Direct projection-invariant subgroup

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 said to be direct projection-invariant if it is invariant under any projection map from the group to a direct factor.

Definition with symbols

A subgroup $H$ of a group $G$ is said to be direct projection-invariant if given any direct factor $L$ of $G$ and the projection map $\pi :G\to L$ , we have $\pi (H)\leq H$ .

Formalisms

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
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This subgroup property can be expressed in the function restriction formalism as the invariance property corresponding to projection on a direct factor, or in other words, as:

projection on a direct factor $\to$ Function

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

Intersection-closedness

YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are intersection-closed | View variations of this property that are not intersection-closed
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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

The whole group is direct projection-invariant as a subgroup of itself. The trivial subgroup isalso direct projection-invariant.