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 ∈ G such that H ∩ H x ∩ 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.