Subgroup whose commutator with any subset is normal

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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]

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Definition

A subgroup of a group is termed a subgroup whose commutator with any subset is normal if the commutator of that subgroup with any subset of the whole group is a normal subgroup of the whole group.

Relation with other properties

Stronger properties

• Transitively normal subgroup: For proof of the implication, refer Commutator of transitively normal subgroup and any subset is normal and for proof of its strictness (i.e. the reverse implication being false) refer Commutator with any subset is normal not implies transitively normal.
  • Direct factor
  • Central factor

Metaproperties

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

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