Unconditionally closed subgroup: Difference between revisions
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A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed an '''unconditionally closed subgroup''' if <math>H</math> is a [[closed subgroup]] of <math>G</math> for ''any'' topology on <math>G</math> that turns <math>G</math> into a [[T0 topological group]]. | A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed an '''unconditionally closed subgroup''' if <math>H</math> is a [[closed subgroup]] of <math>G</math> for ''any'' topology on <math>G</math> that turns <math>G</math> into a [[T0 topological group]]. | ||
==Relation with other properties== | |||
===Corresponding subset property=== | |||
* [[Unconditionally closed subset]] is the "subset" version of the property. | |||
===Stronger properties=== | |||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Weaker than::finite subgroup]] || || || || {{intermediate notions short|unconditionally closed subgroup|finite subgroup}} | |||
|- | |||
| [[Weaker than::c-closed subgroup]] || || || || {{intermediate notions short|unconditionally closed subgroup|c-closed subgroup}} | |||
|- | |||
| [[Weaker than::algebraic subgroup]] || || || || {{intermediate notions short|unconditionally closed subgroup|algebraic subgroup}} | |||
|- | |||
| [[Weaker than::marginal subgroup]] || || || || {{intermediate notions short|unconditionally closed subgroup|marginal subgroup}} | |||
|- | |||
| [[Weaker than::weakly marginal subgroup]] || || || || {{intermediate notions short|unconditionally closed subgroup|weakly marginal subgroup}} | |||
|} | |||
Latest revision as of 01:47, 28 July 2013
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
Definition
A subgroup of a group is termed an unconditionally closed subgroup if is a closed subgroup of for any topology on that turns into a T0 topological group.
Relation with other properties
Corresponding subset property
- Unconditionally closed subset is the "subset" version of the property.
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| finite subgroup | |FULL LIST, MORE INFO | |||
| c-closed subgroup | |FULL LIST, MORE INFO | |||
| algebraic subgroup | |FULL LIST, MORE INFO | |||
| marginal subgroup | |FULL LIST, MORE INFO | |||
| weakly marginal subgroup | |FULL LIST, MORE INFO |