Two Conditions for Infinite Groups to Satisfy Certain Laws

摘要

Let G be an infinite group and m ∈ {2k [ k ∈ N* }. In this paper, we prove that G satisfies the law [xm, ym] = 1 ifand only ifin any two infinite subsets X and Y of G, there exist a ∈ X and b ∈ Y such that [am,bm] = 1. We also prove X1, X2,..., Xn, there exist ai ∈ Xi (I = 1,..., n) such that (am1am2... Amn)2 = 1.