Regular orbits of quasisimple linear groups I

TL;DR

This paper determines the minimal base size and existence of regular orbits for certain linear groups with a unique quasisimple subgroup, especially when the subgroup's quotient is a finite simple Lie type group in cross-characteristic.

Contribution

It establishes the minimal base size and regular orbit existence for groups with a quasisimple subgroup acting irreducibly, extending understanding in the representation theory of linear groups.

Findings

G has a regular orbit on V in most cases.

The minimal base size is explicitly determined for the groups considered.

Identifies specific exceptions where the base size differs.

Abstract

Let $G \leq \mathrm{GL}(V)$ be a group with a unique subnormal quasisimple subgroup $E(G)$ that acts absolutely irreducibly on $V$. A base for $G$ acting on $V$ is a set of vectors with trivial pointwise stabiliser in $G$. In this paper we determine the minimal base size of $G$ when $E(G)/Z(E(G))$ is a finite simple group of Lie type in cross-characteristic. We show that $G$ has a regular orbit on $V$, with specific exceptions, for which we find the base size.