Abelianization

Abelianization of a group G is the quotient G/[G,G], where [G,G] is the commutator subgroup generated by all commutators [g,h] = g^{-1}h^{-1}gh. It is the largest abelian quotient of G: for any homomorphism f: G -> A into an abelian group A, there exists a unique homomorphism f̄: G/[G,G] -> A such that f = f̄ ∘ π, where π: G -> G/[G,G] is the natural projection.

Construction and computation: The commutator subgroup [G,G] is the subgroup generated by all commutators. If G has a presentation <X | R>, then G_ab has presentation <X | R, [x,y] for all x,y in X>, i.e., one imposes commutativity on generators. In particular, the abelianization of a free group F_r is a free abelian group Z^r, and the abelianization of Z is Z.

Examples: The abelianization of the symmetric group S3 is S3/[S3,S3] ≅ C2. More generally, the abelianization of a finite group is its quotient by the subgroup generated by all commutators; a perfect group has trivial abelianization. The abelianization of a direct product satisfies (G × H)_ab ≅ G_ab × H_ab, reflecting functorial behavior.

Relation to other concepts: The first derived subgroup G' equals [G,G], and the abelianization is G/G'. The notion extends to topological groups via the closed commutator subgroup, yielding a topological abelian quotient; it is closely related to the first homology group H_1 for spaces via the fundamental group.