Normal subgroup whose derived subgroup equals its intersection with whole derived subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a normal subgroup whose commutator subgroup equals its intersection with whole commutator subgroup if it satisfies the following equivalent conditions:


 * 1) $$H$$ is a normal subgroup of $$G$$ and $$[H,H] = H \cap [G,G]$$.
 * 2) $$[H,H] = [G,H] = H \cap [G,G]$$.

Stronger properties

 * Weaker than::Direct factor
 * Weaker than::Perfect normal subgroup

Weaker properties

 * Stronger than::Normal subgroup whose focal subgroup equals its commutator subgroup
 * Stronger than::Normal subgroup whose focal subgroup equals its intersection with the commutator subgroup