Cayley graphs of more than one abelian group

TL;DR

This paper investigates when Cayley digraphs of cyclic groups are also isomorphic to Cayley digraphs of other abelian groups, providing conditions and criteria for such isomorphisms.

Contribution

It introduces a reduction method to determine isomorphism of Cayley digraphs across different abelian groups and offers necessary and sufficient conditions for these isomorphisms.

Findings

A reduction criterion involving automorphism groups is established.

Necessary and sufficient conditions for isomorphism are provided.

An easy-to-check necessary condition for such isomorphisms is introduced.

Abstract

We show that for certain integers $n$, the problem of whether or not a Cayley digraph $\Gamma$ of $\mathbb Z_n$ is also isomorphic to a Cayley digraph of some other abelian group $G$ of order $n$ reduces to the question of whether or not a natural subgroup of the full automorphism group contains more than one regular abelian group up to isomorphism (as opposed to the full automorphism group). A necessary and sufficient condition is then given for such circulants to be isomorphic to Cayley digraphs of more than one abelian group, and an easy-to-check necessary condition is provided.