Normality is commutator-closed

Verbal statement
The commutator of two normal subgroups of a group is normal.

Statement with symbols
Suppose $$G$$ is a group and $$H,K$$ are normal subgroups of $$G$$. Then, the commutator $$[H,K]$$ is also a normal subgroup.

Property-theoretic statement
The subgroup property of being a normal subgroup satisfies the subgroup metaproperty of being commutator-closed.

Applications

 * Commutator of normal subgroups is normal closure of commutators of generators

Similar facts
Any endo-invariance property, i.e., any property that can be described as invariance under certain kinds of endomorphisms, is commutator-closed.

Some other instances of this general fact:


 * Characteristicity is commutator-closed
 * Full characteristicity is commutator-closed
 * Strict characteristicity is commutator-closed

Other facts about commutators and normal subgroups

 * Commutator of the whole group and a subset implies normal
 * Commutator of a normal subgroup and a subset implies 2-subnormal
 * Commutator of a 2-subnormal subgroup and a subset implies 3-subnormal

Analogues in other structures
In the variety of Lie rings, the analogue to normality is the notion of an ideal of a Lie ring, and the analogue of the commutator operation is the Lie bracket. The corresponding statement is then: Lie bracket of ideals is ideal.