On soluble subgroups of sporadic groups

TL;DR

This paper proves that for any soluble subgroup of an almost simple sporadic group, there exist conjugates intersecting trivially, establishing a tight bound related to the base size and confirming a conjecture by Vdovin.

Contribution

It demonstrates a precise bound on the intersection of conjugate soluble subgroups in sporadic groups, confirming a conjecture and combining computational and probabilistic methods.

Findings

Bound on the intersection of conjugate soluble subgroups is at most 3.

The result confirms Vdovin's conjecture in the context of sporadic groups.

The proof employs both computational and probabilistic techniques.

Abstract

Let $G$ be an almost simple sporadic group and let $H$ be a soluble subgroup of $G$. In this paper we prove that there exists $x,y \in G$ such that $H \cap H^x \cap H^y=1$, which is equivalent to the bound $b(G,H) \leqslant 3$ with respect to the base size of $G$ on the set of cosets of $H$. This bound is best possible. In this setting, our main result establishes a strong form of a more general conjecture of Vdovin on the intersection of conjugate soluble subgroups of finite groups. The proof uses a combination of computational and probabilistic methods.