The classification of groups in which every product of four elements can be reordered (Q1904086)

If \(n\geq 2\) is an integer, \(P_n\) is defined to be the class of groups \(G\) such that for all \(n\)-tuples \((x_1, \dots, x_n)\) of elements of \(G\) there exists a non-trivial permutation \(\sigma\) of \(\{1, \dots, n\}\) such that \(x_{\sigma(1)}x_{\sigma(2)}\dots x_{\sigma (n)}=x_1 x_2\dots x_n\). Trivially \(P_2\) is the class of abelian groups, and \textit{M. Curzio}, \textit{P. Longobardi} and \textit{M. Maj} proved that \(P_3\) is the class of all groups whose commutator subgroup has order at most 2 [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 74, 136-142 (1983; Zbl 0528.20031)]. In this paper the authors, improving previous results, give a complete classification of groups in the class \(P_4\).