${\mathbb{Z}}_3^8$ is not a CI-group

Abstract

A Cayley graph Cay$(G;S)$ has the CI (Cayley Isomorphism) property if for every isomorphic graph Cay$(G;T)$, there is a group automorphism $\alpha$ of $G$ such that $S^{\alpha }=T$. The DCI (Directed Cayley Isomorphism) property is defined analogously on digraphs. A group $G$ is a CI-group if every Cayley graph on $G$ has the CI property, and is a DCI-group if every Cayley digraph on $G$ has the DCI property. Since a graph is a special type of digraph, this means that every DCI-group is a CI-group, and if a group is not a CI-group then it is not a DCI-group, but there are well-known examples of groups that are CI-groups but not DCI-groups. In 2009, Spiga showed that ${\mathbb{Z}}_3^8$ is not a DCI-group, by producing a digraph that does not have the DCI property. He also showed that ${\mathbb{Z}}_3^5$ is a DCI-group (and therefore also a CI-group). Until recently the question of whether there are elementary abelian $3$-groups that are not CI-groups remained open. In a recent preprint with Dave Witte Morris, we showed that ${\mathbb{Z}}_3^{10}$ is not a CI-group. In this paper we show that with slight modifications, the underlying undirected graph of order $3^8$ described by Spiga is does not have the CI property, so ${\mathbb{Z}}_3^8$ is not a CI-group.

10 pages