The isomorphism problem for Cayley digraphs on groups of prime-squared order

Abstract

Given any prime p, there are two non-isomorphic groups of order p². We determine precisely when a Cayley digraph on one of these groups is isomorphic to a Cayley digraph on the other group. Namely, let X = Cay(G:T) be a Cayley digraph on a group G of order p² with generating set T. We prove that X is isomorphic to a Cayley digraph on both $\text{Z}$_p² and $\text{Z}$_p × $\text{Z}$_p if and only if X is a lexicographic product of two Cayley digraphs of order p. Equivalently, there exists a subgroup H of G of order p such that for every t ϵ TβH, we have tH ⊆ T.