On closed subgroups of precompact groups-Reference-Cited by-同舟云学术

On closed subgroups of precompact groups

Published:2022-09-28 Issue:0 Volume:0 Page:
ISSN:1433-5883
Container-title:Journal of Group Theory
language:en
Short-container-title:

Author:

Hernández Salvador¹,Remus Dieter²,Trigos-Arrieta F. Javier³

Affiliation:

1. Departamento de Matemáticas , Universitat Jaume I , 8029-AP , Castellón , Spain

2. Institut für Mathematik , Universität Paderborn , Warburger Str. 100, 33098 Paderborn , Germany

3. Department of Mathematics , California State University, Bakersfield , Bakersfield , CA 93311 , USA

Abstract

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

C_{S}

denotes the poset of all subgroups of 𝐺 that are 𝑆-closed, ordered by inclusion, does C S

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 }

\{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?

Publisher

Walter de Gruyter GmbH

Subject

Algebra and Number Theory

Link

https://www.degruyter.com/document/doi/10.1515/jgth-2022-0093/pdf

Reference41 articles.

1. S. Berhanu, W. W. Comfort and J. D. Reid, Counting subgroups and topological group topologies, Pacific J. Math. 116 (1985), no. 2, 217–241.

2. E. Boschi and D. Dikranjan, Essential subgroups of topological groups, Comm. Algebra 28 (2000), no. 10, 4941–4970.

3. N. Bourbaki, Elements of Mathematics. General Topology. Part 1, Hermann, Paris, 1966.

4. T. P. Chalebgwa and S. A. Morris, Topological transcendental fields, Axioms 118 (2022), 1–5.

5. M. J. Chasco, X. Domínguez and M. Tkachenko, Duality properties of bounded torsion topological abelian groups, J. Math. Anal. Appl. 448 (2017), no. 2, 968–981.