On soluble subgroups of sporadic groups

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 $b(G,H)⩽3$ 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.

18 pages; to appear in the Israel Journal of Mathematics