Weakly nilpotent groups (Q1174612)

A group \(G\) is called weakly nilpotent of length \(n\) (schwach nilpotent der Länge \(n\)), if for each \(n\)-tuple \((g_ 1,\dots,g_ n)\) of elements of \(G\), there exists a permutation \(\pi\in S_ n\) depending on the tuple such that \([g_{\pi(1)},\dots,g_{\pi(n)}]=1\). It is shown that for a finite weakly nilpotent group \(G\) of length \(n\), the Sylow \(p\)-subgroup is a direct factor of \(G\) for all \(p\geq n\). In particular, if \(n\leq 5\), then \(G\) is soluble. \{Reviewer's remark: A somewhat related property was considered by \textit{P. Longobardi} [Lect. Notes Math. 1398, 110-116 (1989; Zbl 0682.20024)]\}.