Subgroup whose derived subgroup equals its intersection with whole 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 $H\leq G$ is termed a subgroup whose derived subgroup equals its intersection with whole derived subgroup or subgroup whose commutator subgroup equals its intersection with whole commutator subgroup if $[H,H]=H\cap [G,G]$.

Relation with other properties

Stronger properties

• Conjugacy-closed Hall subgroup: For full proof, refer: Conjugacy-closed and Hall implies commutator subgroup equals intersection with whole commutator subgroup
• Perfect subgroup
• Direct factor
• Retract: For full proof, refer: Retract implies commutator subgroup equals intersection with whole commutator subgroup
• Cocentral subgroup: For full proof, refer: Cocentral implies commutator subgroup equals intersection with whole commutator subgroup

Weaker properties

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

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

If $H\leq K\leq G$ are groups such that $[K,K]=K\cap [G,G]$ and $[H,H]=H\cap [K,K]$, then $[H,H]=H\cap [G,G]$. For full proof, refer: Commutator subgroup equals intersection with whole commutator subgroup is transitive

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If $H\leq K\leq G$ such that $[H,H]=H\cap [G,G]$, then we also have $[H,H]=H\cap [K,K]$.

For full proof, refer: Commutator subgroup equals intersection with whole commutator subgroup satisfies intermediate subgroup condition