Direct projection-invariant subgroup

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 $$\pi(H) \le H$$.

Formalisms
This subgroup property can be expressed in the function restriction formalism as the corresponding to projection on a direct factor, or in other words, as:

projection on a direct factor $$\to$$ Function

Stronger properties

 * Fully characteristic subgroup
 * Retraction-invariant subgroup
 * Direct projection-invariant direct factor
 * Direct projection-invariant central factor

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