Finite Abelian Groups with the \(m\)-DCI Property

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.