Classes of topological groups suggested by Galois theory

Soundararajan, T.

doi:10.1016/0022-4049(91)90016-u

Cited by 2 publications

(2 citation statements)

References 19 publications

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“…Then G admits at least c-many totally bounded group topologies τ such that every subgroup of G is closed in (G, τ ). For, G = B( G), the set of compact elements of G. For the proof, take H = (G, τ l ), and G the completion of this H in the proof of [38,Theorem 1.15]. Note that in the proof of [38,Theorem 1.15] it is allowed that the rank of F is zero [Look at the end of the proof of [38,Proposition 1.14].…”

Section: Proposition 65 Let G Be An Abelian Group and S Be A Subgroup...mentioning

confidence: 99%

“…(2) The authors of [26] focus on an infinite compact (Hausdorff) totally disconnected Abelian group (G, τ ) and try to obtain finer totally bounded group topologies τ ′ such that every τ ′ -closed subgroup is τ -closed. (3) In [38], the author focuses on totally bounded topological groups in which every subgroup is closed. (4) In [24,Proposition 3.4], the following is proved: Let G be a totally bounded Abelian group with character group S. If L is a subgroup of S, let L G denote L equipped with the weakest topology that makes the elements of G (acting on L) continuous.…”

Section: Introductionmentioning

confidence: 99%

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On closed subgroups of precompact groups

Hernández

Remus

Trigos-Arrieta

2022

Journal of Group Theory

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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?

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Section: Proposition 65 Let G Be An Abelian Group and S Be A Subgroup...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On closed subgroups of precompact groups

Hernández

Remus

Trigos-Arrieta

2022

Journal of Group Theory

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On closed subgroups of precompact groups

Hernández¹,

Remus²,

Trigos-Arrieta³

2022

Preprint

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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 those 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 has 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?

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Classes of topological groups suggested by Galois theory

Cited by 2 publications

References 19 publications

On closed subgroups of precompact groups

On closed subgroups of precompact groups

On closed subgroups of precompact groups

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