Intersection of conjugate solvable subgroups in finite classical groups

TL;DR

This paper proves that for almost simple groups with socles of classical types, five conjugates of a solvable subgroup with no nontrivial solvable normal subgroup intersect trivially, addressing a problem posed by Vdovin.

Contribution

It provides a positive solution to Vdovin's problem for almost simple groups with classical socles, advancing understanding of subgroup intersections in finite groups.

Findings

Confirmed the conjecture for classical groups

Established bounds on subgroup intersections

Connected the problem to broader group theory conjectures

Abstract

We consider the following problem stated by Vdovin (2010) in the "Kourovka notebook" (Problem 17.41): Let $H$ be a solvable subgroup of a finite group $G$ that has no nontrivial solvable normal subgroups. Do there always exist five conjugates of $H$ whose intersection is trivial? This problem is closely related to a conjecture by Babai, Goodman and Pyber (1997) about an upper bound for the index of a normal solvable subgroup in a finite group. In particular, a positive answer to Vdovin's problem yields that if $G$ has a solvable subgroup of index $n$, then it has a solvable normal subgroup of index at most $n^5$. The problem was reduced by Vdovin (2012) to the case when $G$ is an almost simple group. Let $G$ be an almost simple group with socle isomorphic to a simple linear, unitary or symplectic group. For all such groups $G$ we provide a positive answer to Vdovin's problem.