Generalized Quaternion Groups with the m-DCI Property

A Cayley digraph Cay(G, S) of a finite group G with respect to a subset S of G is said to be a CI-digraph if for every Cayley digraph Cay(G, T) isomorphic to Cay(G, S), there exists an automorphism sigma of G such that S sigma = T. A finite group G is said to have the m-DCI property for some positive integer m if every Cayley digraph Cay(G, S) of G with |S| = m is a CI-digraph, and is said to be a DCIgroup if G has the m-DCI property for all 1 m |G|. Let Q4n be a generalized quaternion group (also called dicyclic group) of order 4n with an integer n 3, and let Q4n have the m-DCI property for some 1 m 2n-1. It is shown in this paper that n is odd, and n is not divisible by p2 for any prime p m-1. Furthermore, if n 3 is a power of a prime p, then Q4n has the m-DCI property if and only if p is odd, and either n = p or 1 m p.