Image-closed 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

A subgroup $H$ of a group $G$ is termed image-closed fully invariant in $G$ if, for any surjective homomorphism $\varphi:G \to K$, $\varphi(H)$ is a fully invariant subgroup of $K$.

Formalisms

In terms of the image condition operator

This property is obtained by applying the image condition operator to the property: fully invariant subgroup
View other properties obtained by applying the image condition operator

Relation with other properties

Stronger properties

• Verbal subgroup
• Normal Sylow subgroup
• Normal Hall subgroup
• Order-containing subgroup

Weaker properties

• Fully invariant subgroup
• Image-closed characteristic subgroup
• Characteristic subgroup

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
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Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

Image condition

YES: This subgroup property satisfies the image condition, i.e., under any surjective homomorphism, the image of a subgroup satisfying the property also satisfies the property
View other subgroup properties satisfying image condition

Intermediate subgroup condition

NO: This subgroup property does not satisfy the intermediate subgroup condition: it is possible to have a subgroup satisfying the property in the whole group but not satisfying the property in some 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 SUBGROUP CONDITION: View other subgroup properties not satisfying intermediate subgroup condition| View facts about intermediate subgroup condition