Digraphs with small automorphism groups that are Cayley on two nonisomorphic groups
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.
 Publication:

arXiv eprints
 Pub Date:
 March 2017
 DOI:
 10.48550/arXiv.1703.02290
 arXiv:
 arXiv:1703.02290
 Bibcode:
 2017arXiv170302290M
 Keywords:

 Mathematics  Combinatorics