Normal subgroup whose focal subgroup equals its intersection with the derived subgroup

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


 * 1) $$H$$ is a defining ingredient::normal subgroup of $$G$$ and $$\operatorname{Foc}_G(H) = H \cap [G,G]$$, i.e., $$H$$ is a defining ingredient::subgroup whose focal subgroup equals its intersection with the commutator subgroup (where the defining ingredient::focal subgroup is the subgroup generated by quotients of elements of $$H$$ that are conjugate in $$G$$).
 * 2) $$[G,H] = H \cap [G,G]$$.

Stronger properties

 * Weaker than::Normal Sylow subgroup
 * Weaker than::Normal Hall subgroup
 * Weaker than::Direct factor