Isomorph-containing subgroup: Difference between revisions
(New page: {{wikilocal}} {{subgroup property}} ==Definition== A subgroup <math>H</math> of a group <math>G</math> is termed an '''isomorph-containing subgroup''' if, whenever <math>K \le G<...) |
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* [[Weaker than::Isomorph-free subgroup]]: For a finite subgroup, and more generally, for a [[co-Hopfian group|co-Hopfian]] subgroup, the two properties are equivalent. | * [[Weaker than::Isomorph-free subgroup]]: For a finite subgroup, and more generally, for a [[co-Hopfian group|co-Hopfian]] subgroup, the two properties are equivalent. | ||
* [[Weaker than::Homomorph-containing subgroup]] | * [[Weaker than::Homomorph-containing subgroup]] | ||
* [[Weaker than::Subhomomorph-containing subgroup]] | |||
* [[Weaker than::Subisomorph-containing subgroup]] | |||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Stronger than::Intermediately I-characteristic subgroup]] | |||
* [[Stronger than::I-characteristic subgroup]] | |||
* [[Stronger than::Intermediately characteristic subgroup]] | |||
* [[Stronger than::Characteristic subgroup]] | * [[Stronger than::Characteristic subgroup]] | ||
* [[Stronger than::Normal subgroup]] | * [[Stronger than::Normal subgroup]] | ||
Revision as of 19:10, 12 December 2008
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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 isomorph-containing subgroup if, whenever is a subgroup of isomorphic to , .
Relation with other properties
Stronger properties
- Isomorph-free subgroup: For a finite subgroup, and more generally, for a co-Hopfian subgroup, the two properties are equivalent.
- Homomorph-containing subgroup
- Subhomomorph-containing subgroup
- Subisomorph-containing subgroup
Weaker properties
- Intermediately I-characteristic subgroup
- I-characteristic subgroup
- Intermediately characteristic subgroup
- Characteristic subgroup
- Normal subgroup
Metaproperties
Trimness
This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties
Intermediate subgroup condition
YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition