Normal subgroup whose focal subgroup equals its derived subgroup

Definition
A normal subgroup whose focal subgroup equals its commutator subgroup is a subgroup $$H$$ of a group $$G$$ satisfying the following equivalent conditions:


 * 1) $$H$$ is a normal subgroup of $$G$$ and $$[H,H] = \operatorname{Foc}_G(H)$$, i.e., $$H$$ is a subgroup whose focal subgroup equals its commutator subgroup.
 * 2) $$[G,H] = [H,H]$$.

Stronger properties

 * Weaker than::Direct factor
 * Weaker than::Central factor
 * Weaker than::Conjugacy-closed normal subgroup
 * Weaker than::Perfect normal subgroup

Weaker properties

 * Stronger than::Normal subgroup
 * Stronger than::Subgroup whose focal subgroup equals its commutator subgroup