# Focal series of a subgroup

## Contents

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:

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

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.