Abelianization

The abelianization of G is its largest abelian quotient,

Gab = G / [G, G],

the quotient by the commutator subgroup [G, G]. Collapsing [G, G] forces all elements to commute, and it is the smallest normal subgroup that does so — so Gᵃᵇ is abelian, and no larger quotient of G is.

Universal property

Every homomorphism from G to an abelian group factors uniquely through the quotient map G → Gᵃᵇ. The commutators [a, b] = a⁻¹b⁻¹ab are exactly what every such map must send to the identity.

Key facts

• Gᵃᵇ is abelian, and [G, G] is the unique smallest normal subgroup with abelian quotient.
• G is abelian ⟺ [G, G] = {e} ⟺ Gᵃᵇ = G.
• G is perfect ⟺ [G, G] = G ⟺ Gᵃᵇ = {e} — no non-trivial abelian quotient.
• |Gᵃᵇ| = [G : G'] equals the number of 1-dimensional complex representations of G.

Examples

More generally Sₙᵃᵇ = C₂ for every n ≥ 2, realized by the sign homomorphism — so every homomorphism from Sₙ to an abelian group is a power of the sign.

See also

• Commutator subgroup — the [G, G] quotiented out.
• Derived series — iterating G ↦ [G, G].
• Abelian group — where Gᵃᵇ lives.
• Wikipedia: Commutator subgroup § Abelianization