# Digraphs with small automorphism groups that are Cayley on two nonisomorphic groups

## TL;DR

This paper investigates the automorphism groups of Cayley digraphs on abelian groups, showing that noncyclic groups of certain orders can have digraphs with small automorphism groups and nonabelian regular subgroups.

## Contribution

It provides new examples of Cayley digraphs on abelian groups with small automorphism groups and nonabelian regular subgroups, contrasting previous results on cyclic groups.

## Key findings

- Cayley digraphs on certain abelian groups can have Cayley index as low as p
- Existence of Cayley digraphs with nonabelian regular subgroups and small automorphism groups
- Contrasts with prior results on cyclic p-groups where Cayley index is superexponential

## Abstract

Let $\Gamma=\mathrm{Cay}(G,S)$ be a Cayley digraph on a group $G$ and let $A=\mathrm{Aut}(\Gamma)$. The Cayley index of $\Gamma$ is $|A:G|$. It has previously been shown that, if $p$ is a prime, $G$ is a cyclic $p$-group and $A$ contains a noncyclic regular subgroup, then the Cayley index of $\Gamma$ is superexponential in $p$. We present evidence suggesting that cyclic groups are exceptional in this respect. Specifically, we establish the contrasting result that, if $p$ is an odd prime and $G$ is abelian but not cyclic, and has order a power of $p$ at least $p^3$, then there is a Cayley digraph $\Gamma$ on $G$ whose Cayley index is just $p$, and whose automorphism group contains a nonabelian regular subgroup.