Base sizes for finite linear groups with solvable stabilisers

TL;DR

This paper proves that for a class of finite linear groups with solvable stabilisers, the base size is at most 5, confirming a conjecture for groups with socle isomorphic to PSL_n(q).

Contribution

It establishes the strong form of Vdovin's conjecture for all almost simple groups with socle PSL_n(q), advancing the understanding of base sizes in finite linear groups.

Findings

Confirmed the conjecture for groups with socle PSL_n(q)

Reduced the problem to classical groups for future work

Progressed towards a complete classification of base sizes

Abstract

Let $G$ be a transitive permutation group on a finite set with solvable point stabiliser and assume that the solvable radical of $G$ is trivial. In 2010, Vdovin conjectured that the base size of $G$ is at most 5. Burness proved this conjecture in the case of primitive $G$. The problem was reduced by Vdovin in 2012 to the case when $G$ is an almost simple group. Now the problem is further reduced to groups of Lie type through work of Baykalov and Burness. In this paper, we prove the strong form of the conjecture for all almost simple groups with socle isomorphic to $\PSL_n(q),$ and the remaining classical groups will be handled in two forthcoming papers.