标题： 关于具有BFC覆盖词值的群组 显示英文标题

标题： On groups with BFC-covered word values

摘要： 对于群G和正整数n，令B_n(G) ={x \in G : |x^G | \le n}。 如果s是正整数且w是一个群字，那么称G相对于字w满足(n,s)-覆盖条件，如果存在G的一个子集S，使得|S|\le s且G的所有w-值都包含在B_n(G)S中。以自然的方式，这个条件出现在对二阶概率幂零群的研究中。 在本文中我们得到以下结果。 设w是一个k个变量的多重交换字，设G是一个相对于字w满足(n,s)-覆盖条件的群。则G有一个可解子群T，使得指数[G : T]和T的导出长度都是(k,n,s)-有界的。 设G是一个相对于字\gamma _k满足(n,s)-覆盖条件的群。 然后 (1) \gamma _{2k-1}(G) 有一个子群 $T$，使得指数 [\gamma _{2k-1}(G) : T] 和 |T'| 都是 (k,n,s)-有界的；并且 (2) G 有一个幂零子群 U，使得指数 [G : U] 和 U 的幂零类都是 (k,n,s)-有界的。 显示英文摘要

摘要： For a group G and a positive integer n write B_n(G) = {x \in G : |x^G | \le n}. If s is a positive integer 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 of G such that |S| \le s and all w-values 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 the index [G : T] and the derived length of T are both (k,n,s)-bounded. Let 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 the index [\gamma_{2k-1}(G) : T] and |T'| are both (k,n,s)-bounded; and (2) G has a nilpotent subgroup U such that the index [G : U] and the nilpotency class of U are both (k,n,s)-bounded.