On the generic family of Cayley graphs of a finite group

TL;DR

This paper introduces a family of Cayley graphs derived from finite groups, characterizes their automorphism groups, and establishes conditions for graph isomorphism based on group isomorphism.

Contribution

It defines a new class of Cayley graphs for Cartesian powers of finite groups and describes their automorphism groups, linking graph isomorphism to group isomorphism.

Findings

Graphs are isomorphic iff the underlying groups are isomorphic.

Automorphism groups are explicitly characterized for abelian and non-abelian groups.

Application to relations with Bergman-Isaacs Theorem on rings with fixed-point-free actions.

Abstract

Let $G$ be a finite group. For each $m>1$ we define the symmetric canonical subset $S=S(m)$ of the Cartesian power $G^m$ and we consider the family of Cayley graphs $\mathscr{G}_m(G)=Cay(G^m,S)$. We describe properties of these graphs and show that for a fixed $m>1$ and groups $G$ and $H$ the graphs $\mathscr{G}_m(G)$ and $\mathscr{G}_m(H)$ are isomorphic if and only if the groups $G$ and $H$ are isomorphic. We describe also the groups of automorphisms $\mathbf{Aut}(\mathscr{G}_m(G))$. It is shown that if $G$ is a non-abelian group, then $\mathbf{Aut}(\mathscr{G}_m(G))\simeq \big(G^m \rtimes \mathbf{Aut}(G)\big)\rtimes D_{m+1}$, where $D_{m+1}$ is the dihedral group of order $2m+2$. If $G$ is an abelian group (with some exceptions for $m=3$), then $\mathbf{Aut}(\mathscr{G}_m(G))\simeq G^m\rtimes \big(\mathbf{Aut}(G)\times S_{m+1}\big)$, where $S_{m+1}$ is the symmetric group of degree $m+1$. As an example of application we discuss relations between Cayley graphs $\mathscr{G}_m(G)$ and Bergman-Isaacs Theorem on rings with fixed-point-free group actions.