# Subgroup whose focal subgroup equals its derived subgroup

From Groupprops

(Redirected from Subgroup whose focal subgroup equals its commutator subgroup)

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 a **subgroup whose focal subgroup equals its commutator subgroup** if we have the following condition. Let denote the focal subgroup of in :

.

Then, we require that:

,

i.e., the focal subgroup of equals its own commutator subgroup.

## Relation with other properties

### Stronger properties

- Direct factor
- Central factor
- Central subgroup
- Conjugacy-closed normal subgroup
- Conjugacy-closed subgroup:
*For proof of the implication, refer Conjugacy-closed implies focal subgroup equals commutator subgroup and for proof of its strictness (i.e. the reverse implication being false) refer Focal subgroup equals commutator subgroup not implies conjugacy-closed*.