×

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.