Homomorph-containing subgroup

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Definition

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

Relation with other properties

Weaker properties

• Fully characteristic subgroup
• Strictly characteristic subgroup
• Characteristic subgroup
• Isomorph-free subgroup in case the subgroup is co-Hopfian as a group: it is not isomorphic to any proper subgroup of itself.

Facts

• The omega subgroups of a group of prime power order are homomorph-containing. Further information: Omega subgroups are homomorph-containing