Potentially fully invariant subgroup

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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]

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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)	Intermediate notions
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
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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