Normal subgroup whose focal subgroup equals its derived subgroup

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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and subgroup whose focal subgroup equals its commutator subgroup
View other subgroup property conjunctions | view all subgroup properties

Definition

A normal subgroup whose focal subgroup equals its commutator subgroup is a subgroup $H$ of a group $G$ satisfying the following equivalent conditions:

1. $H$ is a normal subgroup of $G$ and $[H,H]={\mathrm{F}}_{\mathrm{o}}(H)$, i.e., $H$ is a subgroup whose focal subgroup equals its commutator subgroup.
2. $[G,H]=[H,H]$.

Relation with other properties

Stronger properties

• Direct factor
• Central factor
• Conjugacy-closed normal subgroup
• Perfect normal subgroup

Weaker properties

• Normal subgroup
• Subgroup whose focal subgroup equals its commutator subgroup