Abelian permutation groups with graphical representations

TL;DR

This paper characterizes abelian permutation groups that serve as automorphism groups of coloured graphs and digraphs, focusing on classes of 2-closed and 2*-closed groups, using basic permutation group properties.

Contribution

It provides the first characterization of 2-closed and 2*-closed abelian permutation groups in terms of fundamental group properties.

Findings

Characterization of abelian automorphism groups of coloured graphs and digraphs.

First description of 2-closed and 2*-closed abelian groups.

Use of classical Schur terminology for group classification.

Abstract

In this paper we characterize permutation groups that are automorphism groups of coloured graphs and digraphs and are abelian as abstract groups. This is done in terms of basic permutation group properties. Using Schur's classical terminology, what we provide is characterizations of the classes of 2-closed and $2*$-closed abelian permutation groups. This is the first characterization concerning these classes since they were defined.