# Difference between revisions of "Homomorph-dominating subgroup"

From Groupprops

(New page: {{wikilocal}} {{subgroup property}} ==Definition== A subgroup <math>H</math> of a group <math>G</math> is termed '''homomorph-dominating''' in <math>G</math> if, for any homomorp...) |
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===Weaker properties=== | ===Weaker properties=== | ||

+ | * [[Stronger than::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. | * [[Isomorph-conjugate subgroup]] if the whole group is a [[co-Hopfian group]] -- it is not isomorphic to any proper subgroup of itself. | ||

## Latest revision as of 18:23, 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-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`