# 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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## Contents

## Statement

### Verbal statement

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

### Statement with symbols

Suppose is a group and are normal subgroups of . Then, the commutator 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

### 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.