Finite Abelian Groups with the $m$-DCI Property

Abstract

A Cayley digraph $Cay(G,S)$ of a finite group $G$ is isomorphic to another Cayley digraph $Cay(G,T)$ for each automorphism $\sigma$ of $G$. We will call $Cay(G,S)$ a CI-graph if, for each Cayley digraph $Cay(G,T)$, whenever $Cay(G,S)\cong Cay(G,T)$ there exists an automorphism $\sigma$ of $G$ such that $S^{\sigma }=T$. Further, for a positive integer $m$, if all Cayley digraphs of $G$ of out-valency $m$ are CI-graphs, then $G$ is said to have the $m$-DCI property. This paper shows that for any positive integer $m$, if a finite abelian group $G$ has the $m$-DCI property, then all Sylow subgroups of $G$ are homocyclic.