Focal series of a subgroup

Definition
Suppose $$H$$ is a subgroup of a group $$G$$. The focal series of $$H$$ in $$G$$ is a series $$H = H_0 \ge H_1 \ge H_2 \ge \dots$$ where we define:

$$H_{n+1} = \operatorname{Foc}_G(H)$$.

In other words, each member of the series is the focal subgroup of its predecessor. The focal subgroup is defined as:

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

A subgroup whose focal series terminates at the trivial subgroup is termed a hyperfocal subgroup.