Focal subgroup of a subgroup

Definition
Let $$H$$ be a subgroup of a group $$G$$. We define the focal subgroup of $$H$$ in the following equivalent ways:


 * The subgroup generated by the left quotients of pairs of elements of $$H$$ which are conjugate in $$G$$.
 * The subgroup generated by the right quotients of pairs of elements of $$H$$ which are conjugate in $$G$$.

We use the notation $$\operatorname{Foc}_G(H)$$ or $$H^*_G$$ for the focal subgroup of $$H$$ in $$G$$.

Note that the focal subgroup of $$H$$ in $$G$$ is contained within the commutator $$[H,G]$$, and contains the commutator $$[H,H]$$. In fact, we have the following string of inequalities:

$$[H,H] \le \operatorname{Foc}_G(H) \le H \cap [H,G] \le H \cap [G,G]$$.

Facts

 * A subgroup $$H \le G$$ for which $$[H,H] = \operatorname{Foc}_G(H)$$ is termed a subgroup whose focal subgroup equals its commutator subgroup. Any conjugacy-closed subgroup has this property.
 * A subgroup $$H \le G$$ for which $$\operatorname{Foc}_G(H) = H \cap [G,G]$$ is termed a subgroup whose focal subgroup equals its intersection with the commutator subgroup. Any Sylow subgroup has this property.