Extremely primitive groups and linear spaces

TL;DR

This paper corrects an inaccuracy in the classification of extremely primitive groups and explores their role as automorphism groups of regular linear spaces.

Contribution

It rectifies a previous classification error and investigates the properties of groups acting extremely primitively on points of linear spaces.

Findings

Corrected classification of soluble extremely primitive groups.

Identified properties of groups acting extremely primitively on linear spaces.

Enhanced understanding of the structure of extremely primitive groups.

Abstract

A finite non-regular primitive permutation group $G$ is extremely primitive if a point stabiliser acts primitively on each of its nontrivial orbits. Such groups have been studied for almost a century, finding various applications. The classification of extremely primitive groups was recently completed by Burness and Lee, who relied on an earlier classification of soluble extremely primitive groups by Mann, Praeger and Seress. Unfortunately, there is an inaccuracy in the latter classification. We correct this mistake, and also investigate regular linear spaces which admit groups of automorphisms that are extremely primitive on points.