abelianization
The abelianization of a group is , the quotient (http://planetmath.org/QuotientGroup) of by its derived subgroup.
The abelianization of is the largest abelian quotient of , in the sense that if is a normal subgroup
of then is abelian if and only if .In particular, every abelian quotient of is a homomorphic image
of .
If is an abelian group and is a homomorphism (http://planetmath.org/GroupHomomorphism),then there is a unique homomorphism such that, where is the canonical projection.