Base sizes for primitive groups with soluble stabilisers

Abstract

Let $G$ be a finite primitive permutation group on a set $\mathrm{\Omega }$ with point stabiliser $H$. Recall that a subset of $\mathrm{\Omega }$ is a base for $G$ if its pointwise stabiliser is trivial. We define the base size of $G$, denoted $b\left(G,H\right)$, to be the minimal size of a base for $G$. Determining the base size of a group is a fundamental problem in permutation group theory, with a long history stretching back to the 19th century. Here one of our main motivations is a theorem of Seress from 1996, which states that $b\left(G,H\right)\le 4$ if $G$ is soluble. In this paper we extend Seress’s result by proving that $b\left(G,H\right)\le 5$ for all finite primitive groups $G$ with a soluble point stabiliser $H$. This bound is best possible. We also determine the exact base size for all almost simple groups and we study random bases in this setting. For example, we prove that the probability that $4$ random elements in $\mathrm{\Omega }$ form a base tends to $1$ as $\left|G\right|$ tends to infinity.