# Homomorph-dominating subgroup

From Groupprops

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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 of a group is termed **homomorph-dominating** in if, for any homomorphism , there exists such that .

## Relation with other properties

### Stronger properties

### Weaker properties

- Endomorph-dominating subgroup
- Isomorph-conjugate subgroup if the whole group is a co-Hopfian group -- it is not isomorphic to any proper subgroup of itself.

### 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`