Documentation

Mathlib.GroupTheory.GroupAction.Quotient

Properties of group actions involving quotient groups #

This file proves properties of group actions which use the quotient group construction, notably

class MulAction.QuotientAction {α : Type u} (β : Type v) [Group α] [Monoid β] [MulAction β α] (H : Subgroup α) :

A typeclass for when a MulAction β α descends to the quotient α ⧸ H.

  • inv_mul_mem : ∀ (b : β) {a a' : α}, a⁻¹ * a' H(b a)⁻¹ * b a' H

    The action fulfils a normality condition on products that lie in H. This ensures that the action descends to an action on the quotient α ⧸ H.

Instances
    theorem MulAction.QuotientAction.inv_mul_mem {α : Type u} {β : Type v} :
    ∀ {inst : Group α} {inst_1 : Monoid β} {inst_2 : MulAction β α} {H : Subgroup α} [self : MulAction.QuotientAction β H] (b : β) {a a' : α}, a⁻¹ * a' H(b a)⁻¹ * b a' H

    The action fulfils a normality condition on products that lie in H. This ensures that the action descends to an action on the quotient α ⧸ H.

    class AddAction.QuotientAction {α : Type u} (β : Type v) [AddGroup α] [AddMonoid β] [AddAction β α] (H : AddSubgroup α) :

    A typeclass for when an AddAction β α descends to the quotient α ⧸ H.

    • inv_mul_mem : ∀ (b : β) {a a' : α}, -a + a' H-(b +ᵥ a) + (b +ᵥ a') H

      The action fulfils a normality condition on summands that lie in H. This ensures that the action descends to an action on the quotient α ⧸ H.

    Instances
      theorem AddAction.QuotientAction.inv_mul_mem {α : Type u} {β : Type v} :
      ∀ {inst : AddGroup α} {inst_1 : AddMonoid β} {inst_2 : AddAction β α} {H : AddSubgroup α} [self : AddAction.QuotientAction β H] (b : β) {a a' : α}, -a + a' H-(b +ᵥ a) + (b +ᵥ a') H

      The action fulfils a normality condition on summands that lie in H. This ensures that the action descends to an action on the quotient α ⧸ H.

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      instance AddAction.right_quotientAction {α : Type u} [AddGroup α] (H : AddSubgroup α) :
      AddAction.QuotientAction (↥H.normalizer.op) H
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      instance MulAction.right_quotientAction {α : Type u} [Group α] (H : Subgroup α) :
      MulAction.QuotientAction (↥H.normalizer.op) H
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      instance AddAction.right_quotientAction' {α : Type u} [AddGroup α] (H : AddSubgroup α) [hH : H.Normal] :
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      instance MulAction.right_quotientAction' {α : Type u} [Group α] (H : Subgroup α) [hH : H.Normal] :
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      theorem AddAction.quotient.proof_2 {α : Type u_1} (β : Type u_2) [AddGroup α] [AddMonoid β] [AddAction β α] (H : AddSubgroup α) [AddAction.QuotientAction β H] (q : α H) :
      0 +ᵥ q = q
      theorem AddAction.quotient.proof_3 {α : Type u_1} (β : Type u_2) [AddGroup α] [AddMonoid β] [AddAction β α] (H : AddSubgroup α) [AddAction.QuotientAction β H] (b : β) (b' : β) (q : α H) :
      b + b' +ᵥ q = b +ᵥ (b' +ᵥ q)
      instance AddAction.quotient {α : Type u} (β : Type v) [AddGroup α] [AddMonoid β] [AddAction β α] (H : AddSubgroup α) [AddAction.QuotientAction β H] :
      AddAction β (α H)
      Equations
      theorem AddAction.quotient.proof_1 {α : Type u_1} (β : Type u_2) [AddGroup α] [AddMonoid β] [AddAction β α] (H : AddSubgroup α) [AddAction.QuotientAction β H] (b : β) :
      ∀ (x x_1 : α), Setoid.r x x_1Setoid.r ((fun (x : α) => b +ᵥ x) x) ((fun (x : α) => b +ᵥ x) x_1)
      instance MulAction.quotient {α : Type u} (β : Type v) [Group α] [Monoid β] [MulAction β α] (H : Subgroup α) [MulAction.QuotientAction β H] :
      MulAction β (α H)
      Equations
      @[simp]
      theorem AddAction.Quotient.vadd_mk {α : Type u} {β : Type v} [AddGroup α] [AddMonoid β] [AddAction β α] (H : AddSubgroup α) [AddAction.QuotientAction β H] (b : β) (a : α) :
      b +ᵥ a = (b +ᵥ a)
      @[simp]
      theorem MulAction.Quotient.smul_mk {α : Type u} {β : Type v} [Group α] [Monoid β] [MulAction β α] (H : Subgroup α) [MulAction.QuotientAction β H] (b : β) (a : α) :
      b a = (b a)
      @[simp]
      theorem AddAction.Quotient.vadd_coe {α : Type u} {β : Type v} [AddGroup α] [AddMonoid β] [AddAction β α] (H : AddSubgroup α) [AddAction.QuotientAction β H] (b : β) (a : α) :
      b +ᵥ a = (b +ᵥ a)
      @[simp]
      theorem MulAction.Quotient.smul_coe {α : Type u} {β : Type v} [Group α] [Monoid β] [MulAction β α] (H : Subgroup α) [MulAction.QuotientAction β H] (b : β) (a : α) :
      b a = (b a)
      @[simp]
      theorem AddAction.Quotient.mk_vadd_out' {α : Type u} {β : Type v} [AddGroup α] [AddMonoid β] [AddAction β α] (H : AddSubgroup α) [AddAction.QuotientAction β H] (b : β) (q : α H) :
      (b +ᵥ Quotient.out' q) = b +ᵥ q
      @[simp]
      theorem MulAction.Quotient.mk_smul_out' {α : Type u} {β : Type v} [Group α] [Monoid β] [MulAction β α] (H : Subgroup α) [MulAction.QuotientAction β H] (b : β) (q : α H) :
      (b Quotient.out' q) = b q
      theorem AddAction.Quotient.coe_vadd_out' {α : Type u} {β : Type v} [AddGroup α] [AddMonoid β] [AddAction β α] (H : AddSubgroup α) [AddAction.QuotientAction β H] (b : β) (q : α H) :
      (b +ᵥ Quotient.out' q) = b +ᵥ q
      theorem MulAction.Quotient.coe_smul_out' {α : Type u} {β : Type v} [Group α] [Monoid β] [MulAction β α] (H : Subgroup α) [MulAction.QuotientAction β H] (b : β) (q : α H) :
      (b Quotient.out' q) = b q
      theorem QuotientGroup.out'_conj_pow_minimalPeriod_mem {α : Type u} [Group α] (H : Subgroup α) (a : α) (q : α H) :
      (Quotient.out' q)⁻¹ * a ^ Function.minimalPeriod (fun (x : α H) => a x) q * Quotient.out' q H
      def MulActionHom.toQuotient {α : Type u} [Group α] (H : Subgroup α) :
      α →ₑ[id] α H

      The canonical map to the left cosets.

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        @[simp]
        theorem MulActionHom.toQuotient_apply {α : Type u} [Group α] (H : Subgroup α) (g : α) :
        instance MulAction.mulLeftCosetsCompSubtypeVal {α : Type u} [Group α] (H : Subgroup α) (I : Subgroup α) :
        MulAction (↥I) (α H)
        Equations
        def AddAction.ofQuotientStabilizer (α : Type u) {β : Type v} [AddGroup α] [AddAction α β] (x : β) (g : α AddAction.stabilizer α x) :
        β

        The canonical map from the quotient of the stabilizer to the set.

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          theorem AddAction.ofQuotientStabilizer.proof_1 (α : Type u_1) {β : Type u_2} [AddGroup α] [AddAction α β] (x : β) (g1 : α) (g2 : α) (H : Setoid.r g1 g2) :
          g1 +ᵥ x = g2 +ᵥ x
          def MulAction.ofQuotientStabilizer (α : Type u) {β : Type v} [Group α] [MulAction α β] (x : β) (g : α MulAction.stabilizer α x) :
          β

          The canonical map from the quotient of the stabilizer to the set.

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            @[simp]
            theorem AddAction.ofQuotientStabilizer_mk (α : Type u) {β : Type v} [AddGroup α] [AddAction α β] (x : β) (g : α) :
            @[simp]
            theorem MulAction.ofQuotientStabilizer_mk (α : Type u) {β : Type v} [Group α] [MulAction α β] (x : β) (g : α) :
            theorem AddAction.ofQuotientStabilizer_vadd (α : Type u) {β : Type v} [AddGroup α] [AddAction α β] (x : β) (g : α) (g' : α AddAction.stabilizer α x) :
            theorem MulAction.ofQuotientStabilizer_smul (α : Type u) {β : Type v} [Group α] [MulAction α β] (x : β) (g : α) (g' : α MulAction.stabilizer α x) :
            noncomputable def AddAction.orbitEquivQuotientStabilizer (α : Type u) {β : Type v} [AddGroup α] [AddAction α β] (b : β) :

            Orbit-stabilizer theorem.

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              noncomputable def MulAction.orbitEquivQuotientStabilizer (α : Type u) {β : Type v} [Group α] [MulAction α β] (b : β) :

              Orbit-stabilizer theorem.

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                noncomputable def AddAction.orbitProdStabilizerEquivAddGroup (α : Type u) {β : Type v} [AddGroup α] [AddAction α β] (b : β) :

                Orbit-stabilizer theorem.

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                  noncomputable def MulAction.orbitProdStabilizerEquivGroup (α : Type u) {β : Type v} [Group α] [MulAction α β] (b : β) :

                  Orbit-stabilizer theorem.

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                    Orbit-stabilizer theorem.

                    @[simp]
                    theorem AddAction.orbitEquivQuotientStabilizer_symm_apply (α : Type u) {β : Type v} [AddGroup α] [AddAction α β] (b : β) (a : α) :
                    @[simp]
                    theorem MulAction.orbitEquivQuotientStabilizer_symm_apply (α : Type u) {β : Type v} [Group α] [MulAction α β] (b : β) (a : α) :
                    ((MulAction.orbitEquivQuotientStabilizer α b).symm a) = a b
                    @[simp]
                    @[simp]
                    theorem MulAction.stabilizer_quotient {G : Type u_1} [Group G] (H : Subgroup G) :
                    noncomputable def AddAction.selfEquivSigmaOrbitsQuotientStabilizer' (α : Type u) (β : Type v) [AddGroup α] [AddAction α β] {φ : Quotient (AddAction.orbitRel α β)β} (hφ : Function.LeftInverse Quotient.mk'' φ) :
                    β (ω : Quotient (AddAction.orbitRel α β)) × α AddAction.stabilizer α (φ ω)

                    Class formula : given G an additive group acting on X and φ a function mapping each orbit of X under this action (that is, each element of the quotient of X by the relation orbit_rel G X) to an element in this orbit, this gives a (noncomputable) bijection between X and the disjoint union of G/Stab(φ(ω)) over all orbits ω. In most cases you'll want φ to be Quotient.out', so we provide AddAction.selfEquivSigmaOrbitsQuotientStabilizer' as a special case.

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                      noncomputable def MulAction.selfEquivSigmaOrbitsQuotientStabilizer' (α : Type u) (β : Type v) [Group α] [MulAction α β] {φ : Quotient (MulAction.orbitRel α β)β} (hφ : Function.LeftInverse Quotient.mk'' φ) :
                      β (ω : Quotient (MulAction.orbitRel α β)) × α MulAction.stabilizer α (φ ω)

                      Class formula : given G a group acting on X and φ a function mapping each orbit of X under this action (that is, each element of the quotient of X by the relation orbitRel G X) to an element in this orbit, this gives a (noncomputable) bijection between X and the disjoint union of G/Stab(φ(ω)) over all orbits ω. In most cases you'll want φ to be Quotient.out', so we provide MulAction.selfEquivSigmaOrbitsQuotientStabilizer' as a special case.

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                        theorem AddAction.card_eq_sum_card_addGroup_sub_card_stabilizer' (α : Type u) (β : Type v) [AddGroup α] [AddAction α β] [Fintype α] [Fintype β] [Fintype (Quotient (AddAction.orbitRel α β))] [(b : β) → Fintype (AddAction.stabilizer α b)] {φ : Quotient (AddAction.orbitRel α β)β} (hφ : Function.LeftInverse Quotient.mk'' φ) :

                        Class formula for a finite group acting on a finite type. See AddAction.card_eq_sum_card_addGroup_div_card_stabilizer for a specialized version using Quotient.out'.

                        theorem MulAction.card_eq_sum_card_group_div_card_stabilizer' (α : Type u) (β : Type v) [Group α] [MulAction α β] [Fintype α] [Fintype β] [Fintype (Quotient (MulAction.orbitRel α β))] [(b : β) → Fintype (MulAction.stabilizer α b)] {φ : Quotient (MulAction.orbitRel α β)β} (hφ : Function.LeftInverse Quotient.mk'' φ) :

                        Class formula for a finite group acting on a finite type. See MulAction.card_eq_sum_card_group_div_card_stabilizer for a specialized version using Quotient.out'.

                        noncomputable def AddAction.selfEquivSigmaOrbitsQuotientStabilizer (α : Type u) (β : Type v) [AddGroup α] [AddAction α β] :
                        β (ω : Quotient (AddAction.orbitRel α β)) × α AddAction.stabilizer α ω.out'

                        Class formula. This is a special case of AddAction.self_equiv_sigma_orbits_quotient_stabilizer' with φ = Quotient.out'.

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                          noncomputable def MulAction.selfEquivSigmaOrbitsQuotientStabilizer (α : Type u) (β : Type v) [Group α] [MulAction α β] :
                          β (ω : Quotient (MulAction.orbitRel α β)) × α MulAction.stabilizer α ω.out'

                          Class formula. This is a special case of MulAction.self_equiv_sigma_orbits_quotient_stabilizer' with φ = Quotient.out'.

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                            Class formula for a finite group acting on a finite type.

                            Class formula for a finite group acting on a finite type.

                            noncomputable def AddAction.sigmaFixedByEquivOrbitsProdAddGroup (α : Type u) (β : Type v) [AddGroup α] [AddAction α β] :
                            (a : α) × (AddAction.fixedBy β a) Quotient (AddAction.orbitRel α β) × α

                            Burnside's lemma : a (noncomputable) bijection between the disjoint union of all {x ∈ X | g • x = x} for g ∈ G and the product G × X/G, where G is an additive group acting on X and X/Gdenotes the quotient of X by the relation orbitRel G X.

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                              theorem AddAction.sigmaFixedByEquivOrbitsProdAddGroup.proof_1 (α : Type u_1) (β : Type u_2) [AddGroup α] [AddAction α β] :
                              ∀ (x : α × β), x.1 +ᵥ x.2 = x.2 x.1 +ᵥ x.2 = x.2
                              noncomputable def MulAction.sigmaFixedByEquivOrbitsProdGroup (α : Type u) (β : Type v) [Group α] [MulAction α β] :
                              (a : α) × (MulAction.fixedBy β a) Quotient (MulAction.orbitRel α β) × α

                              Burnside's lemma : a (noncomputable) bijection between the disjoint union of all {x ∈ X | g • x = x} for g ∈ G and the product G × X/G, where G is a group acting on X and X/G denotes the quotient of X by the relation orbitRel G X.

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                                Burnside's lemma : given a finite additive group G acting on a set X, the average number of elements fixed by each g ∈ G is the number of orbits.

                                Burnside's lemma : given a finite group G acting on a set X, the average number of elements fixed by each g ∈ G is the number of orbits.

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                                theorem AddAction.equivAddSubgroupOrbitsSetoidComap.proof_4 {α : Type u_2} {β : Type u_1} [AddGroup α] [AddAction α β] (H : AddSubgroup α) (ω : Quotient (AddAction.orbitRel α β)) (a : { x : β // x Quotient.mk (AddAction.orbitRel α β) ⁻¹' {ω} }) (b : { x : β // x Quotient.mk (AddAction.orbitRel α β) ⁻¹' {ω} }) (h : Setoid.r a b) :
                                (fun (x : { x : β // x Quotient.mk (AddAction.orbitRel α β) ⁻¹' {ω} }) => x, ) a = (fun (x : { x : β // x Quotient.mk (AddAction.orbitRel α β) ⁻¹' {ω} }) => x, ) b

                                A bijection between the quotient of the action of an additive subgroup H on an orbit, and a corresponding quotient expressed in terms of Setoid.comap Subtype.val.

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                                  theorem AddAction.equivAddSubgroupOrbitsSetoidComap.proof_5 {α : Type u_2} {β : Type u_1} [AddGroup α] [AddAction α β] (H : AddSubgroup α) (ω : Quotient (AddAction.orbitRel α β)) :
                                  Function.LeftInverse (fun (q : Quotient (Setoid.comap Subtype.val (AddAction.orbitRel (↥H) β))) => q.liftOn' (fun (x : { x : β // x Quotient.mk (AddAction.orbitRel α β) ⁻¹' {ω} }) => x, ) ) fun (q : AddAction.orbitRel.Quotient H (AddAction.orbitRel.Quotient.orbit ω)) => Quotient.liftOn' q (fun (x : (AddAction.orbitRel.Quotient.orbit ω)) => x, )
                                  theorem AddAction.equivAddSubgroupOrbitsSetoidComap.proof_6 {α : Type u_2} {β : Type u_1} [AddGroup α] [AddAction α β] (H : AddSubgroup α) (ω : Quotient (AddAction.orbitRel α β)) :
                                  Function.RightInverse (fun (q : Quotient (Setoid.comap Subtype.val (AddAction.orbitRel (↥H) β))) => q.liftOn' (fun (x : { x : β // x Quotient.mk (AddAction.orbitRel α β) ⁻¹' {ω} }) => x, ) ) fun (q : AddAction.orbitRel.Quotient H (AddAction.orbitRel.Quotient.orbit ω)) => Quotient.liftOn' q (fun (x : (AddAction.orbitRel.Quotient.orbit ω)) => x, )
                                  theorem AddAction.equivAddSubgroupOrbitsSetoidComap.proof_2 {α : Type u_2} {β : Type u_1} [AddGroup α] [AddAction α β] (H : AddSubgroup α) (ω : Quotient (AddAction.orbitRel α β)) (a : (AddAction.orbitRel.Quotient.orbit ω)) (b : (AddAction.orbitRel.Quotient.orbit ω)) (h : Setoid.r a b) :
                                  (fun (x : (AddAction.orbitRel.Quotient.orbit ω)) => x, ) a = (fun (x : (AddAction.orbitRel.Quotient.orbit ω)) => x, ) b

                                  A bijection between the quotient of the action of a subgroup H on an orbit, and a corresponding quotient expressed in terms of Setoid.comap Subtype.val.

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                                    A bijection between the orbits under the action of an additive subgroup H on β, and the orbits under the action of H on each orbit under the action of G.

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                                      A bijection between the orbits under the action of a subgroup H on β, and the orbits under the action of H on each orbit under the action of G.

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                                        noncomputable def AddAction.equivAddSubgroupOrbitsQuotientAddGroup {α : Type u} {β : Type v} [AddGroup α] [AddAction α β] (x : β) [AddAction.IsPretransitive α β] (free : ∀ (y : β), AddAction.stabilizer α y = ) (H : AddSubgroup α) :

                                        Given an additive group acting freely and transitively, an equivalence between the orbits under the action of an additive subgroup and the quotient group.

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                                          theorem AddAction.equivAddSubgroupOrbitsQuotientAddGroup.proof_3 {α : Type u_1} {β : Type u_2} [AddGroup α] [AddAction α β] (x : β) (H : AddSubgroup α) (g₁ : α) (g₂ : α) (h : Setoid.r g₁ g₂) :
                                          (fun (g : α) => -g +ᵥ x) g₁ = (fun (g : α) => -g +ᵥ x) g₂
                                          theorem AddAction.equivAddSubgroupOrbitsQuotientAddGroup.proof_4 {α : Type u_1} {β : Type u_2} [AddGroup α] [AddAction α β] (x : β) [AddAction.IsPretransitive α β] (free : ∀ (y : β), AddAction.stabilizer α y = ) (H : AddSubgroup α) (y : AddAction.orbitRel.Quotient (↥H) β) :
                                          (fun (q : α H) => Quotient.liftOn' q (fun (g : α) => -g +ᵥ x) ) ((fun (q : AddAction.orbitRel.Quotient (↥H) β) => Quotient.liftOn' q (fun (y : β) => .choose) ) y) = y
                                          theorem AddAction.equivAddSubgroupOrbitsQuotientAddGroup.proof_5 {α : Type u_1} {β : Type u_2} [AddGroup α] [AddAction α β] (x : β) [AddAction.IsPretransitive α β] (free : ∀ (y : β), AddAction.stabilizer α y = ) (H : AddSubgroup α) (g : α H) :
                                          (fun (q : AddAction.orbitRel.Quotient (↥H) β) => Quotient.liftOn' q (fun (y : β) => .choose) ) ((fun (q : α H) => Quotient.liftOn' q (fun (g : α) => -g +ᵥ x) ) g) = g
                                          theorem AddAction.equivAddSubgroupOrbitsQuotientAddGroup.proof_2 {α : Type u_2} {β : Type u_1} [AddGroup α] [AddAction α β] (x : β) [AddAction.IsPretransitive α β] (free : ∀ (y : β), AddAction.stabilizer α y = ) (H : AddSubgroup α) (y₁ : β) (y₂ : β) (h : Setoid.r y₁ y₂) :
                                          (fun (y : β) => .choose) y₁ = (fun (y : β) => .choose) y₂
                                          theorem AddAction.equivAddSubgroupOrbitsQuotientAddGroup.proof_1 {α : Type u_1} {β : Type u_2} [AddGroup α] [AddAction α β] (x : β) [AddAction.IsPretransitive α β] (y : β) :
                                          ∃ (m : α), m +ᵥ y = x
                                          noncomputable def MulAction.equivSubgroupOrbitsQuotientGroup {α : Type u} {β : Type v} [Group α] [MulAction α β] (x : β) [MulAction.IsPretransitive α β] (free : ∀ (y : β), MulAction.stabilizer α y = ) (H : Subgroup α) :

                                          Given a group acting freely and transitively, an equivalence between the orbits under the action of a subgroup and the quotient group.

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                                            theorem Subgroup.normalCore_eq_ker {G : Type u_1} [Group G] (H : Subgroup G) :
                                            H.normalCore = (MulAction.toPermHom G (G H)).ker

                                            Cosets of the centralizer of an element embed into the set of commutators.

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                                              noncomputable def Subgroup.quotientCenterEmbedding {G : Type u_1} [Group G] {S : Set G} (hS : Subgroup.closure S = ) :
                                              G Subgroup.center G S(commutatorSet G)

                                              If G is generated by S, then the quotient by the center embeds into S-indexed sequences of commutators.

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                                                theorem Subgroup.quotientCenterEmbedding_apply {G : Type u_1} [Group G] {S : Set G} (hS : Subgroup.closure S = ) (g : G) (s : S) :
                                                (Subgroup.quotientCenterEmbedding hS) (↑g) s = g, s,