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, S-sigma) 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) congruent to Cay(G, T) there exists an automorphism a of G such that S-sigma = T. Further, for a positive integer m, if all Cayley digraphs of G of out-valency in 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.