Automorphism groups of a class of cubic Cayley graphs on symmetric groups

TL;DR

This paper determines the automorphism groups of a specific class of cubic Cayley graphs on symmetric groups, showing they are normal and explicitly describing their automorphism groups for sufficiently large n.

Contribution

It proves that for n ≥ 13, these Cayley graphs are normal and characterizes their full automorphism groups explicitly.

Findings

$ ext{Aut}( ext{Cay}(S_n,S))$ is isomorphic to $S_n times bZ_2$ for $n geq 13$

The Cayley graphs are normal for all $n geq 13$

Explicit automorphism group structure provided for large $n$

Abstract

Let $S_n$ denote the symmetric group of degree $n$ with $n\geq 3$. Set $S=\{c_n=(1\ 2\ldots \ n),c_n^{-1},(1\ 2)\}$. Let $\Gamma_n=\mathrm{Cay}(S_n,S)$ be the Cayley graph on $S_n$ with respect to $S$. In this paper, we show that $\Gamma_n$ ($n\geq 13$) is a normal Cayley graph, and that the full automorphism group of $\Gamma_n$ is equal to $\mathrm{Aut}(\Gamma_n)=R(S_n)\rtimes \langle\mathrm{Inn}(\phi)\rangle\cong S_n\rtimes \mathbb{Z}_2$, where $R(S_n)$ is the right regular representation of $S_n$, $\phi=(1\ 2)(3\ n)(4\ n-1)(5\ n-2)\cdots$ $(\in S_n)$, and $\mathrm{Inn}(\phi)$ is the inner isomorphism of $S_n$ induced by $\phi$.