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A permutational property of groups. (English) Zbl 0544.20036
If \(n\) is an integer greater than 1, a group \(G\) is said to have the property \({\mathcal P}_ n\) if every ordered product of \(n\) elements of \(G\) can be rewritten in at least one way, i.e. to each n-tuple \((x_ 1,x_ 2,...,x_ n)\) with \(x_ i\in G\), there corresponds a nontrivial element \(\sigma\) of the symmetric group \(S_ n\) such that \(x_ 1x_ 2...x_ n=x_{\sigma(1)}x_{\sigma(2)}...x_{\sigma(n)}.\) Obviously \({\mathcal P}_ 2\) is commutativity, while \({\mathcal P}_ 3\), \({\mathcal P}_ 4\), etc. are successively weaker properties. A group has the property \({\mathcal P}\) if is satisfies \({\mathcal P}_ n\) for some \(n>1\). A complete description is given of groups with the property \({\mathcal P}\); these are precisely the groups which are finite-by-abelian-by-finite.

MSC:
20F24 FC-groups and their generalizations
20E07 Subgroup theorems; subgroup growth
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20F05 Generators, relations, and presentations of groups
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[1] M. Curzio, P. Longobardi andM. Maj, Su di un problema combinatorio in teoria dei gruppi. Atti Acc. Lincei Rend. fis. VIII,74, 136-142 (1983). · Zbl 0528.20031
[2] B. H. Neumann, Groups covered by finitely many cosets. Publ. Math. Debrecen3, 227-242 (1954). · Zbl 0057.25603
[3] A. Restivo andC. Reutenauer, On the Burnside problem for semigroups. J. Algebra89, 102-104 (1984). · Zbl 0545.20051
[4] D. J. S.Robinson, Finiteness conditions and generalized soluble groups, Part 1. Berlin 1972. · Zbl 0243.20032
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