Subgroup whose focal subgroup equals its derived subgroup

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Definition

A subgroup $H$ of a group $G$ is termed a subgroup whose focal subgroup equals its commutator subgroup if we have the following condition. Let $\operatorname {Foc} _{G}(H)$ denote the focal subgroup of $H$ in $G$:

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

Then, we require that:

$\operatorname {Foc} _{G}(H)=[H,H]$,

i.e., the focal subgroup of $H$ equals its own commutator subgroup.

Relation with other properties

Stronger properties

• Direct factor
• Central factor
• Central subgroup
• Conjugacy-closed normal subgroup
• Conjugacy-closed subgroup: For proof of the implication, refer Conjugacy-closed implies focal subgroup equals commutator subgroup and for proof of its strictness (i.e. the reverse implication being false) refer Focal subgroup equals commutator subgroup not implies conjugacy-closed.