Abelianization

The Abelianization of a group is the quotient group , together with the canonical group homomorphism . Its group kernel is the commutator subgroup , the unique smallest normal subgroup of for which 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.