# Abelianization

## Definition

### Abelianization as a group

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

1. It is the quotient of the group by its commutator subgroup: in other words, it is the group $G/[G,G]$.
2. It is the quotient of $G$ by the relation $xy = yx$.
3. It is an Abelian group $A$ such that there exists a surjective homomorphism $f:G \to A$ with the following property. Whenever $\varphi:G \to H$ is a homomorphism and $H$ is an Abelian group, there is a unique homomorphism $\psi: A \to H$ such that $\varphi = \psi \circ f$.

### Abelianization as a homomorphism

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

1. It is the quotient map $G \to G/[G,G]$, where the kernel, $[G,G]$, is the commutator subgroup of $G$.
2. It is a homomorphism $f: G \to A$ to an Abelian group $A$ with the following property. Whenever $\varphi:G \to H$ is a homomorphism and $H$ is an Abelian group, there is a unique homomorphism $\psi: A \to H$ such that $\varphi = \psi \circ f$.

### Abelianization as a functor

The Abelianization is a functor $\operatorname{Ab}$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 $\operatorname{Ab}$ (where $\operatorname{Ab}$ 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.