# Subgroup whose derived subgroup equals its intersection with whole derived subgroup

From Groupprops

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]

## Contents

## Definition

A subgroup is termed a **subgroup whose derived subgroup equals its intersection with whole derived subgroup** or **subgroup whose commutator subgroup equals its intersection with whole commutator subgroup** if .

## Relation with other properties

### Stronger properties

- Conjugacy-closed Hall subgroup:
`For full proof, refer: Conjugacy-closed and Hall implies commutator subgroup equals intersection with whole commutator subgroup` - Perfect subgroup
- Direct factor
- Retract:
`For full proof, refer: Retract implies commutator subgroup equals intersection with whole commutator subgroup` - Cocentral subgroup:
`For full proof, refer: Cocentral implies commutator subgroup equals intersection with whole commutator subgroup`

### Weaker properties

- Subgroup whose focal subgroup equals its intersection with the commutator subgroup
- Subgroup whose focal subgroup equals its commutator 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

If are groups such that and , then . `For full proof, refer: Commutator subgroup equals intersection with whole commutator subgroup 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

If such that , then we also have .

`For full proof, refer: Commutator subgroup equals intersection with whole commutator subgroup satisfies intermediate subgroup condition`