In general, groups are not Abelian. However, there is always a group homomorphism h:G->G' to an Abelian group, and this homomorphism is called Abelianization. The homomorphism is abstractly described by its kernel, the commutator subgroup [G, G], which is the unique smallest normal subgroup of G such that the quotient group G' = G/[G, G] 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.

Abelian | commutator subgroup | group | homomorphism