On the Saxl graph of a permutation group

TL;DR

This paper introduces the Saxl graph of a permutation group, exploring its properties like connectivity, diameter, and Hamiltonicity, and presents conjectures and results for various classes of groups.

Contribution

It defines the Saxl graph for permutation groups and investigates its structural properties, providing new insights and conjectures in group theory and graph theory.

Findings

Conjecture: primitive groups with base size 2 have Saxl graph diameter at most 2

Proven the conjecture for certain almost simple groups like $S_n$ and $A_n$

Constructed examples of imprimitive groups with disconnected or large-diameter Saxl graphs

Abstract

Let $G$ be a permutation group on a set $\Omega$. A subset of $\Omega$ is a base for $G$ if its pointwise stabiliser in $G$ is trivial. In this paper we introduce and study an associated graph $\Sigma(G)$, which we call the Saxl graph of $G$. The vertices of $\Sigma(G)$ are the points of $\Omega$, and two vertices are adjacent if they form a base for $G$. This graph encodes some interesting properties of the permutation group. We investigate the connectivity of $\Sigma(G)$ for a finite transitive group $G$, as well as its diameter, Hamiltonicity, clique and independence numbers, and we present several open problems. For instance, we conjecture that if $G$ is a primitive group with a base of size $2$, then the diameter of $\Sigma(G)$ is at most $2$. Using a probabilistic approach, we establish the conjecture for some families of almost simple groups. For example, the conjecture holds when $G=S_n$ or $A_n$ (with $n>12$) and the point stabiliser of $G$ is a primitive subgroup. In contrast, we can construct imprimitive groups whose Saxl graph is disconnected with arbitrarily many connected components, or connected with arbitrarily large diameter.