Subgroup whose derived subgroup equals its intersection with whole derived subgroup

From Groupprops
Jump to: navigation, search
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]

Definition

A subgroup H \le 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

Weaker properties

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 \le K \le 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 \le K \le 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