Relative minimality and co-minimality of subgroups in topological groups
type: Item
instance of: scholarly article
title: Relative minimality and co-minimality of subgroups in topological groups
zbMATH Open document ID: 1190.22002
author: Dikran Dikranjan
author: Michael Megrelishvili
published in: Topology and its Applications
publication date: 2009-12-03T00:00:00Z
review text: A Hausdorff topological group is called minimal if there does not exist any strictly coarser Hausdorff group topology. Recently it was shown by \textit{V. V. Uspenskij} [Topology Appl. 155, No. 14, 1580--1606 (2008; Zbl 1166.22002)] that every Hausdorff group can be embedded in a minimal group. The second named author proved that every Hausdorff group is a group retract of a minimal group [Topology Appl. 155, No. 17--18, 2105--2127 (2008; Zbl 1153.22002)]. In the paper under review the authors investigate minimal group topologies on semidirect products. More precisely, given a continuous biadditive mapping \(w:(E,\sigma)\times (F,\tau)\to (A,\nu)\), they define the generalized Heisenberg group as follows: \(H(w)=(A\times E)\lambda F\) where \[ (a_1,x_1,f_1)\cdot (a_2,x_2,f_2)= (a_1+a_2+f_1(x_2),x_1+x_2,f_1+f_2). \] They prove: \(H(w)\) is a minimal group iff \(A\) is a minimal group and \(w\) is a minimal biadditive mapping which means that for any Hausdorff group topologies \(\sigma_1\subseteq \sigma\) and \(\tau_1\subseteq \tau\) the continuity of \(w:(E,\sigma_1)\times (F,\tau_1)\to (A,\nu)\) implies \(\sigma_1=\sigma\) and \(\tau_1=\tau\). In this context they study relative minimality and co-minimality, where a subset \(X\) of a Hausdorff group \((G,\sigma)\) is called relatively minimal if any coarser Hausdorff group topology \(\sigma_1\) on \(G\) induces the same topology as \(\sigma\) on \(X\). A subgroup \(H\) of \((G,\sigma)\) is called co-minimal if for any coarser Hausdorff group topology \(\sigma_1\) on \(G\) the quotient topologies induced by \(\sigma\) and \(\sigma_1\) on the quotient space \(G/H\) are the same. Many applications and examples are given.
reviewed by: Lydia Außenhofer
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54D25
Mathematics Subject Classification ID: 20K45
Mathematics Subject Classification ID: 20K99
Mathematics Subject Classification ID: 54H11
zbMATH DE Number: 5641745
zbMATH Keywords: Heisenberg group
zbMATH Keywords: minimal group
zbMATH Keywords: relatively minimal subgroup
zbMATH Keywords: co-minimal subgroup
MaRDI profile type: Publication
full work available at URL: j.topol.2009.04.048
OpenAlex ID: W2056097324
cites work: Locally minimal topological groups. I
cites work: Q4814351
cites work: Examples of locally compact non-compact minimal topological groups
cites work: Recent advances in minimal topological groups
cites work: Arnautov's problems on semitopological isomorphisms
cites work: Q4819379
cites work: Q3885938
cites work: Transversal group topologies on non-Abelian groups
cites work: Q5645625
cites work: On the product of two (totally) minimal topological groups and the three- space-problem
cites work: Group representations and construction of minimal topological groups
cites work: Q4214879
cites work: Q4662374
cites work: Every topological group is a group retract of a minimal group
cites work: Q3196983
cites work: On Lie groups in varieties of topological groups
cites work: Über den Satz von Wiener und lokalkompakte Gruppen
cites work: Minimality in topological groups and Heisenberg type groups
cites work: Complete minimal and totally minimal groups
cites work: An example of a Q-minimal precompact topological group containing a nonminimal closed normal subgroup
cites work: Minimal topological groups
cites work: On subgroups of minimal topological groups
DOI: 10.1016/J.TOPOL.2009.04.048
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description: scientific article; zbMATH DE number 5641745