Normal Cayley digraphs of generalized quaternion groups with CI-property

Abstract

A Cayley digraph Cay(G,S) of a finite group G with respect to a subset S of G, where S does not contain the identity 1 of G, is said to be a CI-digraph, if Cay(G,S)≅Cay(G,T) implies that G has an automorphism mapping S to T. The group G is called a DCI-group or an NDCI-group if all Cayley digraphs or normal Cayley digraphs of G are CI-digraphs. We prove in this paper that a generalized quaternion group Q4n of order 4n is an NDCI-group if and only if n=2 or n is odd. As a result, we show that if Q4n is a DCI-group then n=2 or n is odd-square-free.