On $n$-partite digraphical representations of finite groups

TL;DR

This paper classifies all finite groups that can be represented as automorphism groups of regular n-partite digraphs with specific symmetry properties, for every positive integer n.

Contribution

It provides a complete classification of finite groups admitting n-partite digraphical representations for all positive integers n.

Findings

Complete classification for all finite groups and n-partite digraphical representations

Identification of structural conditions for groups to admit such representations

Extension of previous work on graph representations of groups

Abstract

A group $G$ admits an \textbf{\em $n$-partite digraphical representation} if there exists a regular $n$-partite digraph $\Gamma$ such that the automorphism group $\mathrm{Aut}(\Gamma)$ of $\Gamma$ satisfies the following properties: $\mathrm{Aut}(\Gamma)$ is isomorphic to $G$, $\mathrm{Aut}(\Gamma)$ acts semiregularly on the vertices of $\Gamma$ and the orbits of $\mathrm{Aut}(\Gamma)$ on the vertex set of $\Gamma$ form a partition into $n$ parts giving a structure of $n$-partite digraph to $\Gamma$. In this paper, for every positive integer $n$, we classify the finite groups admitting an $n$-partite digraphical representation.