Subgroup whose focal subgroup equals its derived subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a subgroup whose focal subgroup equals its commutator subgroup if we have the following condition. Let $$\operatorname{Foc}_G(H)$$ denote the focal subgroup of $$H$$ in $$G$$:

$$\operatorname{Foc}_G(H) = \langle xy^{-1} \mid x,y \in H, \exists g \in G, gxg^{-1} = y \rangle$$.

Then, we require that:

$$\operatorname{Foc}_G(H) = [H,H]$$,

i.e., the focal subgroup of $$H$$ equals its own defining ingredient::commutator subgroup.

Stronger properties

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