On CI-property of normal Cayley digraphs over abelian groups

A Cayley digraph Γ over a finite group G is said to be CI if for every Cayley digraph Γ^ over G isomorphic to Γ, there is an isomorphism from Γ to Γ^ which is at the same time an automorphism of G. In the present paper, we study a CI-property of normal Cayley digraphs over abelian groups, i.e. such Cayley digraphs Γ that the group Gᵣ of all right translations of G is normal in Aut(Γ). At first, we reduce the case of an arbitrary abelian group to the case of an abelian p-group. Further, we obtain several results on CI-property of normal Cayley digraphs over abelian p-groups. In particular, we prove that every normal Cayley digraph over an abelian p-group of order at most p⁵, where p is an odd prime, is CI.