Regular orbits of sporadic simple groups

Abstract

Given a finite group G and a faithful irreducible FG-module V where F has prime order, does G have a regular orbit on V? This problem is equivalent to determining which primitive permutation groups of affine type have a base of size 2. Let G be a covering group of an almost simple group whose socle T is sporadic, and let V be a faithful irreducible FG-module where F has prime order dividing |G|. We classify the pairs (G,V) for which G has no regular orbit on V, and determine the minimal base size of G in its action on V. To obtain this classification, for each non-trivial g∈G/Z(G), we compute the minimal number of T-conjugates of g generating 〈T,g〉.