Base sizes for finite linear groups with solvable stabilisers

TL;DR

This work tackles the base-size problem for finite transitive groups with solvable stabilisers, focusing on the conjectured bound $b_S(G)\le 5$ when the solvable radical is trivial. The authors develop a two-pronged strategy: (i) probabilistic fixed-point ratio methods to handle almost simple groups with socles $\mathrm{PSL}_n(q)$, and (ii) a thorough matrix-group (GL/ΓL) analysis of irreducible solvable subgroups, augmented by explicit Singer-cycle constructions and computational checks for small instances. They prove the key result Reg$_S(G,5)\ge 5$ (hence $b_S(G)\le 5$) for almost simple groups with socle $\mathrm{PSL}_n(q)$, providing a strong form of Vdovin’s conjecture in the Lie-type setting and laying groundwork for the remaining classical groups in forthcoming papers. The methods combine probabilistic, constructive, and computational tools, and have implications for normal-subgroup index bounds and broader conjectures of Babai–Goodman–Pyber in the solvable setting. The paper also clarifies and corrects aspects of previous statements, situating the PSL$_n(q)$ case as a foundational step in a three-paper program addressing all finite classical groups.

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.

Theorems & Definitions (89)

• Theorem 1.1
• Remark 1.2
• Lemma 1.3: bay
• Theorem 1.4
• Theorem A1
• Theorem A2
• Theorem 1.6: zen
• Theorem 1.8
• Lemma 2.2
• proof