Potentially fully invariant subgroup: Difference between revisions

Latest revision as of 22:03, 30 May 2020

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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 $H$ of a group $G$ is termed potentially fully invariant in $G$ if there exists a group $K$ containing $G$ such that $H$ is fully invariant in $K$ .

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

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)
fully invariant subgroup	invariant under all endomorphisms	(by definition)	(cyclic normal subgroup examples)
verbal subgroup	image of a word map	(via fully invariant)	(via fully invariant)
potentially verbal subgroup	can be verbal inside a bigger group	follows from verbal implies fully invariant	(unclear)
normal-potentially fully invariant subgroup	can be fully invariant in a bigger group in which the original ambient group is normal	(by definition)	(unclear)
central subgroup of finite group		central implies potentially fully invariant in finite	any non-abelian group as a subgroup of itself
cyclic normal subgroup of a finite group		(via homocyclic normal)
homocyclic normal subgroup of a finite group		homocyclic normal implies potentially fully invariant in finite
fully normalized potentially fully invariant subgroup	also a fully normalized subgroup

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal subgroup		potentially fully invariant implies normal	normal not implies potentially fully invariant

Incomparable properties

Characteristic subgroup: See characteristic not implies potentially fully invariant

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

@@ Line 1: / Line 1: @@
+{{wikilocal}}
 {{subgroup property}}
+{{variationof|fully invariant subgroup}}
-{{wikilocal}}
-{{variationof|full characteristicity}}
 ==Definition==
@@ Line 9: / Line 7: @@
 ===Symbol-free definition===
-A subgroup of a group is termed '''potentially fully characteristic''' if there is an embedding of the bigger group in some group such that, in that embedding the subgroup becomes [[fully characteristic subgroup|fully characteristic]].
+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 [[defining ingredient::fully invariant subgroup|fully invariant]].
 ===Definition with symbols===
-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 characteristic subgroup|fully characteristic]] in <math>K</math>.
+A subgroup <math>H</math> of a group <math>G</math> is termed '''potentially fully invariant''' 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>.
-===Definition in terms of the potentially operator===
+==Formalisms==
+{{obtainedbyapplyingthe|potentially operator|fully characteristic subgroup}}
-The property of being potentially fully characteristic is obtained by applying the [[potentially operator]] to the property of being [[fully characteristic subgroup|fully characteristic]]. The potentially operator is an idempotent ascendant monotone operator.
+The property of being potentially fully invariant is obtained by applying the [[potentially operator]] to the property of being [[fully invariant subgroup|fully invariant]]. The potentially operator is an idempotent ascendant monotone operator.
 ==Relation with other properties==
@@ Line 23: / Line 22: @@
 ===Stronger properties===
-* [[Fully characteristic subgroup]]
+{| class="sortable" border="1"
-* [[Strongly potentially fully characteristic subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::fully invariant subgroup]] || invariant under all [[endomorphism]]s || (by definition) || (cyclic normal subgroup examples) ||
+|-
+| [[Weaker than::verbal subgroup]] || image of a [[word map]] || ([[verbal implies fully invariant|via fully invariant]]) || (via fully invariant) ||
+|-
+| [[Weaker than::potentially verbal subgroup]] || can be verbal inside a bigger group || follows from [[verbal implies fully invariant]] || (unclear) ||
+|-
+| [[Weaker than::normal-potentially fully invariant subgroup]] || can be fully invariant in a bigger group in which the original ambient group is normal || (by definition) || (unclear) ||
+|-
+| [[Weaker than::central subgroup of finite group]] || || [[central implies potentially fully invariant in finite]]|| any non-abelian group as a subgroup of itself||
+|-
+| [[cyclic normal subgroup]] of a [[finite group]] || || (via homocyclic normal) || ||
+|-
+| [[homocyclic normal subgroup]] of a [[finite group]] || || [[homocyclic normal implies potentially fully invariant in finite]] || ||
+|-
+| [[Weaker than::fully normalized potentially fully invariant subgroup]] || also a [[fully normalized subgroup]] || || ||
+|}
 ===Weaker properties===
-* [[Potentially characteristic subgroup]]
+{| class="sortable" border="1"
-* [[Extensible automorphism-invariant subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Normal subgroup]]
+|-
+| [[Stronger than::normal subgroup]] || || [[potentially fully invariant implies normal]] || [[normal not implies potentially fully invariant]] ||
+|}
-===Conjecture of equalling normality===
+===Incomparable properties===
-{{conjecturedtoequal|normality}}
+* [[Characteristic subgroup]]: See [[characteristic not implies potentially fully invariant]]
-Since the potentially operator is an idempotent monotone ascendant operator, and the property of being [[normal subgroup|normal]] is a fixed point of this operator, every potentially fully characteristic subgroup  is normal. The converse question: ''is every normal subgroup potentially fully characteristic?'' has not yet been answered.
 ==Metaproperties==
 {{intransitive}}
-{{fillin}}
-{{intersection-closed-open}}
-Is the intersection of two potentially characteristic subgroups potentially characteristic?
-==Property operators==
-===Left transiter===
-Every fully characteristic subgroup of a potentially fully characteristic subgroup is potentially fully characteristic. In fact, the same supergroup works.
-That is, suppose <math>M \le G \le H</math> with <math>M</math> fully characteristic in <math>G</math> and <math>G</math> potentially fully characteristic in <math>H</math>. Then, there exists a group <math>K</math> containing <math>H</math> such that both <math>G</math> and <math>H</math> are [[fully characteristic subgroup|fully characteristic]] in <math>K</math>. Then, we also have that <math>M</math> is fully characteristic in <math>K</math>, and hence <math>M</math> is potentially fully characteristic in <math>H</math>.