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.