Groups with a covering condition on commutators

Given a group G and positive integers k,n, we let Bn=Bn(G) denote the set of all elements with at most n conjugates in G. We say that G satisfies the (k,n)-covering condition for commutators if there is a subset S⊆G such that |S|≤k and all commutators of G are contained in SBn. The importance of groups satisfying this condition was revealed in the recent study of probabilistically nilpotent finite groups of class two. The main result obtained in this paper is the following theorem. Let G be a group satisfying the (k,n)-covering condition for commutators. Then G′ contains a characteristic subgroup W such that [G′:W] and |W′| are both (k,n)-bounded. This extends several earlier results of similar flavor.