Abelianization

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 defining ingredient::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.