On CI-property of normal Cayley digraphs over abelian groups

Abstract

A Cayley digraph [Formula: see text] over a finite group [Formula: see text] is said to be CI if for every Cayley digraph [Formula: see text] over [Formula: see text] isomorphic to [Formula: see text], there is an isomorphism from [Formula: see text] to [Formula: see text] which is at the same time an automorphism of [Formula: see text]. In this paper, we study a CI-property of normal Cayley digraphs over abelian groups, i.e. such Cayley digraphs [Formula: see text] that the group [Formula: see text] of all right translations of [Formula: see text] is normal in [Formula: see text]. At first, we reduce the case of an arbitrary abelian group to the case of an abelian [Formula: see text]-group. Further, we obtain several results on CI-property of normal Cayley digraphs over abelian [Formula: see text]-groups. In particular, we prove that every normal Cayley digraph over an abelian [Formula: see text]-group of order at most p 5 , where [Formula: see text] is an odd prime, is CI.