Cayley digraphs of 2-genetic groups of odd prime-power order
Cayley digraphs of 2-genetic groups of odd prime-power order
复制标题
奇数次幂阶 2 遗传群的凯莱有向图
DOI：
10.1016/j.jcta.2016.05.001
复制
发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Zhou Jin-Xin
中科院分区：
文献类型：
--
作者：
Wang Yi;Feng Yan-Quan;Zhou Jin-Xin
如果一个群的每个正规子群都可以由两个元素生成，则该群被称为2-遗传的。设G是pn阶非交换2-遗传群，其中p为奇素数，n为正整数.本文研究奇阶pn的非交换2-遗传群G的连通Cayley有向图Cay（G，S），在Aut（G，S）={α∈ Aut（G）}的情形下，确定了它们的全自同构群A= Aut（Cay（G，S））|S α= S}是p′-群。证明了要么Cay（G，S）正规，即G的右正则表示在A中正规，要么p= 3，5，7，11且A的最大正规p-子群O p（A）的阶为pn − 1且ASL（2，p）≤ A/Φ（O p（A））≤ AGL（2，p）.进一步，分别对p= 3，5，7，11构造了一个阶数和度数最小的非正规Cayley有向图.特别地，上述p= 3，7，11的非正规Cayley有向图的基础图是半弧传递的，它们是第一个构造a阶素幂半弧传递非正规Cayley图的图.
A group is called 2-genetic if each normal subgroup of the group can be generated by two elements. Let G be a non-abelian 2-genetic group of order p n for an odd prime p and a positive integer n. In this paper, we investigate connected Cayley digraphs Cay (G, S) for non-abelian 2-genetic groups G of odd order p n, and determine their full automorphism groups A= Aut (Cay (G, S)) in the case when Aut (G, S)={α∈ Aut (G)| S α= S} is a p′-group. It is shown that either Cay (G, S) is normal, that is, the right regular representation of G is normal in A, or p= 3, 5, 7, 11 and the largest normal p-subgroup O p (A) of A has order p n− 1 with ASL (2, p)≤ A/Φ (O p (A))≤ AGL (2, p). Furthermore, a non-normal Cayley digraph with smallest order and smallest valency is constructed for each p= 3, 5, 7, 11, respectively. In particular, the underlying graphs of the above non-normal Cayley digraphs for p= 3, 7, 11 are half-arc-transitive, and they are the first constructions of half-arc-transitive non-normal Cayley graphs of order a prime-power.