DOI: 10.1515/jgth-2022-0093

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# On closed subgroups of precompact groups

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

Abelian group

Mathematical analysis

Group (periodic table)

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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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>C</m:mi> <m:mi>S</m:mi> </m:msub> </m:math> C_{S} denotes the poset of all subgroups of 𝐺 that are 𝑆-closed, ordered by inclusion, does <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>C</m:mi> <m:mi>S</m:mi> </m:msub> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mi>e</m:mi> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> \{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?