Subgroup whose focal subgroup equals its intersection with the derived subgroup

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Definition

Suppose $G$ is a group and $H$ is a subgroup. Denote by ${\mathrm{F}}_{\mathrm{o}}(H)$ the focal subgroup of $H$ in $G$:

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

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

${\mathrm{F}}_{\mathrm{o}}(H)=H\cap [G,G]$.

Note that in general, we only have the containment ${\mathrm{F}}_{\mathrm{o}}(H)\le H\cap [G,G]$.

Relation with other properties

Stronger properties

• Sylow subgroup: For full proof, refer: Focal subgroup theorem
• Direct factor
• Normal subgroup whose focal subgroup equals its intersection with the commutator subgroup