# Commutator of two subgroups

## Contents

## Definition

### Symbol-free definition

The **commutator** of two subgroups of a group is defined as the subgroup generated by commutators between elements in the two subgroups.

### Definition with symbols

Suppose is a group and and are subgroups of . The **commutator** of the subgroups and , denoted , is defined as:

where:

is the commutator of the elements and .

Note that there are two conventions for commutators; in some other conventions:

.

Whatever the convention, the set of commutators is the same; the commutator of and in the former convention equals the commutator of and in the latter convention.

## Facts

### Commutator, closure and join

If are subgroups, let denote the closure of under the action of . Define analogously. We then have:

- is a normal subgroup inside . In fact, , where normalizes .
- is a normal subgroup inside . In fact, where normalizes .
- is a normal subgroup inside . Both and are normal inside , with .

`For full proof, refer: Commutator of two subgroups is normal in join`

### Normalizing characterized in terms of commutators

For subgroups , is contained in the normalizer of if and only if . (In particular, is normal if and only if ).

Similarly, is contained in the normalizer of if and only if . Thus, the subgroups and normalize each other iff . In particular, if both subgroups are normal, their commutator is contained in their intersection.

### Permuting subgroups characterized in terms of commutators

Subgroups are permuting subgroups if and only if ; in other words, the commutator of the subgroups is contained in their product.

### Normal closure and quotient

The commutator of two subgroups need not, in general, be a normal subgroup. The normal closure of the commutator of two subgroups is of greater interest. If denotes the normal closure of for subgroups of , then the images of and in commute element-wise. Conversely, any normal subgroup for which the images of and commute element-wise in the quotient, must be contained in .

However, in the special case when both and are normal, the commutator of the subgroups is also normal. `Further information: Commutator of normal subgroups is normal`