On groups with BFC-covered word values

For a group G and a positive integer n write B-n(G) = { x is an element of G : | x(G) | <= 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 subset of G such that |S| <= s and all wvalues of G are contained in B-n(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.) Let k >= 1 and G be a group satisfying the (n, s)-covering condition with respect to the word gamma(k) . Then (1) gamma(2k -1)(G) has a subgroup T such that [ gamma(2k-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.) (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.