Abelianization

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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.