On base sizes for primitive groups of product type

TL;DR

This paper systematically studies base sizes of primitive groups of product type, providing formulas, classifications, and applications, thereby extending understanding beyond the well-studied primitive groups.

Contribution

It determines base sizes for product type primitive groups with soluble point stabilisers and classifies groups with a unique regular suborbit, extending prior work.

Findings

Derived base size formulas for specific product type groups.

Classified groups with a unique regular suborbit.

Extended results on base sizes for general product type primitive groups.

Abstract

Let $G \leqslant {\rm Sym}(\Omega)$ be a finite permutation group and recall that the base size of $G$ is the minimal size of a subset of $\Omega$ with trivial pointwise stabiliser. There is an extensive literature on base sizes for primitive groups, but there are very few results for primitive groups of product type. In this paper, we initiate a systematic study of bases in this setting. Our first main result determines the base size of every product type primitive group of the form $L \wr P \leqslant {\rm Sym}(\Omega)$ with soluble point stabilisers, where $\Omega = \Gamma^k$, $L \leqslant {\rm Sym}(\Gamma)$ and $P \leqslant S_k$ is transitive. This extends recent work of Burness on almost simple primitive groups. We also obtain an expression for the number of regular suborbits of any product type group of the form $L \wr P$ and we classify the groups with a unique regular suborbit under the assumption that $P$ is primitive, which involves extending earlier results due to Seress and Dolfi. We present applications on the Saxl graphs of base-two product type groups and we conclude by establishing several new results on base sizes for general product type primitive groups.