# Subgroup whose derived subgroup equals its intersection with whole derived 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 $H \le G$ 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 $[H,H] = H \cap [G,G]$.

## 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 transitive
ABOUT 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 $H \le K \le G$ are groups such that $[K,K] = K \cap [G,G]$ and $[H,H] = H \cap [K,K]$, then $[H,H] = H \cap [G,G]$. 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 condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If $H \le K \le G$ such that $[H,H] = H \cap [G,G]$, then we also have $[H,H] = H \cap [K,K]$.