Subgroup whose focal subgroup equals its 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]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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 ${\mathrm{F}}_{\mathrm{o}}(H)$ denote the focal subgroup of $H$ in $G$:

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

Then, we require that:

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

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.