The classification of groups in which every product of four elements can be reordered. (English) Zbl 0838.20038

Longobardi, Patrizia; Maj, Mercede; Stonehewer, Stewart

Rend. Semin. Mat. Univ. Padova 93, 7-26 (1995).

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 \(P_4\).

Reviewer: S.Franciosi (Napoli)

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)

Keywords:

metabelian groups; derived subgroup; permutation property \(P_ 4\)

Citations:

Zbl 0528.20031

References:

[1]	M. Bianchi - R. Brandl - A. Gillio Berta Mauri , On the 4-permutational property , Arch. Math. , 48 ( 1987 ), pp. 281 - 285 . MR 884558 | Zbl 0623.20022 · Zbl 0623.20022
[2]	M. Curzio - P. LONGOBARDI - M. MAJ, Su di un problema combinatorio in teoria dei gruppi, Atti Acc. Lincei Rend. Sci. Mat. Fis . Nat. , 74 ( 1983 ), pp. 136 - 142 . MR 739397 | Zbl 0528.20031 · Zbl 0528.20031
[3]	M. Curzio - P. LONGOBARDI - M. MAJ - D. J. S. ROBINSON, On a permutational property of groups , Arch. Math. , 44 ( 1985 ), pp. 385 - 389 . MR 792360 | Zbl 0544.20036 · Zbl 0544.20036
[4]	G. Higman , Rewriting products of group elements, Lectures given in Urbana in 1985 (unpublished).
[5]	A.G. Kurosh , The Theory of Groups , 2 nd edition ( 2 vols.), Chelsea , New York ( 1960 ). MR 109842 · Zbl 0094.24501
[6]	P. Longobardi - M. MAJ, On groups in which every product of four elements can be reordered , Arch. Math. , 49 ( 1987 ), pp. 273 - 276 . MR 913155 | Zbl 0607.20017 · Zbl 0607.20017
[7]	P. Longobardi - S. Stonehewer , Finite 2-groups of class 2 in which every product of four elements can be reordered , Illinois J ., 35 ( 1991 ), pp. 198 - 219 . Article | MR 1091438 | Zbl 0698.20013 · Zbl 0698.20013
[8]	P. Longobardi - M. Maj - S. Stonehewer , The classification of groups in which every product of four elements can be reordered , preprint, Warwick University . · Zbl 0838.20038
[9]	M. Maj - S. Stonehewer , Non-nilpotent groups in which every product of four elements can be reordered , Canadian J. , 42 ( 1990 ), pp. 1053 - 1066 . MR 1099457 | Zbl 0727.20027 · Zbl 0727.20027