Regular orbits of sporadic simple groups

摘要

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 vertical bar G vertical bar. 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 is an element of G/Z(G), we compute the minimal number of T-conjugates of g generating T, g . (C) 2018 Elsevier Inc. All rights reserved.