On the generalised Saxl graphs of permutation groups

TL;DR

This paper introduces a generalized Saxl graph for permutation groups, capturing base extension properties, and investigates its structural features like completeness and arc-transitivity, especially for primitive groups.

Contribution

It extends the Saxl graph concept to encode base extension information for groups with base size at least two, and explores its properties in primitive groups.

Findings

Analyzes completeness and arc-transitivity of the generalized graph.

Examines the generalized graph in the context of primitive groups.

Investigates the extension of the Common Neighbour Conjecture.

Abstract

A base for a finite permutation group $G \le \mathrm{Sym}(\Omega)$ is a subset of $\Omega$ with trivial pointwise stabiliser in $G$, and the base size of $G$ is the smallest size of a base for $G$. Motivated by the interest in groups of base size two, Burness and Giudici introduced the notion of the Saxl graph. This graph has vertex set $\Omega$, with edges between elements if they form a base for $G$. We define a generalisation of this graph that encodes useful information about $G$ whenever $b(G) \ge 2$: here, the edges are the pairs of elements of $\Omega$ that can be extended to bases of size $b(G)$. In particular, for primitive groups, we investigate the completeness and arc-transitivity of the generalised graph, and the generalisation of Burness and Giudici's Common Neighbour Conjecture on the original Saxl graph.