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Non-nilpotent groups in which every product of four elements can be reordered. (English) Zbl 0727.20027
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.

##### 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
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