# Isomorph-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]

## Definition

QUICK PHRASES: contains all isomorphic subgroups, weakly closed in any ambient group

A subgroup of a group is termed an **isomorph-containing subgroup** if it satisfies the following equivalent conditions:

- Whenever is a subgroup of isomorphic to , .
- If is a subgroup of , is weakly closed in with respect to .

### Equivalence of definitions

`Further information: Isomorph-containing iff weakly closed in any ambient group`

## 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

- Isomorph-free subgroup: For a finite subgroup, and more generally, for a co-Hopfian subgroup, the two properties are equivalent.
- Homomorph-containing subgroup: Also related:
- Fully invariant direct factor:
`For full proof, refer: Equivalence of definitions of fully invariant direct factor`

### Weaker properties

- Characteristic subgroup:
*For proof of the implication, refer Isomorph-containing implies characteristic and for proof of its strictness (i.e. the reverse implication being false) refer Characteristic not implies isomorph-containing*. Also related: - Normal-isomorph-containing subgroup

## Metaproperties

### 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

`For full proof, refer: Isomorph-containment is not transitive`

### 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

### 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