# Formula for commutator of element and product of two elements

From Groupprops

## Contents

## Statement

### With the left-action convention

Suppose is a group and are elements of . Then:

.

and:

Here and .

### With the right-action convention

Suppose is a group and are elements of . Then:

and:

here and .

## Related facts

### Applications

- Class two implies commutator map is endomorphism
- Commutator map is homomorphism if commutator is in centralizer
- Subgroup normalizes its commutator with any subset

## Proof

### With the left-action convention

**Given**: A group , elements

**To prove**: and where and .

**Proof**: We have:

and:

### With the right-action convention

**Given**: A group , elements

**To prove**: and where and .

**Proof**: We have:

and: