Subisomorph-containing subgroup: Difference between revisions
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==Definition== | ==Definition== | ||
A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''subisomorph-containing''' if whenever <math>K</math> is a subgroup of <math> | A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''subisomorph-containing''' if whenever <math>K</math> is a subgroup of <math>H</math> and <math>L</math> is a subgroup of <math>G</math> such that <math>K</math> and <math>L</math> are [[isomorphic groups|isomorphic]], then <math>L</math> is also a subgroup of <math>H</math>. | ||
==Relation with other properties== | ==Relation with other properties== | ||
Latest revision as of 14:06, 1 June 2020
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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]
If the ambient group is a finite group, this property is equivalent to the property: variety-containing subgroup
View other properties finitarily equivalent to variety-containing subgroup | View other variations of variety-containing subgroup |
Definition
A subgroup of a group is termed subisomorph-containing if whenever is a subgroup of and is a subgroup of such that and are isomorphic, then is also a subgroup of .
Relation with other properties
In groups with specific properties
- Finite group and periodic group: In a finite group and more generally in a periodic group, the notion of subisomorph-containing subgroup coincides with the notions of subhomomorph-containing subgroup and variety-containing subgroup. Further information: Equivalence of definitions of variety-containing subgroup of finite group, Equivalence of definitions of variety-containing subgroup of periodic group
- Group of prime power order: For groups of prime power order, subisomorph-containing subgroups must be omega subgroups of group of prime power order, though the converse, while true for regular p-groups, is not always true. For full proof, refer: Variety-containing implies omega subgroup in group of prime power order, omega subgroups are variety-containing in regular p-group, omega subgroups not are variety-containing
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| variety-containing subgroup | contains every subgroup of the whole group in the variety generated by it | |FULL LIST, MORE INFO | ||
| subhomomorph-containing subgroup | contains every subgroup of the whole group isomorphic to a subquotient of it | |FULL LIST, MORE INFO | ||
| normal Sylow subgroup | normal subgroup of prime power order whose order and index are relatively prime | |FULL LIST, MORE INFO | ||
| normal Hall subgroup | subgroup whose order and index are relatively prime | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| isomorph-containing subgroup | contains every isomorphic subgroup | |FULL LIST, MORE INFO | ||
| injective endomorphism-invariant subgroup | invariant under all injective endomorphisms | |FULL LIST, MORE INFO | ||
| intermediately injective endomorphism-invariant subgroup | injective endomorphism-invariant in every intermediate subgroup | |FULL LIST, MORE INFO | ||
| characteristic subgroup | invariant under every automorphism | |FULL LIST, MORE INFO | ||
| intermediately characteristic subgroup | characteristic in every intermediate subgroup | |FULL LIST, MORE INFO | ||
| transfer-closed characteristic subgroup | intersection with any subgroup is characteristic in that subgroup | |FULL LIST, MORE INFO |