Two Conditions for Infinite Groups to Satisfy Certain Laws

Let G be an infinite group and m {2^k | k N^*}. In this paper, we prove that G satisfies the law x_m, y_m] = 1 if and only if in any two infinite subsets X and Y of G, there exist a X and b Y such that a^m,b^m] = 1. We also prove that G satisfies the law (x₁^mx₂^m x_n^m)² = 1 if and only if in any n infinite subsets X₁, X₂,..., X_n, there exist a_i X_i (i = 1,..., n) such that (a₁^ma₂^m a_n^m)² = 1.2000 Mathematics Subject Classification: 20F99