Abelian group

Abelianization

The abelianization of a group is the largest abelian quotient of . group For a group is abelianized by taking the quotient with the Commutator subgroup .

Main theorem

Let . The quotient group is abelian iff . group

Proof

is abelian iff for all , and the latter holds iff for all .

Universal property

Abelianization has a unique extension to a functor from Category of groups into Category of abelian groups so that the projection becomes a natural transformation . This is done by defining with the help of the following universal property:

is abelian. If is an abelian group and is a homomorphism, then there exists a unique such that , i.e. the following diagram commutes

Proof

is abelian by construction. By properties of the Kernel of a homomorphism into an abelian group, the universal property is equivalent to that of the quotient group.

This of course forms a Free-forgetful adjunction

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