On groups in which every product of four elements can be reordered
A group G belongs to the class \(P_ n\) (n an integer \(>1)\) if and only if to each n-tuple \((x_ 1,x_ 2,...,x_ n)\) of elements of G there corresponds a non-trivial element \(\sigma =\sigma (x_ 1,x_ 2,...,x_ n)\) of the symmetric group \(S_ n\) such that \(x_ 1x_ 2...x_ n=x_{\sigma (1)}x_{\sigma (2)}...x_{\sigma (n)}\). If P is the class of groups belonging to some of the \(P_ n\), it was shown by \textit{M. Curzio}, \textit{D. J. S. Robinson} and the authors [Arch. Math. 44, 385-389 (1985; Zbl 0544.20036)] that the groups \(G\in P\) are exactly the finite- by-abelian-by-finite groups. Clearly \(P_ 2\) is the class of abelian groups, while \(G\in P_ 3\) if and only if \(| G'| \leq 2\) [\textit{M. Curzio} and the authors, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 74, 136- 142 (1983; Zbl 0528.20031)]. G. Higman has proved that a finite group \(G\in P_ 4\) of odd order is nilpotent of class \(\leq 2\). \textit{M. Bianchi}, \textit{R. Brandl} and \textit{A. Gillio Berta Mauri} [''On the four permutational property for groups'', Arch. Math. (to appear)] have shown that a group G in \(P_ 4\) is soluble and that the commutator subgroup of a finite group G in \(P_ 4\) is nilpotent. In this paper is is shown the following Theorem: If a group G belongs to \(P_ 4\), then G is metabelian. Remark that a group G with an abelian subgroup of index 2 belongs to \(P_ 4\); hence every abelian group occurs as the derived group of some group in \(P_ 4\). Therefore there are neither limits to the order of G' nor further restrictions on its structure.
scientific article; zbMATH DE number 3981425