Free groups in the smallest ideal of \(\beta G\)
type: Item
instance of: scholarly article
title: Free groups in the smallest ideal of \(\beta G\)
zbMATH Open document ID: 1168.22003
author: Yevhen Zelenyuk
author: Yuliya Zelenyuk
published in: Semigroup Forum
publication date: 2009-04-27T00:00:00Z
review text: The paper under review is devoted to the proof of the following statement (Theorem 1.1): If \(G\) is an infinite group with cardinality \(\kappa\) which is embeddable into a direct sum of countable subgroups, then the structural group of the smallest two-sided ideal of the semigroup \(\beta G\) (the Stone-Čech compactification of \(G\)) contains copies of the free group on \(2^{2^{\kappa}}\) generators. (As the authors recall, the result holds in particular for all infinite Abelian groups.) This generalizes or refines previous results by \textit{N.~Hindman} and \textit{J.~ Pym} [Semigroup Forum 30, 177--193 (1984; Zbl 0536.22005)] and \textit{N.~Hindman} and \textit{D.~Strauss} [Algebra in the Stone-Čech compactification: theory and applications (De Gruyter Expositions in Mathematics 27, Walter de Gruyter, Berlin) (1998; Zbl 0918.22001)].
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 20M99
Mathematics Subject Classification ID: 22A15
Mathematics Subject Classification ID: 54D35
zbMATH DE Number: 5547498
zbMATH Keywords: Stone-Čech compactification
zbMATH Keywords: free group
zbMATH Keywords: local homomorphism
zbMATH Keywords: smallest ideal
reviewed by: X. Domínguez
MaRDI profile type: Publication
full work available at URL: s00233-008-9126-9
OpenAlex ID: W2108720865
cites work: Q5609457
cites work: Free groups and semigroups in \(\beta\) N
cites work: Algebra in the Stone-Čech compactification: theory and applications
cites work: Extremal topologies on a semigroup
cites work: On the ultrafilter semigroup of a topological group
cites work: Local Homomorphisms of Topological Groups
DOI: 10.1007/S00233-008-9126-9
description: scientific article; zbMATH DE number 5547498