Intermediately fully invariant subgroup: Difference between revisions

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===Symbol-free definition===
===Symbol-free definition===


A [[subgroup]] of a [[group]] is termed '''intermediately fully characteristic''' if it is [[defining ingredient::fully characteristic subgroup|fully characteristic]] in every intermediate subgroup of the group containing it.
A [[subgroup]] of a [[group]] is termed '''intermediately fully invariant''' or '''intermediately fully characteristic''' if it is [[defining ingredient::fully invariant subgroup|fully invariant]] in every intermediate subgroup of the group containing it.


===Definition with symbols===
===Definition with symbols===


A subgroup <math>H</math> of a group <math>G</math> is termed '''intermediately fully characteristic''' in <math>G</math> if, for any intermediate subgroup <math>K</math> of <math>G</math>, <math>H</math> is fully characteristic in <math>K</math>: for any [[endomorphism]] <math>\varphi</math> of <math>K</math>, <math>\varphi(H) \le H</math>.
A subgroup <math>H</math> of a group <math>G</math> is termed '''intermediately fully invariant''' or '''intermediately fully characteristic''' in <math>G</math> if, for any intermediate subgroup <math>K</math> of <math>G</math>, <math>H</math> is fully characteristic in <math>K</math>: for any [[endomorphism]] <math>\varphi</math> of <math>K</math>, <math>\varphi(H) \le H</math>.


==Relation with other properties==
==Relation with other properties==
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* [[Weaker than::Subhomomorph-containing subgroup]]
* [[Weaker than::Subhomomorph-containing subgroup]]
* [[Weaker than::Variety-containing subgroup]]
* [[Weaker than::Variety-containing subgroup]]
* [[Weaker than::Normal Sylow subgroup]]
* [[Weaker than::Normal Hall subgroup]]
* [[Weaker than::Normal subgroup having no nontrivial homomorphism to its quotient group]]


===Weaker properties===
===Weaker properties===


* [[Stronger than::Fully characteristic subgroup]]
* [[Stronger than::Fully invariant subgroup]]
* [[Stronger than::Intermediately characteristic subgroup]]
* [[Stronger than::Intermediately characteristic subgroup]]
* [[Stronger than::Characteristic subgroup]]
* [[Stronger than::Characteristic subgroup]]
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{{intransitive}}
{{intransitive}}


An intermediately fully characteristic subgroup of an intermediately fully characteristic subgroup need not be intermediately fully characteristic. {{proofat|[[Intermediate full characteristicity is not transitive]]}}
An intermediately fully characteristic subgroup of an intermediately fully characteristic subgroup need not be intermediately fully characteristic.  


{{join-closed}}
{{proofat|[[Intermediate full invariance is not transitive]]}}


An arbitrary join of intermediately fully characteristic subgroups is intermediately fully characteristic. This follows from the fact that the intermediately operator preserves the property of being closed under joins. {{proofat|[[Intermediate full characteristicity is strongly join-closed]]}}
{{join-closed|strongly}}
 
An arbitrary join of intermediately fully characteristic subgroups is intermediately fully characteristic. This follows from the fact that the intermediately operator preserves the property of being closed under joins.  
 
{{proofat|[[Intermediate full invariance is strongly join-closed]]}}
 
{{further|[[Intermediately operator preserves join-closedness]], [[Full invariance is strongly join-closed]]}}
 
{{intsubcondn}}
{{quot-transitive}}
{{proofat|[[Intermediate full invariance is quotient-transitive]]}}
 
{{further|[[Intermediately operator preserves quotient-transitivity]], [[Full invariance is quotient-transitive]]}}

Latest revision as of 01:49, 14 August 2009

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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

Symbol-free definition

A subgroup of a group is termed intermediately fully invariant or intermediately fully characteristic if it is fully invariant in every intermediate subgroup of the group containing it.

Definition with symbols

A subgroup H of a group G is termed intermediately fully invariant or intermediately fully characteristic in G if, for any intermediate subgroup K of G, H is fully characteristic in K: for any endomorphism φ of K, φ(H)≤H.

Relation with other properties

Stronger properties

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

An intermediately fully characteristic subgroup of an intermediately fully characteristic subgroup need not be intermediately fully characteristic.

For full proof, refer: Intermediate full invariance is not transitive

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
In fact, since the property is also true for the trivial subgroup in any group, it is a strongly join-closed subgroup property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

An arbitrary join of intermediately fully characteristic subgroups is intermediately fully characteristic. This follows from the fact that the intermediately operator preserves the property of being closed under joins.

For full proof, refer: Intermediate full invariance is strongly join-closed

Further information: Intermediately operator preserves join-closedness, Full invariance is strongly join-closed

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

Quotient-transitivity

This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
View a complete list of quotient-transitive subgroup properties

For full proof, refer: Intermediate full invariance is quotient-transitive

Further information: Intermediately operator preserves quotient-transitivity, Full invariance is quotient-transitive

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