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

• commutator: defines the commutator of a group G as a subgroup of G.
• 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 commutator (G : Type u) [Group G] : Subgroup 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

• commutator G = ⁅⊤, ⊤⁆

instance instNormalCommutator (G : Type u) [Group G] : (commutator G).Normal

theorem commutator_def (G : Type u) [Group G] : commutator G = ⁅⊤, ⊤⁆

theorem commutator_eq_closure (G : Type u) [Group G] : commutator G = Subgroup.closure (commutatorSet G)

theorem commutator_eq_normalClosure (G : Type u) [Group G] : commutator G = Subgroup.normalClosure (commutatorSet G)

theorem Subgroup.map_subtype_commutator {G : Type u} [Group G] (H : Subgroup G) : map H.subtype (_root_.commutator ↥H) = ⁅H, H⁆

instance commutator_characteristic (G : Type u) [Group G] : (commutator G).Characteristic

instance instFGSubtypeMemSubgroupCommutatorOfFiniteElemCommutatorSet (G : Type u) [Group G] [Finite ↑(commutatorSet G)] : Group.FG ↥(commutator G)

theorem rank_commutator_le_card (G : Type u) [Group G] [Finite ↑(commutatorSet G)] : Group.rank ↥(commutator G) ≤ Nat.card ↑(commutatorSet G)

theorem commutator_centralizer_commutator_le_center (G : Type u) [Group G] : ⁅Subgroup.centralizer ↑(commutator G), Subgroup.centralizer ↑(commutator G)⁆ ≤ Subgroup.center G

theorem mem_commutatorSet_of_isConj_sq (G : Type u) [Group G] {g : G} (hg : IsConj g (g ^ 2)) : g ∈ commutatorSet G

If g is conjugate to g ^ 2, then g is a commutator

theorem map_commutator_eq (G : Type u) [Group G] {H : Type u_1} [Group H] (f : G →* H) : Subgroup.map f (commutator G) = ⁅f.range, f.range⁆

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

Equations

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

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

Equations

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

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

Equations

• Abelianization.instUnique G = Quotient.instUniqueQuotient (QuotientGroup.leftRel (commutator G))

instance Abelianization.instFintypeOfDecidablePredMemSubgroupCommutator (G : Type u) [Group G] [Fintype G] [DecidablePred fun (x : G) => x ∈ commutator G] : Fintype (Abelianization G)

Equations

• Abelianization.instFintypeOfDecidablePredMemSubgroupCommutator G = QuotientGroup.fintype (commutator G)

instance Abelianization.instFinite (G : Type u) [Group G] [Finite G] : Finite (Abelianization G)

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

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 (Abelianization.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

def commutatorRepresentatives (G : Type u) [Group G] : Set (G × G)

Representatives (g₁, g₂) : G × G of commutators ⁅g₁, g₂⁆ ∈ G.

Equations

• commutatorRepresentatives G = Set.range fun (g : ↑(commutatorSet G)) => (Exists.choose ⋯, ⋯.choose)

instance instFiniteElemProdCommutatorRepresentativesOfCommutatorSet (G : Type u) [Group G] [Finite ↑(commutatorSet G)] : Finite ↑(commutatorRepresentatives G)

def closureCommutatorRepresentatives (G : Type u) [Group G] : Subgroup G

Subgroup generated by representatives g₁ g₂ : G of commutators ⁅g₁, g₂⁆ ∈ G.

Equations

• closureCommutatorRepresentatives G = Subgroup.closure (Prod.fst '' commutatorRepresentatives G ∪ Prod.snd '' commutatorRepresentatives G)

instance closureCommutatorRepresentatives_fg (G : Type u) [Group G] [Finite ↑(commutatorSet G)] : Group.FG ↥(closureCommutatorRepresentatives G)

theorem rank_closureCommutatorRepresentatives_le (G : Type u) [Group G] [Finite ↑(commutatorSet G)] : Group.rank ↥(closureCommutatorRepresentatives G) ≤ 2 * Nat.card ↑(commutatorSet G)

theorem image_commutatorSet_closureCommutatorRepresentatives (G : Type u) [Group G] : ⇑(closureCommutatorRepresentatives G).subtype '' commutatorSet ↥(closureCommutatorRepresentatives G) = commutatorSet G

theorem card_commutatorSet_closureCommutatorRepresentatives (G : Type u) [Group G] : Nat.card ↑(commutatorSet ↥(closureCommutatorRepresentatives G)) = Nat.card ↑(commutatorSet G)

theorem card_commutator_closureCommutatorRepresentatives (G : Type u) [Group G] : Nat.card ↥(commutator ↥(closureCommutatorRepresentatives G)) = Nat.card ↥(commutator G)

instance instFiniteElemSubtypeMemSubgroupClosureCommutatorRepresentativesCommutatorSet (G : Type u) [Group G] [Finite ↑(commutatorSet G)] : Finite ↑(commutatorSet ↥(closureCommutatorRepresentatives G))

theorem Subgroup.Normal.quotient_commutative_iff_commutator_le {G : Type u} [Group G] {N : Subgroup G} [N.Normal] : (Std.Commutative fun (x1 x2 : G ⧸ N) => x1 * x2) ↔ _root_.commutator G ≤ N

theorem Subgroup.Normal.commutator_le_of_self_sup_commutative_eq_top {G : Type u} [Group G] {N : Subgroup G} [N.Normal] {H : Subgroup G} (hHN : N ⊔ H = ⊤) (hH : IsMulCommutative ↥H) : _root_.commutator G ≤ N

If N is a normal subgroup of G and H a commutative subgroup such that H ⊔ N = ⊤, then N contains commutator G.