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 Mathlib/Algebra/Category/Grp/Adjunctions.lean.

Main definitions

• Abelianization: defines the abelianization of a group G as the quotient of a group by its commutator subgroup.
• Abelianization.map: lifts a group homomorphism to a homomorphism between the abelianizations
• MulEquiv.abelianizationCongr: Equivalent groups have equivalent abelianizations

def Abelianization (G : Type u) [Group G] : Type u

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

Equations

• Abelianization G = (G ⧸ commutator G)

@[instance_reducible]

instance Abelianization.commGroup (G : Type u) [Group G] : CommGroup (Abelianization G)

Equations

• Abelianization.commGroup G = { toGroup := QuotientGroup.Quotient.group (commutator G), mul_comm := ⋯ }

@[instance_reducible]

instance Abelianization.instInhabited (G : Type u) [Group G] : Inhabited (Abelianization G)

Equations

• Abelianization.instInhabited G = { default := 1 }

def Abelianization.of {G : Type u} [Group G] : G →* Abelianization G

of is the canonical projection from G to its abelianization.

Equations

• Abelianization.of = { toFun := QuotientGroup.mk, map_one' := ⋯, map_mul' := ⋯ }

@[simp]

theorem Abelianization.mk_eq_of {G : Type u} [Group G] (a : G) : Quot.mk (⇑(QuotientGroup.leftRel (commutator G))) a = of a

@[simp]

theorem Abelianization.ker_of (G : Type u) [Group G] : of.ker = commutator G

theorem Abelianization.commutator_subset_ker {G : Type u} [Group G] {A : Type v} [CommGroup A] (f : G →* A) : commutator G ≤ f.ker

def Abelianization.lift {G : Type u} [Group G] {A : Type v} [CommGroup A] : (G →* A) ≃ (Abelianization G →* 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

• One or more equations did not get rendered due to their size.

@[simp]

theorem Abelianization.lift_apply_of {G : Type u} [Group G] {A : Type v} [CommGroup A] (f : G →* A) (x : G) : (lift f) (of x) = f x

theorem Abelianization.coe_lift_symm {G : Type u} [Group G] {A : Type v} [CommGroup A] : ⇑lift.symm = fun (x : Abelianization G →* A) => x.comp of

@[simp]

theorem Abelianization.lift_symm_apply {G : Type u} [Group G] {A : Type v} [CommGroup A] (f : Abelianization G →* A) : lift.symm f = f.comp of

theorem Abelianization.lift_unique {G : Type u} [Group G] {A : Type v} [CommGroup A] (f : G →* A) (φ : Abelianization G →* A) (hφ : ∀ (x : G), φ (of x) = f x) {x : Abelianization G} : φ x = (lift f) x

@[simp]

theorem Abelianization.lift_of {G : Type u} [Group G] : lift of = MonoidHom.id (Abelianization G)

theorem Abelianization.hom_ext {G : Type u} [Group G] {A : Type v} [Monoid A] (φ ψ : Abelianization G →* A) (h : φ.comp of = ψ.comp of) : φ = ψ

See note [partially-applied ext lemmas].

theorem Abelianization.hom_ext_iff {G : Type u} [Group G] {A : Type v} [Monoid A] {φ ψ : Abelianization G →* A} : φ = ψ ↔ φ.comp of = ψ.comp of

def Abelianization.map {G : Type u} [Group G] {H : Type v} [Group H] (f : G →* H) : Abelianization G →* Abelianization H

The map operation of the Abelianization functor

Equations

• Abelianization.map f = Abelianization.lift (Abelianization.of.comp f)

@[simp]

theorem Abelianization.lift_of_comp {G : Type u} [Group G] {H : Type v} [Group H] (f : G →* H) : lift (of.comp f) = map f

Use map as the preferred simp normal form.

@[simp]

theorem Abelianization.map_of {G : Type u} [Group G] {H : Type v} [Group H] (f : G →* H) (x : G) : (map f) (of x) = of (f x)

@[simp]

theorem Abelianization.map_id {G : Type u} [Group G] : map (MonoidHom.id G) = MonoidHom.id (Abelianization G)

@[simp]

theorem Abelianization.map_comp {G : Type u} [Group G] {H : Type v} [Group H] (f : G →* H) {I : Type w} [Group I] (g : H →* I) : (map g).comp (map f) = map (g.comp f)

@[simp]

theorem Abelianization.map_map_apply {G : Type u} [Group G] {H : Type v} [Group H] (f : G →* H) {I : Type w} [Group I] {g : H →* I} {x : Abelianization G} : (map g) ((map f) x) = (map (g.comp f)) x

def MulEquiv.abelianizationCongr {G : Type u} [Group G] {H : Type v} [Group H] (e : G ≃* H) : Abelianization G ≃* Abelianization H

Equivalent groups have equivalent abelianizations

Equations

• e.abelianizationCongr = { toFun := ⇑(Abelianization.map e.toMonoidHom), invFun := ⇑(Abelianization.map e.symm.toMonoidHom), left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯ }

@[simp]

theorem abelianizationCongr_of {G : Type u} [Group G] {H : Type v} [Group H] (e : G ≃* H) (x : G) : e.abelianizationCongr (Abelianization.of x) = Abelianization.of (e x)

@[simp]

theorem abelianizationCongr_refl {G : Type u} [Group G] : (MulEquiv.refl G).abelianizationCongr = MulEquiv.refl (Abelianization G)

@[simp]

theorem abelianizationCongr_symm {G : Type u} [Group G] {H : Type v} [Group H] (e : G ≃* H) : e.abelianizationCongr.symm = e.symm.abelianizationCongr

@[simp]

theorem abelianizationCongr_trans {G : Type u} [Group G] {H : Type v} [Group H] {I : Type v} [Group I] (e : G ≃* H) (e₂ : H ≃* I) : e.abelianizationCongr.trans e₂.abelianizationCongr = (e.trans e₂).abelianizationCongr

def Abelianization.equivOfComm {H : Type u_1} [CommGroup H] : H ≃* Abelianization H

An Abelian group is equivalent to its own abelianization.

Equations

• Abelianization.equivOfComm = { toFun := ⇑Abelianization.of, invFun := ⇑(Abelianization.lift (MonoidHom.id H)), left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯ }

@[simp]

theorem Abelianization.equivOfComm_apply {H : Type u_1} [CommGroup H] (a : H) : equivOfComm a = of a

@[simp]

theorem Abelianization.equivOfComm_symm_apply {H : Type u_1} [CommGroup H] (a : Abelianization H) : equivOfComm.symm a = (lift (MonoidHom.id H)) a

@[instance_reducible]

instance instUniqueAbelianization (G : Type u) [Group G] [Unique G] : Unique (Abelianization G)

Equations

• instUniqueAbelianization G = Quotient.instUniqueQuotient (QuotientGroup.leftRel (commutator G))