A property equivalent to \(n\)-permutability for infinite groups (Q1965259)

A group \(G\) is called infinite restricted \((\infty,n)\)-permutable if \[ X_1X_2\cdots X_n\cap\bigcup_{\substack{\sigma\in S_n\\ \sigma\neq 1}} X_{\sigma(1)}X_{\sigma(2)}\cdots X_{\sigma(n)}\neq 0 \] for all families of \(n\) infinite subsets \(X_i\) of \(G\). This notion arises from previous work by various authors. The main result is: Theorem. Every infinite restricted \((\infty,n)\)-permutable group is \(n\)-permutable. Here \(n\)-permutable means that for every \(n\)-tuple of group elements \((x_1,x_2,\dots,x_n)\) there exists \(\sigma\neq 1\) in \(S_n\) such that \(x_1x_2\cdots x_n=x_{\sigma(1)}x_{\sigma(2)}\cdots x_{\sigma(n)}\).