On groups with BFC-covered word values

Abstract

For a group G and a positive integer n write Bn(G)={x∈G:|xG|≤n}. If s≥1 and w is a group word, say that G satisfies the (n,s)-covering condition with respect to the word w if there exists a subset S⊆G such that |S|≤s and all w-values of G are contained in Bn(G)S. In a natural way, this condition emerged in the study of probabilistically nilpotent groups of class two. In this paper we obtain the following results.Let w be a multilinear commutator word on k variables and let G be a group satisfying the(n,s)-covering condition with respect to the word w. Then G has a soluble subgroup T such that[G:T]and the derived length of T are both(k,n,s)-bounded. (Theorem 1.1.)Letk≥1and G be a group satisfying the(n,s)-covering condition with respect to the wordγk. Then (1)γ2k−1(G)has a subgroup T such that[γ2k−1(G):T]and|T′|are both(k,n,s)-bounded; and (2) G has a nilpotent subgroup U such that[G:U]and the nilpotency class of U are both(k,n,s)-bounded. (Theorem 1.2.)