# Subhomomorph-containing subgroup

From Groupprops

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]

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with sub-homomorph-containing subgroup

## Definition

A subgroup of a group is termed a **subhomomorph-containing subgroup** if for any subgroup and any homomorphism of groups , we have .

## Relation with other properties

### Stronger properties

### Weaker properties

- Homomorph-containing subgroup
- Subisomorph-containing subgroup
- Subhomomorph-dominating subgroup
- Transfer-closed fully invariant subgroup
- Intermediately fully invariant subgroup
- Fully invariant subgroup
- Intermediately strictly characteristic subgroup
- Strictly characteristic subgroup
- Transfer-closed characteristic subgroup
- Intermediately characteristic subgroup
- Characteristic subgroup

## Metaproperties

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

Suppose are groups such that is a subhomomorph-containing subgroup of and is a subhomomorph-containing subgroup of . Then, is a subhomomorph-containing subgroup of . `For full proof, refer: Subhomomorph-containment is transitive`

### Intermediate subgroup condition

YES:This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

Suppose are groups such that is a subhomomorph-containing subgroup of . Then, is a subhomomorph-containing subgroup of . `For full proof, refer: Subhomomorph-containment satisfies intermediate subgroup condition`