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)
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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

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:

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.