On the generalised Saxl graphs of permutation groups

TL;DR

This work extends the Saxl-graph framework to general base sizes $b(G)\ge 2$ by defining the generalised Saxl graph $\Sigma(G)$ and investigating its structure for primitive groups. It develops a probabilistic-graph-theoretic toolkit, uses orbital graphs, and performs detailed case analyses across diagonal-type, almost simple, affine, and wreath-product families to study completeness, arc-transitivity, and a generalized Common Neighbour property. Key contributions include (i) complete or near-complete classifications of semi-Frobenius diagonal-type groups and arc-transitivity criteria for diagonal types, (ii) base-size and neighbour-property results for almost simple groups with sporadic and $L_2(q)$ socles, and soluble stabilisers, as well as (iii) a clear criterion for affine groups and a full treatment of primitive wreath products with explicit reg$(G)$-based conditions. The results significantly advance understanding of how base sizes govern the connectivity and symmetry of generalized Saxl graphs, with concrete classifications and verifiable cases across major O'Nan–Scott families, and establish new connections to probabilistic base-size bounds and orbital graphs.

Abstract

A base for a finite permutation group $G \le \mathrm{Sym}(Ω)$ is a subset of $Ω$ 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 $Ω$, 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 $Ω$ 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.

Theorems & Definitions (112)

• Definition 1.1
• Conjecture 1.2: Common Neighbour Conjecture
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Theorem 1.9
• Lemma 2.1