Maximal Cocliques in the Generating Graphs of the Alternating and Symmetric Groups

TL;DR

This paper characterizes when intransitive maximal subgroups form maximal cocliques in the generating graphs of symmetric and alternating groups, and proves a conjecture relating element neighborhoods to maximal subgroups for prime n.

Contribution

It determines conditions for intransitive maximal subgroups to be maximal cocliques and proves a conjecture linking element neighborhoods to maximal subgroups in these groups.

Findings

Intransitive maximal subgroups can form maximal cocliques under specific conditions.

Two elements share the same neighborhood in the generating graph iff they are in the same maximal subgroups.

The conjecture holds for prime n not of a specific form involving prime powers.

Abstract

The generating graph $\Gamma(G)$ of a finite group $G$ has vertex set the non-identity elements of $G$, with two elements connected exactly when they generate $G$. A coclique in a graph is an empty induced subgraph, so a coclique in $\Gamma(G)$ is a subset of $G$ such that no pair of elements generate $G$. A coclique is maximal if it is contained in no larger coclique. It is easy to see that the non-identity elements of a maximal subgroup of $G$ form a coclique in $\Gamma(G)$, but this coclique need not be maximal. In this paper we determine when the intransitive maximal subgroups of $\textrm{S}_n$ and $\textrm{A}_n$ are maximal cocliques in the generating graph. In addition, we prove a conjecture of Cameron, Lucchini, and Roney-Dougal [3] in the case of $G = \textrm{A}_n$ and $\textrm{S}_n$, when n is prime and $n \neq \frac{(q^d -1)}{(q-1)}$ for all prime powers $q$ and $d \geq 2$. Namely, we show that two elements of $G$ have identical sets of neighbours in $\Gamma(G)$ if and only if they belong to exactly the same maximal subgroups.