A property equivalent to permutability for groups (Q1284630)

Let \(m\) and \(n\) be positive integers. The author calls a group \(G\) restricted \((m,n)\)-permutable if \[ X_1X_2\cdots X_n\cap\bigcup_{\sigma\in S_m\setminus 1} X_{\sigma(1)}X_{\sigma(2)}\cdots X_{\sigma(n)} \] is non-empty for all \(m\)-element subsets \(X_i\) of \(G\). When \(m=1\), this is the same as \(n\)-permutable, i.e., given \(x_1,x_2,\dots,x_n\in G\), there is a permutation \(\sigma\neq 1\) such that \(x_1x_2\cdots x_n=x_{\sigma(1)}x_{\sigma(2)}\cdots x_{\sigma(n)}\). The following main result is proved. Theorem: If a group \(G\) is restricted \((m,n)\)-permutable, then \(G\) is finite-by-abelian-by-finite. When \(m=1\), this is a theorem of Curzio, Longobardi, Maj and the reviewer (which is used in the proof).