The exponent of the primitive Cayley digraphs on finite Abelian groups

Abstract

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, S) to be primitive when S contains the central elements of G. As an immediate consequence we obtain that a Cayley digraph X ( G, S) on an Abelian group is primitive if and only if S −1 S is a generating set for G. Moreover, it is shown that if a Cayley digraph X ( G, S) on an Abelian group is primitive, then its exponent either is n − l, [ n 2 ], [ n 2 ] − 1 or is not exceeding [ n 3 ] + 1 . Finally, we also characterize those Cayley digraphs 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 circulant matrices.