A Property Equivalent to n-Permutability for Infinite Groups

Let n be an integer greater than 1. A group G is said to be n-permutable whenever for every n-tuple (x1,…,xn) of elements of G there exists a non-identity permutation σ of {1,…,n} such that x1•••xn = xσ(1)•••xσ(n). In this paper we prove that an infinite group G is n-permutable if and only if for every n infinite subsets X1,…,Xn of G there exists a non-identity permutation σ on {1,…,n} such that X1•••Xn Xσ(1)•••Xσ(n) ≠ .