Abelianization

In general, groups are not Abelian. However, there is always a group homomorphism to an Abelian group, and this homomorphism is called Abelianization. The homomorphism is abstractly described by its kernel, the commutator subgroup , which is the unique smallest normal subgroup of such that the quotient group is Abelian. Roughly speaking, in any expression, every product becomes commutative after Abelianization. As a consequence, some previously unequal expressions may become equal, or even represent the identity element.

For example, in the eight-element quaternion group , the commutator subgroup is . The Abelianization of is a copy of , and for instance, in the Abelianization.