Graphs of Solvable Groups Providence : American Mathematical Society,c2022 9781470453831 327 $aCover -- 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. 330 $a"A characterization is given for the factorizations of almost simple groups with a solvable factor. It turns out that there are only several infinite families of these nontrivial factorizations, and an almost simple group with such a factorization cannot have socle exceptional Lie type or orthogonal of minus type. The characterization is then applied to study s-arc-transitive Cayley graphs of solvable groups, leading to a striking corollary that, except for cycles, a non-bipartite connected 3-arc-transitive Cayley graph of a finite solvable group is necessarily a normal cover of the Petersen graph or the Hoffman-Singleton graph"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society 606 $aFinite groups 606 $aGroup theory 606 $aGroup theory and generalizations -- Abstract finite groups -- Products of subgroups$2msc 606 $aGroup theory and generalizations -- Abstract finite groups -- Simple groups: alternating groups and groups of Lie type$2msc 606 $aGroup theory and generalizations -- Abstract finite groups -- Simple groups: sporadic groups$2msc 606 $aCombinatorics -- Algebraic combinatorics -- Group actions on combinatorial structures$2msc 615 0$aFinite groups. 615 0$aGroup theory. 615 7$aGroup theory and generalizations -- Abstract finite groups -- Products of subgroups. 615 7$aGroup theory and generalizations -- Abstract finite groups -- Simple groups: alternating groups and groups of Lie type. 615 7$aGroup theory and generalizations -- Abstract finite groups -- Simple groups: sporadic groups. 615 7$aCombinatorics -- Algebraic combinatorics -- Group actions on combinatorial structures. 676 $a512/.23 676 $a512.23 686 $a20D40$a20D06$a20D08$a05E18$2msc 700 $aLi$b Cai Heng$01801534 701 $aXia$b Binzhou$01801535 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910970915503321 996 $aFactorizations of Almost Simple Groups with a Solvable Factor, and Cayley Graphs of Solvable Groups$94346814 997 $aUNINA