Minimal and precompact group topologies on free groups
type: Item
instance of: scholarly article
title: Minimal and precompact group topologies on free groups
zbMATH Open document ID: 0726.22001
DOI: 10.1016/0022-4049(91)90014-S
author: Dieter Remus
published in: Journal of Pure and Applied Algebra
publication date: 1991-01-01T00:00:00Z
review text: A topological group G is called minimal if its topology is a minimal element of the set of all Hausdorff group topologies on G. If G/N is minimal for every closed normal subgroup N of G, then G is called totally minimal. It was proved by \textit{D. Shakhmatov} [Mat. Zametki 38, 908-914 (1985); English transl.: Math. Notes 38, 1003-1006 (1985; Zbl 0594.22001)] that on every free group G of infinite rank there exists a totally minimal group topology \(\tau\). Since in this construction (G,\(\tau\)) has no group completion, it is natural to ask whether there exist totally minimal group topologies \(\tau\) such that (G,\(\tau\)) has a group completion. A more difficult problem is to construct a precompact totally minimal group topology \(\tau\). In the first part of the paper under review the author shows that on very countable free group there exist infinitely many nonisomorphic precompact totally minimal group topologies. The second part of the paper is devoted mainly to the existence of certain types of \(T_ 2\)-precompact group topologies on free groups and the study of the number of these topologies. It is proved, in particular, that every free group F with cardinality \(\alpha\) admits a \(T_ 2\)- precompact group topology \(\tau\) such that \(w(F,\tau)=\log \alpha\).
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54A10
Mathematics Subject Classification ID: 54D10
zbMATH DE Number: 4198341
zbMATH Keywords: Hausdorff group topologies
zbMATH Keywords: group completion
zbMATH Keywords: totally minimal group topologies
zbMATH Keywords: countable free group
zbMATH Keywords: precompact totally minimal group topologies
zbMATH Keywords: \(T_ 2\)-precompact group topologies
reviewed by: Luchezar Stoyanov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4198341