# Abelianization

## Contents

## Definition

### Abelianization as a group

The **Abelianization** of a group is defined in the following equivalent ways:

- It is the quotient of the group by its commutator subgroup: in other words, it is the group .
- It is the quotient of by the relation .
- It is an Abelian group such that there exists a surjective homomorphism with the following property. Whenever is a homomorphism and is an Abelian group, there is a unique homomorphism such that .

### Abelianization as a homomorphism

The **Abelianization** of a group is defined in the following equivalent ways:

- It is the quotient map , where the kernel, , is the commutator subgroup of .
- It is a homomorphism to an Abelian group with the following property. Whenever is a homomorphism and is an Abelian group, there is a unique homomorphism such that .

### Abelianization as a functor

The Abelianization is a functor from the category of groups to the subcategory which is the category of Abelian groups, along with a natural transformation from the identity functor on the category of groups to the functor (where is viewed as a self-functor on the category of Abelian groups), defined as follows:

- On objects: It sends each group to the quotient group by its commutator subgroup. The corresponding natural transformation is the quotient map.
- On morphisms: The morphism is the unique one so that the quotient map described here
*is*a natural transformation.

## Related terminology

- Abelian group: A group such that the quotient map to is Abelianization is the identity map.
- Perfect group: A group whose Abelianization is trivial.