Normal Minimal Cayley Digraphs of Abelian Groups

A Cayley digraph X = Cay(G, S) is said to be normal for G if the regular representation R(G) of G is normal in the full automorphism group Aut(X) of X. A characterization of normal minimal Cayley digraphs for abelian groups is given. In addition, the abelian groups, all of whose minimal Cayley digraphs are normal, are characterized.