Normal edge-transitive Cayley graphs and Frattini-like subgroups

被引:3
|
作者
Khosravi, Behnam [1 ]
Praeger, Cheryl E. [2 ]
机构
[1] Inst Adv Studies Basic Sci IASBS, Dept Math, Zanjan 4513766731, Iran
[2] Univ Western Australia, Dept Math & Stat, 35 Stirling Highway, Perth, WA 6009, Australia
基金
英国工程与自然科学研究理事会;
关键词
Normal edge-transitive Cayley graphs; Frattini subgroup; Graph constructions; Automorphisms of groups; NON-ABELIAN GROUPS;
D O I
10.1016/j.jalgebra.2021.03.035
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
For a finite group G and an inverse-closed generating set C of G, let Aut(G; C) consist of those automorphisms of G which leave C invariant. We define an Aut(G; C)-invariant normal subgroup Phi(G; C) of G which has the property that, for any Aut(G; C)-invariant normal set of generators for G, if we remove from it all the elements of Phi(G; C), then the remaining set is still an Aut(G; C)-invariant normal generating set for G. The subgroup Phi(G; C) contains the Frattini subgroup Phi(G) but the inclusion may be proper. The Cayley graph Cay(G, C) is normal edge-transitive if Aut(G; C) acts transitively on the pairs {c, c(-1)} from C. We show that, for a normal edge-transitive Cayley graph Cay(G, C), its quotient modulo Phi(G; C) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever we have Phi(G; C) trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all 4-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi. (c) 2021 Elsevier Inc. All rights reserved.
引用
收藏
页码:473 / 498
页数:26
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