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.
See alsoAbelian, Commutator Subgroup, Group, Homomorphism
Portions of this entry contributed by Todd Rowland