# Intersection of conjugate solvable subgroups in symmetric groups

## TL;DR

This paper proves that for certain solvable subgroups within almost simple symmetric groups, there exist conjugates whose intersection is trivial, advancing understanding of subgroup intersections in group theory.

## Contribution

It establishes the existence of conjugates of solvable subgroups in almost simple groups with trivial intersection, a novel result in the structure of symmetric groups.

## Key findings

- Existence of conjugates with trivial intersection
- Applicable to solvable subgroups in symmetric groups
- Enhances understanding of subgroup intersection properties

## Abstract

It is shown that for a solvable subgroup $G$ of an almost simple group $S$ which socle is isomorphic to $A_n$ $ (n\ge5)$ there are $x,y,z,t \in S$ such that $G \cap G^x \cap G^y \cap G^z \cap G^t =1.$