Residually finite \(\mathbf{PSP}\)-groups (Q1296425)

Let \(G\) be a group and let \(n\) be an integer \(>1\). Then \(G\) is called a \(\mathbf{PSP}_n\)-group if, for every \(n\)-tuple of subgroups \((H_1,H_2,\dots,H_n)\), there is a non-trivial permutation \(\pi\) in \(S_n\) such that \(H_1H_2\cdots H_n=H_{\pi(1)}H_{\pi(2)}\cdots H_{\pi(n)}\). If \(G\) satisfies \(\mathbf{PSP}_n\) for some \(n\), then \(G\) is a \(\mathbf{PSP}\)-group. The property \(\mathbf{PSP}\), and a similar one for tuples of elements, has been studied by numerous authors over the last 15 years. In this short note it is shown that a finitely generated residually finite group is \(\mathbf{PSP}\) if and only if it is finite-by-Abelian.