Intersections of conjugate solvable subgroups in classical groups of Lie type

Abstract

In this paper we obtain a partial solution to Problem 17.41 from "Kourovka notebook": if $S$ is a solvable subgroup of a finite group $G$ with trivial solvable radical, do there exist five conjugate of $S$ with trivial intersection. We consider the case, when $G$ is a finite almost simple group with simple socle isomorphic to $L_n^\epsilon(q)$ and $S$ is a solvable subgroup of special type