# Commutator-closed subgroup property

From Groupprops

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

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property

View a complete list of subgroup metaproperties

View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metapropertyVIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

## Definition

### Symbol-free definition

A subgroup property is termed **commutator-closed** if the commutator of any two subgroups, each of which has property in the whole group, also has property in the whole group.

### Definition with symbols

A subgroup property is termed **commutator-closed** if, given any group and subgroups such that and both satisfy property in , the commutator also satisfies property in .

## Relation with other metaproperties

### Stronger metaproperties

- Endo-invariance property:
`For full proof, refer: Endo-invariance implies commutator-closed`