# Difference between revisions of "Homomorph-containing subgroup"

From Groupprops

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===Weaker properties=== | ===Weaker properties=== | ||

+ | * [[Stronger than::Intermediately fully characteristic subgroup]] | ||

* [[Stronger than::Fully characteristic subgroup]] | * [[Stronger than::Fully characteristic subgroup]] | ||

* [[Stronger than::Strictly characteristic subgroup]] | * [[Stronger than::Strictly characteristic subgroup]] | ||

+ | * [[Stronger than::Intermediately characteristic subgroup]] | ||

* [[Stronger than::Characteristic subgroup]] | * [[Stronger than::Characteristic subgroup]] | ||

* [[Isomorph-free subgroup]] in case the subgroup is [[co-Hopfian group|co-Hopfian as a group]]: it is not isomorphic to any proper subgroup of itself. | * [[Isomorph-free subgroup]] in case the subgroup is [[co-Hopfian group|co-Hopfian as a group]]: it is not isomorphic to any proper subgroup of itself. |

## Revision as of 11:35, 19 September 2008

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

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 , the image is contained in .

## Relation with other properties

### Weaker properties

- Intermediately fully characteristic subgroup
- Fully characteristic subgroup
- Strictly characteristic subgroup
- Intermediately 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`