Soluble Groups with a Condition on Infinite Subsets

• 来源期刊: 代数集刊（英文版）
• 关键词: nilpotent-by-finite groups finitely generated soluble groups infinite subsets
• 年，卷（期）: 2002,（4）
• 页码范围: 427-432
• 页数: 6页
• 语种: 英文

摘要

We prove that a finitely generated soluble group G is nilpotent-by-finite (respectively, finite-by-nilpotent) if every infinite subset of G contains two elements x,y generating a nilpotent-by-finite (respectively, finite-by-nilpotent)group.Moreover,for a positive integer k,if we suppose <x,y>is (nipotent of class k)-by-finite (respectively, finite-by-(nilpotent of class k)), then there is an integer c =c(k) such that G is (nilpotent of class c)-by-finite (respectively, finiteby-(nilpotent of class c)).