Potentially fully invariant subgroup: Difference between revisions
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A subgroup <math>H</math> of a group <math>G</math> is termed ''potentially fully characteristic'' in <math>G</math> if there exists a group <math>K</math> containing <math>G</math> such that <math>H</math> is [[fully invariant subgroup|fully invariant]] in <math>K</math>. | A subgroup <math>H</math> of a group <math>G</math> is termed ''potentially fully characteristic'' in <math>G</math> if there exists a group <math>K</math> containing <math>G</math> such that <math>H</math> is [[fully invariant subgroup|fully invariant]] in <math>K</math>. | ||
=Formalisms== | ==Formalisms== | ||
{{obtainedbyapplyingthe|potentially operator|fully characteristic subgroup}} | {{obtainedbyapplyingthe|potentially operator|fully characteristic subgroup}} | ||
Revision as of 15:30, 8 July 2011
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]
This is a variation of fully invariant subgroup|Find other variations of fully invariant subgroup |
Definition
Symbol-free definition
A subgroup of a group is termed potentially fully invariant if there is an embedding of the bigger group in some group such that, in that embedding the subgroup becomes fully invariant.
Definition with symbols
A subgroup of a group is termed potentially fully characteristic in if there exists a group containing such that is fully invariant in .
Formalisms
In terms of the potentially operator
This property is obtained by applying the potentially operator to the property: fully characteristic subgroup
View other properties obtained by applying the potentially operator
The property of being potentially fully invariant is obtained by applying the potentially operator to the property of being fully invariant. The potentially operator is an idempotent ascendant monotone operator.
Relation with other properties
Stronger properties
- Fully invariant subgroup
- Verbal subgroup
- Potentially verbal subgroup
- Normal-potentially fully invariant subgroup
- Central subgroup of finite group: For full proof, refer: central implies potentially fully invariant in finite
- Cyclic normal subgroup of a finite group: For full proof, refer: cyclic normal implies potentially fully invariant in finite
- Homocyclic normal subgroup: For full proof, refer: homocyclic normal implies potentially fully invariant in finite
Weaker properties
Metaproperties
Transitivity
NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity