On soluble subgroups of sporadic groups

Let G be an almost simple sporadic group and let H be a soluble subgroup of G. In this paper we prove that H ∩ Hx ∩ Hy = 1 for some elements x, y ∈ G, which is equivalent to the bound b(G, H) ⩽ 3 with respect to the base size for the natural action 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.