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The classification of groups in which every product of four elements can be reordered. (English) Zbl 0838.20038
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 M. Curzio, P. Longobardi and 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 \(P_4\).

MSC:
20F16 Solvable groups, supersolvable groups
20F05 Generators, relations, and presentations of groups
20F12 Commutator calculus
20E34 General structure theorems for groups
20F24 FC-groups and their generalizations
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
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References:
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