THE EXPONENT OF THE PRIMITIVE CAYLEY DIGRAPHS ON FINITE ABELIAN-GROUPS

Let G be finite group and let S be a subset of G. We prove a necessary and sufficient condition for the Cayley digraph X-(G,X-S) to be primi tive when S contains the central elements of G. As an immediate conseq uence we obtain that a Cayley digraph X-(G,X-S) on an Abelian group is primitive if and only if S-1S is a generating set for G. Moreover, it is shown that if a Cayley digraph X-(G,X-S) on an Abelian group is pr imitive, then its exponent either is n-1, [n/2], [n/2] - 1 or is not e xceeding [n/3] + 1. Finally, we also characterize those Cayley digraph s on Abelian groups with exponent n -1,[n/2],[n/2] - 1. In particular, we generalize a number of well-known results for the primitive circul ant matrices.