Maj, M.; Stonehewer, S. E. Non-nilpotent groups in which every product of four elements can be reordered. (English) Zbl 0727.20027 Can. J. Math. 42, No. 6, 1053-1066 (1990). Groups in which every product of four elements can be reordered to yield the same product are known to be metabelian [see P. Longobardi and M. Maj, Arch. Math. 49, 273-276 (1987; Zbl 0607.20017)]. In the present paper it is shown that the group G satisfies the above property if and only if one of the following holds: (i) G has an abelian subgroup of index 2; (ii) G is nilpotent of class \(\leq 4\) and G has the above property; (iii) \(G'\cong V_ 4\); (iv) \(G=B<a,x>\), where \(B\leq Z(G)\), \(o(a)=5\) and \(a^ x=a^ 2\). The nilpotent case (ii) will be dealt with in subsequent work. Reviewer: R.Brandl (Würzburg) Cited in 8 Documents MSC: 20F12 Commutator calculus 20F16 Solvable groups, supersolvable groups 20E10 Quasivarieties and varieties of groups 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20E07 Subgroup theorems; subgroup growth 20F05 Generators, relations, and presentations of groups Keywords:permutation property; property \(P_ n\); rewritable groups; product of four elements; metabelian; abelian subgroup of index 2 Citations:Zbl 0621.20022; Zbl 0607.20017 PDFBibTeX XMLCite \textit{M. Maj} and \textit{S. E. Stonehewer}, Can. J. Math. 42, No. 6, 1053--1066 (1990; Zbl 0727.20027) Full Text: DOI