Homomorph-containing subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed homomorph-containing if for any $$\varphi \in \operatorname{Hom}(H,G)$$ (i.e., any homomorphism of groups from $$H$$ to $$G$$), the image $$\varphi(H)$$ is contained in $$H$$.

Extreme examples

 * Every group is homomorph-containing as a subgroup of itself.
 * The trivial subgroup is homomorph-containing in any group.

Important classes of examples

 * Normal Sylow subgroups and normal Hall subgroups are homomorph-containing.
 * Subgroups defined as the subgroup generated by elements of specific orders, are all homomorph-containing subgroups. The omega subgroups of a group of prime power order are such examples.
 * The perfect core of a group is a homomorph-containing subgroup.

See also the section in this page.

Metaproperties
Here is a summary: