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Mathlib.GroupTheory.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 #

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.

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

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

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      of is the canonical projection from G to its abelianization.

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        @[simp]
        theorem Abelianization.mk_eq_of {G : Type u} [Group G] (a : G) :
        Quot.mk Setoid.r a = Abelianization.of a
        def Abelianization.lift {G : Type u} [Group G] {A : Type v} [CommGroup 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.

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          @[simp]
          theorem Abelianization.lift.of {G : Type u} [Group G] {A : Type v} [CommGroup A] (f : G →* A) (x : G) :
          ↑(Abelianization.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 = ↑(Abelianization.lift f) x
          @[simp]
          theorem Abelianization.lift_of {G : Type u} [Group G] :
          Abelianization.lift Abelianization.of = MonoidHom.id (Abelianization G)
          theorem Abelianization.hom_ext {G : Type u} [Group G] {A : Type v} [Monoid A] (φ : Abelianization G →* A) (ψ : Abelianization G →* A) (h : MonoidHom.comp φ Abelianization.of = MonoidHom.comp ψ Abelianization.of) :
          φ = ψ

          See note [partially-applied ext lemmas].

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

          The map operation of the Abelianization functor

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

            Equivalent groups have equivalent abelianizations

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              @[simp]
              theorem abelianizationCongr_of {G : Type u} [Group G] {H : Type v} [Group H] (e : G ≃* H) (x : G) :
              ↑(MulEquiv.abelianizationCongr e) (Abelianization.of x) = Abelianization.of (e x)
              @[simp]
              theorem Abelianization.equivOfComm_symm_apply {H : Type u_1} [CommGroup H] (a : Abelianization H) :
              ↑(MulEquiv.symm Abelianization.equivOfComm) a = ↑(Abelianization.lift (MonoidHom.id H)) a
              @[simp]
              theorem Abelianization.equivOfComm_apply {H : Type u_1} [CommGroup H] (a : H) :
              Abelianization.equivOfComm a = Abelianization.of a

              An Abelian group is equivalent to its own abelianization.

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

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

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

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