Cayley digraphs with normal adjacency matrices

In this paper we give a criterion for the adjacency matrix of a Cayley digraph to be normal in terms of the Cayley subset S. It is shown with the use of this result that the adjacency matrix of every Cayley digraph on a finite group G is normal iff G is either abelian or has the form Q8×Z2n for some non-negative integer n, where Q8 is the quaternion group and Z2n is the abelian group of order 2n and exponent 2.