# Difference between revisions of "Homomorph-dominating 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]

## Definition

A subgroup $H$ of a group $G$ is termed homomorph-dominating in $G$ if, for any homomorphism $\varphi \in \operatorname{Hom}(H,G)$, there exists $g \in G$ such that $\varphi(H) \le gHg^{-1}$.

## Relation with other properties

### Conjunction with other properties

A homomorph-containing subgroup is precisely the same as a subgroup that is both normal and homomorph-dominating. For full proof, refer: Homomorph-dominating and normal equals homomorph-containing