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

TL;DR

This paper classifies finite groups generated by at most two elements that admit an oriented $m$-semiregular representation of valency three, extending previous work on valency two and focusing on a specific group class.

Contribution

It provides a classification of finite 2-generated groups admitting an oriented $m$-semiregular representation of valency three for $m \,\geq\, 2$, filling a gap in the existing literature.

Findings

Classified finite 2-generated groups with OmSR of valency 3

Extended understanding of group actions on digraphs

Builds on prior classifications for valency 2

Abstract

Let $G$ be a group and $m$ a positive integer. We say an $m$-Cayley digraph $\Gamma$ over $G$ is a digraph that admits a group of automorphisms isomorphic to $G$ acting semiregularly on the vertex set with $m$ orbits. The digraph $\Sigma$ is $k$-regular if there exists a non-negative integer $k$ such that every vertex has out-valency and in-valency equal to $k$. All digraphs considered in this paper are regular. We say that $G$ admits an oriented $m$-semiregular representation (abbreviated as OmSR) if there exists a regular $m$-Cayley digraph $\Gamma$ over $G$ such that $\Gamma$ is oriented and its automorphism group is isomorphic to $G$. In particular, an O1SR is called an ORR. Xia et al. \cite{x2} provided a classification of finite simple groups admitting an ORR of valency 2. Furthermore, in 2022, Du et al. \cite{du2} proved that most finite simple groups admit an OmSR of valency 2 for $m \geq 2$, except for a few exceptional cases. In this paper, we classify the finite groups generated by at most two elements that admit an OmSR of valency 3 for $m \geq 2$.