mathlib documentation

group_theory.abelianization

The abelianization of a group #

This file defines the commutator and the abelianization of a group. It furthermore prepares for the result that the abelianization is left adjoint to the forgetful functor from abelian groups to groups, which can be found in algebra/category/Group/adjunctions.

Main definitions #

@[instance]
def commutator.normal (G : Type u) [group G] :
def commutator (G : Type u) [group G] :

The commutator subgroup of a group G is the normal subgroup generated by the commutators [p,q]=p*q*p⁻¹*q⁻¹.

Equations
def abelianization (G : Type u) [group G] :
Type u

The abelianization of G is the quotient of G by its commutator subgroup.

Equations
@[instance]
Equations
def abelianization.of {G : Type u} [group G] :

of is the canonical projection from G to its abelianization.

Equations
theorem abelianization.commutator_subset_ker {G : Type u} [group G] {A : Type v} [comm_group A] (f : G →* A) :
def abelianization.lift {G : Type u} [group G] {A : Type v} [comm_group A] :

If f : G → A is a group homomorphism to an abelian group, then lift f is the unique map from the abelianization of a G to A that factors through f.

Equations
@[simp]
theorem abelianization.lift.of {G : Type u} [group G] {A : Type v} [comm_group A] (f : G →* A) (x : G) :
theorem abelianization.lift.unique {G : Type u} [group G] {A : Type v} [comm_group A] (f : G →* A) (φ : abelianization G →* A) (hφ : ∀ (x : G), φ (abelianization.of x) = f x) {x : abelianization G} :
@[ext]
theorem abelianization.hom_ext {G : Type u} [group G] {A : Type v} [monoid A] (φ ψ : abelianization G →* A) (h : φ.comp abelianization.of = ψ.comp abelianization.of) :
φ = ψ

See note [partially-applied ext lemmas].