Abelianization
From Groupprops
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.