Regular orbits of quasisimple linear groups II

TL;DR

This paper investigates the minimal base size of certain linear groups with a quasisimple subgroup, aiming to classify those that have a regular orbit on the vector space.

Contribution

It provides a classification of groups with a regular orbit on the vector space, focusing on groups with a quasisimple subgroup of type PSL in defining characteristic.

Findings

Identifies conditions for groups to have a regular orbit

Classifies groups with minimal base size 1 in this setting

Advances understanding of group actions on vector spaces

Abstract

Let $V$ be a finite-dimensional vector space over a finite field, and suppose $G \leq \Gamma \mathrm{L}(V)$ is a group with a unique subnormal quasisimple subgroup $E(G)$ that is absolutely irreducible on $V$. A base for $G$ is a set of vectors $B\subseteq V$ with pointwise stabiliser $G_B=1$. If $G$ has a base of size 1, we say that it has a regular orbit on $V$. In this paper we investigate the minimal base size of groups $G$ with $E(G)/Z(E(G)) \cong \mathrm{PSL}_n(q)$ in defining characteristic, with an aim of classifying those with a regular orbit on $V$.