A property equivalent to \(n\)-permutability for infinite groups
type: Item
instance of: scholarly article
title: A property equivalent to \(n\)-permutability for infinite groups
zbMATH Open document ID: 0944.20025
author: Alireza Abdollahi
author: Aliakbar Mohammadi Hassanabadi
author: Bijan Taeri
published in: Journal of Algebra
publication date: 2000-05-28T00:00:00Z
review text: A group \(G\) is called infinite restricted \((\infty,n)\)-permutable if \[ X_1X_2\cdots X_n\cap\bigcup_{\substack{\sigma\in S_n\\ \sigma\neq 1}} X_{\sigma(1)}X_{\sigma(2)}\cdots X_{\sigma(n)}\neq 0 \] for all families of \(n\) infinite subsets \(X_i\) of \(G\). This notion arises from previous work by various authors. The main result is: Theorem. Every infinite restricted \((\infty,n)\)-permutable group is \(n\)-permutable. Here \(n\)-permutable means that for every \(n\)-tuple of group elements \((x_1,x_2,\dots,x_n)\) there exists \(\sigma\neq 1\) in \(S_n\) such that \(x_1x_2\cdots x_n=x_{\sigma(1)}x_{\sigma(2)}\cdots x_{\sigma(n)}\).
Mathematics Subject Classification ID: 20F99
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F12
zbMATH DE Number: 1400109
zbMATH Keywords: permutable subsets
zbMATH Keywords: permutation properties
reviewed by: Derek J. S. Robinson
MaRDI profile type: Publication
full work available at URL: jabr.1999.7996
OpenAlex ID: W2001489736
cites work: Rewriting products of group elements. I
cites work: Rewriting products of group elements. II
cites work: A permutational property of groups
cites work: Infinite groups in a given variety and ramsey's theorem
cites work: The classification of groups in which every product of four elements can be reordered
cites work: On the derived length of groups with some permutational property
cites work: Remarks on the Upper Centralc Series of a Group
cites work: A property equivalent to permutability for groups
cites work: Criteria for Commutativity in Large Groups
cites work: Groups Covered By Permutable Subsets
cites work: Q4177758
cites work: Q3961021
DOI: 10.1006/JABR.1999.7996
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description: scientific article; zbMATH DE number 1400109