Non-Cayley-Isomorphic Cayley graphs from non-Cayley-Isomorphic Cayley digraphs

Abstract

A finite group G is said to be a non-DCI group if there exist subsets S1 and S2 of G, such that the associated Cayley digraphs C−→ay(G; S1) and C−→ay(G; S2) are isomorphic, but no automorphism of G carries S1 to S2. Furthermore, G is said to be a non-CI group if the subsets S1 and S2 can be chosen to be closed under inverses, so we have undirected Cayley graphs Cay(G; S1) and Cay(G; S2). We show that if p is a prime number, and the elementary abelian p- group (Zp)r is a non-DCI group, then (Zp)r+3 is a non-CI group. In most cases, we can also show that (Zp)r+2 is a non-CI group. In particular, from Pablo Spiga’s proof that (Z3)8 is a non-DCI group, we conclude that (Z3)10 is a non-CI group. This is the first example of a non-CI elementary abelian 3-group.