Normality is commutator-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., normal subgroup) satisfying a subgroup metaproperty (i.e., commutator-closed subgroup property)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about normal subgroup |Get facts that use property satisfaction of normal subgroup | Get facts that use property satisfaction of normal subgroup|Get more facts about commutator-closed subgroup property

Statement

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.

Related facts

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.

Further information: Endo-invariance implies 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.