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 characteristic 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

• Fully invariant subgroup
• Verbal subgroup
• Potentially verbal subgroup
• Normal-potentially fully invariant subgroup
• Central subgroup: For full proof, refer: central implies potentially fully invariant
• Cyclic normal subgroup: For full proof, refer: cyclic normal implies potentially fully invariant
• Homocyclic normal subgroup: For full proof, refer: homocyclic normal implies potentially fully invariant

Weaker properties

• Normal subgroup

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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