Groups with a covering condition on commutators

Given a group G and positive integers [Formula: see text], we let [Formula: see text] denote the set of all elements with at most n conjugates in [Formula: see text] We say that G satisfies the [Formula: see text]-covering condition for commutators if there is a subset [Formula: see text] such that [Formula: see text] and all commutators of G are contained in [Formula: see text]. 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 [Formula: see text]-covering condition for commutators. Then [Formula: see text] contains a characteristic subgroup W such that [Formula: see text] and [Formula: see text] are both [Formula: see text]-bounded. This extends several earlier results of similar flavor.