On closed subgroups of precompact groups

Salvador Hernández; Dieter Remus; F. Javier Trigos-Arrieta

doi:10.1515/jgth-2022-0093

Article

On closed subgroups of precompact groups

Salvador Hernández , Dieter Remus and F. Javier Trigos-Arrieta

Published/Copyright: September 28, 2022

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From the journal Journal of Group Theory Volume 26 Issue 3

Abstract

It is a theorem of W. W. Comfort and K. A. Ross that if 𝐺 is a subgroup of a compact Abelian group and 𝑆 denotes the continuous homomorphisms from 𝐺 to the one-dimensional torus, then the topology on 𝐺 is the initial topology given by 𝑆. Assume that 𝐻 is a subgroup of 𝐺. We study how the choice of 𝑆 affects the topological placement and properties of 𝐻 in 𝐺. Among other results, we have made significant progress toward the solution of the following specific questions. How many totally bounded group topologies does 𝐺 admit such that 𝐻 is a closed (dense) subgroup? If C S denotes the poset of all subgroups of 𝐺 that are 𝑆-closed, ordered by inclusion, does C S have a greatest (resp. smallest) element? We say that a totally bounded (topological, resp.) group is an SC group (topologically simple, resp.) if all its subgroups are closed (if 𝐺 and { e } are its only possible closed normal subgroups, resp.) In addition, we investigate the following questions. How many SC-(topologically simple totally bounded, resp.) group topologies does an arbitrary Abelian group 𝐺 admit?

Dedicated to María Jesús Chasco on the occasion of her 65th birthday

Funding source: Ministerio de Economía y Competitividad

Award Identifier / Grant number: MTM/PID2019-106529GB-I00

Funding source: Universitat Jaume I

Award Identifier / Grant number: UJI-B2019-06

Funding statement: The first author’s research was partially supported by the Spanish Ministerio de Economía y Competitividad, grant MTM/PID2019-106529GB-I00 (AEI/FEDER, EU) and by the Universitat Jaume I, grant UJI-B2019-06.

Acknowledgements

We are very grateful to the referee for her/his helpful comments.

Communicated by: George Willis

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Received: 2022-05-24

Revised: 2022-07-23

Published Online: 2022-09-28

Published in Print: 2023-05-01

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