A Property Equivalent to n-Permutability for Infinite Groups

Abstract

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)≠∅. © 1999 Academic Press.