A permutational property of groups
type: Item
instance of: scholarly article
title: A permutational property of groups
zbMATH Open document ID: 0544.20036
DOI: 10.1007/BF01229319
author: Mario Curzio
author: M. Maj
author: Derek J. S. Robinson
author: P. Longobardi
published in: Archiv der Mathematik
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20F24
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 3865574
zbMATH Keywords: finite-by-abelian-by-finite groups
zbMATH Keywords: ordered product of n elements
MaRDI profile type: Publication
cites work: Q3310057
cites work: Q5830671
cites work: On the Burnside problem for semigroups
cites work: Q5654235
description: scientific article; zbMATH DE number 3865574