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

TL;DR

This paper explores the properties of non-Cayley-Isomorphic Cayley graphs and digraphs, establishing new relationships between non-DCI and non-CI groups, especially for elementary abelian p-groups, with specific examples for p=3.

Contribution

It proves that certain non-DCI elementary abelian p-groups imply the existence of non-CI groups with higher rank, providing the first example for elementary abelian 3-groups.

Findings

If $(Z_p)^r$ is non-DCI, then $(Z_p)^{r+3}$ is non-CI.

Most cases show $(Z_p)^{r+2}$ is non-CI.

$(Z_3)^{10}$ is a non-CI group, first example for p=3.

Abstract

A finite group $G$ is a "non-DCI group" if there exist subsets $S_1$ and $S_2$ of $G$, such that the associated Cayley digraphs $C\overrightarrow{ay}(G;S_1)$ and $C\overrightarrow{ay}(G;S_2)$ are isomorphic, but no automorphism of $G$ carries $S_1$ to $S_2$. Furthermore, $G$ is a "non-CI group" if the subsets $S_1$ and $S_2$ can be chosen to be closed under inverses, so we have undirected Cayley graphs $Cay(G;S_1)$ and $Cay(G;S_2)$. We show that if $p$ is a prime number, and the elementary abelian $p$-group $(\mathbb{Z}_p)^r$ is a non-DCI group, then $(\mathbb{Z}_p)^{r+3}$ is a non-CI group. In most cases, we can also show that $(\mathbb{Z}_p)^{r+2}$ is a non-CI group. In particular, from Pablo Spiga's proof that $(\mathbb{Z}_p)^8$ is a non-DCI group, we conclude that $(\mathbb{Z}_3)^{10}$ is a non-CI group. This is the first example of a non-CI elementary abelian $3$-group.