# Subgroup whose focal subgroup equals its intersection with the derived subgroup

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

Suppose $G$ is a group and $H$ is a subgroup. Denote by $\operatorname{Foc}_G(H)$ the focal subgroup of $H$ in $G$:

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

$H$ is termed a subgroup whose focal subgroup equals its intersection with the commutator subgroup if we have:

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

Note that in general, we only have the containment $\operatorname{Foc}_G(H) \le H \cap [G,G]$.