# Homomorph-containing 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-containing** if for any **Failed to parse (unknown function "\operatornanem"): \varphi \in \operatornanem{Hom}(H,G)**
, the image is contained in .

## 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`