Focal series of a subgroup

This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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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:

$H_{n+1}={\mathrm{F}}_{\mathrm{o}}(H)$.

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

${\mathrm{F}}_{\mathrm{o}}(K)=⟨x{y}^{-1}\mid x,y\in K,\mathrm{\exists }g\in G,gx{g}^{-1}=y⟩$.

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