关于有限群关于二价的定向$m$ -半正则表示
On oriented $m$-semiregular representations of finite groups about valency two
摘要：
给定一个群$G$，{ \em $m$ -Cayley有向图$\G$ over $G$ }是一个有向图，它有一组同构于$G$的自同构，它们半正则地作用在具有$m$轨道的顶点集上。如果存在$G$上的正则$m$-Cayley有向图$\G$，使得$\G$是定向的，并且它的自同构群与$G$同构，则称$G$允许{\em定向的$m$ -半正则表示}（简称O $m$ SR）.特别地，O $1$ SR也称为ORR。Verret和Xia在[Ars Math. Contemp. 22（2022），\#P1.07]。令$m\geq 2$为整数。本文证明了除四个小阶群外，所有至多由两个元素生成的有限群都允许O $m$ SR为二价.从而得到了允许O $m$ SR为二价的有限单群的分类.
英文摘要：
Given a group $G$, an {\em $m$-Cayley digraph $\G$ over $G$} is a digraph that has a group of automorphisms isomorphic to $G$ acting semiregularly on the vertex set with $m$ orbits. We say that $G$ admits an {\em oriented $m$-semiregular representation} (O$m$SR for short), if there exists a regular $m$-Cayley digraph $\G$ over $G$ such that $\G$ is oriented and its automorphism group is isomorphic to $G$. In particular, O$1$SR is also named as ORR. Verret and Xia gave a classification of finite simple groups admitting an ORR of valency two in [Ars Math. Contemp. 22 (2022), \#P1.07]. Let $m\geq 2$ be an integer. In this paper, we show that all finite groups generated by at most two elements admit an O$m$SR of valency two except four groups of small orders. Consequently, a classification of finite simple groups admitting an O$m$SR of valency two is obtained.
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