On oriented $m$-semiregular representations of finite groups about valency two

TL;DR

This paper proves that most finite groups generated by up to two elements have an oriented $m$-semiregular representation of valency two, extending previous classifications for simple groups and identifying exceptions.

Contribution

It generalizes the classification of finite simple groups admitting an ORR of valency two to all groups generated by at most two elements, with four small exceptions.

Findings

Most finite groups generated by ≤2 elements admit an O$m$SR of valency two.

A complete classification of simple groups with such representations is provided.

Four small-order groups are exceptions to the general result.

Abstract

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.