Criteria for laws between infinite subsets of infinite groups

Abstract

A theorem of B. H. Neumman shows that infinite group in which every two infinite subsets there exist two commuting elements, is abelian.In this paper, we prove that if in an infinite group G, every two infinite subsets X and Y , there exist a ∈ X and bMoreover, and using this result, we also prove that an infinite group satisfies the law (x n 1 1 x n 2 2 . . . x nr r ) 2 = 1 if and only if in any r infinite subsets X 1 , . . . , X r , of G there exist ai ∈ Xi(i = 1, . . . , r) such that (a n 1 1 . . . a nr r ) 2 = 1, where n1, . . . , nr ∈ {2 k /k ∈ N * } and r ≥ 2.