Normal subgroup whose derived subgroup equals its intersection with whole derived subgroup

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Normal subgroup whose derived subgroup equals its intersection with whole derived subgroup, all facts related to Normal subgroup whose derived subgroup equals its intersection with whole derived subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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

Definition

A subgroup $H$ of a group $G$ is termed a normal subgroup whose commutator subgroup equals its intersection with whole commutator subgroup if it satisfies the following equivalent conditions:

1. $H$ is a normal subgroup of $G$ and $[H,H]=H\cap [G,G]$.
2. $[H,H]=[G,H]=H\cap [G,G]$.

Relation with other properties

Stronger properties

• Direct factor
• Perfect normal subgroup

Weaker properties

• Normal subgroup whose focal subgroup equals its commutator subgroup
• Normal subgroup whose focal subgroup equals its intersection with the commutator subgroup