Pentavalent symmetric graphs admitting transitive non-abelian characteristically simple groups

TL;DR

This paper investigates the normality of automorphism groups in connected pentavalent symmetric graphs, establishing conditions under which these graphs are G-normal when G is a product of simple groups, with exceptions linked to specific simple groups.

Contribution

It proves that G-normality in pentavalent symmetric graphs extends from simple groups to their direct products, identifying exceptions related to certain simple groups.

Findings

G-normality holds for most simple groups in pentavalent symmetric graphs.

Exceptions occur for 57, 20, and 17 specific simple groups in different graph symmetry contexts.

The results unify the understanding of automorphism group normality in these graphs.

Abstract

Let $\Gamma$ be a graph and let $G$ be a group of automorphisms of $\Gamma$. The graph $\Gamma$ is called $G$-normal if $G$ is normal in the automorphism group of $\Gamma$. Let $T$ be a finite non-abelian simple group and let $G = T^l$ with $l\geq 1$. In this paper we prove that if every connected pentavalent symmetric $T$-vertex-transitive graph is $T$-normal, then every connected pentavalent symmetric $G$-vertex-transitive graph is $G$-normal. This result, among others, implies that every connected pentavalent symmetric $G$-vertex-transitive graph is $G$-normal except $T$ is one of $57$ simple groups. Furthermore, every connected pentavalent symmetric $G$-regular graph is $G$-normal except $T$ is one of $20$ simple groups, and every connected pentavalent $G$-symmetric graph is $G$-normal except $T$ is one of $17$ simple groups.