# Homomorph-containing subgroup

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 .

## Examples

VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this propertyVIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

## Relation with other properties

### Stronger properties

- Order-containing subgroup
- Subhomomorph-containing subgroup
- Variety-containing subgroup
- Normal Sylow subgroup
- Normal Hall subgroup
- Fully invariant direct factor
- Left-transitively homomorph-containing subgroup
- Right-transitively homomorph-containing subgroup

### Weaker properties

## Facts

- The omega subgroups of a group of prime power order are homomorph-containing.
`Further information: Omega subgroups are homomorph-containing`

## Metaproperties

BEWARE!This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Here is a summary:

### Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).

View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

For any group , the trivial subgroup and the whole group are both homomorph-containing.

### Transitivity

NO:This subgroup property isnottransitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole groupABOUT 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 subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

We can have subgroups such that is a homomorph-containing subgroup of and is a homomorph-containing subgroup of but is not a homomorph-containing subgroup of . `For full proof, refer: Homomorph-containment is not 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

If and is a homomorph-containing subgroup of , is also a homomorph-containing subgroup of . `For full proof, refer: Homomorph-containment satisfies intermediate subgroup condition`

### Join-closedness

YES:This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closedABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

If , are all homomorph-containing subgroups of , then so is the join of subgroups . `For full proof, refer: Homomorph-containment is strongly join-closed`

### Quotient-transitivity

This subgroup property is quotient-transitive: the corresponding quotient property is transitive.

View a complete list of quotient-transitive subgroup properties

If are groups such that is a homomorph-containing subgroup of and is a homomorph-containing subgroup of , then is a homomorph-containing subgroup of . `For full proof, refer: Homomorph-containment is quotient-transitive`