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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

as well as their analogues for additive groups.

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
    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
      instance MulAction.right_quotientAction {α : Type u} [Group α] (H : Subgroup α) :
      QuotientAction (↥H.normalizer.op) H
      instance AddAction.right_quotientAction {α : Type u} [AddGroup α] (H : AddSubgroup α) :
      QuotientAction (↥H.normalizer.op) H
      instance MulAction.right_quotientAction' {α : Type u} [Group α] (H : Subgroup α) [hH : H.Normal] :
      instance AddAction.right_quotientAction' {α : Type u} [AddGroup α] (H : AddSubgroup α) [hH : H.Normal] :
      instance MulAction.quotient {α : Type u} (β : Type v) [Group α] [Monoid β] [MulAction β α] (H : Subgroup α) [QuotientAction β H] :
      MulAction β (α H)
      Equations
      instance AddAction.quotient {α : Type u} (β : Type v) [AddGroup α] [AddMonoid β] [AddAction β α] (H : AddSubgroup α) [QuotientAction β H] :
      AddAction β (α H)
      Equations
      @[simp]
      theorem MulAction.Quotient.smul_mk {α : Type u} {β : Type v} [Group α] [Monoid β] [MulAction β α] (H : Subgroup α) [QuotientAction β H] (b : β) (a : α) :
      b a = (b a)
      @[simp]
      theorem AddAction.Quotient.vadd_mk {α : Type u} {β : Type v} [AddGroup α] [AddMonoid β] [AddAction β α] (H : AddSubgroup α) [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) [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) [QuotientAction β H] (b : β) (a : α) :
      b +ᵥ a = (b +ᵥ a)
      @[simp]
      theorem MulAction.Quotient.mk_smul_out {α : Type u} {β : Type v} [Group α] [Monoid β] [MulAction β α] (H : Subgroup α) [QuotientAction β H] (b : β) (q : α H) :
      (b Quotient.out q) = b q
      @[simp]
      theorem AddAction.Quotient.mk_vadd_out {α : Type u} {β : Type v} [AddGroup α] [AddMonoid β] [AddAction β α] (H : AddSubgroup α) [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) [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) [QuotientAction β H] (b : β) (q : α H) :
      (b +ᵥ Quotient.out q) = b +ᵥ 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
      @[deprecated MulAction.Quotient.mk_smul_out (since := "2024-10-19")]
      theorem MulAction.Quotient.mk_smul_out' {α : Type u} {β : Type v} [Group α] [Monoid β] [MulAction β α] (H : Subgroup α) [QuotientAction β H] (b : β) (q : α H) :
      (b Quotient.out q) = b q

      Alias of MulAction.Quotient.mk_smul_out.

      @[deprecated AddAction.Quotient.mk_vadd_out (since := "2024-10-19")]
      theorem AddAction.Quotient.mk_vadd_out' {α : Type u} {β : Type v} [AddGroup α] [AddMonoid β] [AddAction β α] (H : AddSubgroup α) [QuotientAction β H] (b : β) (q : α H) :
      (b +ᵥ Quotient.out q) = b +ᵥ q
      @[deprecated MulAction.Quotient.coe_smul_out (since := "2024-10-19")]
      theorem MulAction.Quotient.coe_smul_out' {α : Type u} {β : Type v} [Group α] [Monoid β] [MulAction β α] (H : Subgroup α) [QuotientAction β H] (b : β) (q : α H) :
      (b Quotient.out q) = b q

      Alias of MulAction.Quotient.coe_smul_out.

      @[deprecated AddAction.Quotient.coe_vadd_out (since := "2024-10-19")]
      theorem AddAction.Quotient.coe_vadd_out' {α : Type u} {β : Type v} [AddGroup α] [AddMonoid β] [AddAction β α] (H : AddSubgroup α) [QuotientAction β H] (b : β) (q : α H) :
      (b +ᵥ Quotient.out q) = b +ᵥ q
      @[deprecated QuotientGroup.out_conj_pow_minimalPeriod_mem (since := "2024-10-19")]
      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

      Alias of QuotientGroup.out_conj_pow_minimalPeriod_mem.

      def MulActionHom.toQuotient {α : Type u} [Group α] (H : Subgroup α) :
      α →ₑ[id] α H

      The canonical map to the left cosets.

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

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

        Equations
        Instances For
          def AddAction.ofQuotientStabilizer (α : Type u) {β : Type v} [AddGroup α] [AddAction α β] (x : β) (g : α stabilizer α x) :
          β

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

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

            Orbit-stabilizer theorem.

            Equations
            Instances For
              noncomputable def AddAction.orbitEquivQuotientStabilizer (α : Type u) {β : Type v} [AddGroup α] [AddAction α β] (b : β) :
              (orbit α b) α stabilizer α b

              Orbit-stabilizer theorem.

              Equations
              Instances For
                noncomputable def MulAction.orbitProdStabilizerEquivGroup (α : Type u) {β : Type v} [Group α] [MulAction α β] (b : β) :
                (orbit α b) × (stabilizer α b) α

                Orbit-stabilizer theorem.

                Equations
                • One or more equations did not get rendered due to their size.
                Instances For
                  noncomputable def AddAction.orbitProdStabilizerEquivAddGroup (α : Type u) {β : Type v} [AddGroup α] [AddAction α β] (b : β) :
                  (orbit α b) × (stabilizer α b) α

                  Orbit-stabilizer theorem.

                  Equations
                  • One or more equations did not get rendered due to their size.
                  Instances For
                    theorem MulAction.card_orbit_mul_card_stabilizer_eq_card_group (α : Type u) {β : Type v} [Group α] [MulAction α β] (b : β) [Fintype α] [Fintype (orbit α b)] [Fintype (stabilizer α b)] :

                    Orbit-stabilizer theorem.

                    theorem AddAction.card_orbit_mul_card_stabilizer_eq_card_addGroup (α : Type u) {β : Type v} [AddGroup α] [AddAction α β] (b : β) [Fintype α] [Fintype (orbit α b)] [Fintype (stabilizer α b)] :

                    Orbit-stabilizer theorem.

                    @[simp]
                    theorem MulAction.orbitEquivQuotientStabilizer_symm_apply (α : Type u) {β : Type v} [Group α] [MulAction α β] (b : β) (a : α) :
                    ((orbitEquivQuotientStabilizer α b).symm a) = a b
                    @[simp]
                    theorem AddAction.orbitEquivQuotientStabilizer_symm_apply (α : Type u) {β : Type v} [AddGroup α] [AddAction α β] (b : β) (a : α) :
                    ((orbitEquivQuotientStabilizer α b).symm a) = a +ᵥ b
                    @[simp]
                    theorem MulAction.stabilizer_quotient {G : Type u_1} [Group G] (H : Subgroup G) :
                    stabilizer G 1 = H
                    @[simp]
                    theorem AddAction.stabilizer_quotient {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
                    stabilizer G 0 = H
                    noncomputable def MulAction.selfEquivSigmaOrbitsQuotientStabilizer' (α : Type u) (β : Type v) [Group α] [MulAction α β] {φ : Quotient (orbitRel α β)β} (hφ : Function.LeftInverse Quotient.mk'' φ) :
                    β (ω : Quotient (orbitRel α β)) × α 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.

                    Equations
                    • One or more equations did not get rendered due to their size.
                    Instances For
                      noncomputable def AddAction.selfEquivSigmaOrbitsQuotientStabilizer' (α : Type u) (β : Type v) [AddGroup α] [AddAction α β] {φ : Quotient (orbitRel α β)β} (hφ : Function.LeftInverse Quotient.mk'' φ) :
                      β (ω : Quotient (orbitRel α β)) × α 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.

                      Equations
                      • One or more equations did not get rendered due to their size.
                      Instances For
                        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 (orbitRel α β))] [(b : β) → Fintype (stabilizer α b)] {φ : Quotient (orbitRel α β)β} (hφ : Function.LeftInverse Quotient.mk'' φ) :
                        Fintype.card β = ω : Quotient (orbitRel α β), Fintype.card α / Fintype.card (stabilizer α (φ ω))

                        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.

                        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 (orbitRel α β))] [(b : β) → Fintype (stabilizer α b)] {φ : Quotient (orbitRel α β)β} (hφ : Function.LeftInverse Quotient.mk'' φ) :
                        Fintype.card β = ω : Quotient (orbitRel α β), Fintype.card α / Fintype.card (stabilizer α (φ ω))

                        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.

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

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

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

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

                          Equations
                          Instances For
                            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 (orbitRel α β))] [(b : β) → Fintype (stabilizer α b)] :
                            Fintype.card β = ω : Quotient (orbitRel α β), Fintype.card α / Fintype.card (stabilizer α ω.out)

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

                            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 (orbitRel α β))] [(b : β) → Fintype (stabilizer α b)] :
                            Fintype.card β = ω : Quotient (orbitRel α β), Fintype.card α / Fintype.card (stabilizer α ω.out)

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

                            noncomputable def MulAction.sigmaFixedByEquivOrbitsProdGroup (α : Type u) (β : Type v) [Group α] [MulAction α β] :
                            (a : α) × (fixedBy β a) Quotient (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.

                            Equations
                            • One or more equations did not get rendered due to their size.
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                              noncomputable def AddAction.sigmaFixedByEquivOrbitsProdAddGroup (α : Type u) (β : Type v) [AddGroup α] [AddAction α β] :
                              (a : α) × (fixedBy β a) Quotient (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.

                              Equations
                              • One or more equations did not get rendered due to their size.
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                                theorem MulAction.sum_card_fixedBy_eq_card_orbits_mul_card_group (α : Type u) (β : Type v) [Group α] [MulAction α β] [Fintype α] [(a : α) → Fintype (fixedBy β a)] [Fintype (Quotient (orbitRel α β))] :
                                a : α, Fintype.card (fixedBy β a) = Fintype.card (Quotient (orbitRel α β)) * Fintype.card α

                                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.

                                theorem AddAction.sum_card_fixedBy_eq_card_orbits_mul_card_addGroup (α : Type u) (β : Type v) [AddGroup α] [AddAction α β] [Fintype α] [(a : α) → Fintype (fixedBy β a)] [Fintype (Quotient (orbitRel α β))] :
                                a : α, Fintype.card (fixedBy β a) = Fintype.card (Quotient (orbitRel α β)) * Fintype.card α

                                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.

                                noncomputable def MulAction.equivSubgroupOrbitsSetoidComap {α : Type u} {β : Type v} [Group α] [MulAction α β] (H : Subgroup α) (ω : Quotient (orbitRel α β)) :

                                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.

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

                                  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.

                                  Equations
                                  • One or more equations did not get rendered due to their size.
                                  Instances For
                                    noncomputable def MulAction.equivSubgroupOrbits {α : Type u} (β : Type v) [Group α] [MulAction α β] (H : Subgroup α) :

                                    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.

                                    Equations
                                    • One or more equations did not get rendered due to their size.
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                                      noncomputable def AddAction.equivAddSubgroupOrbits {α : Type u} (β : Type v) [AddGroup α] [AddAction α β] (H : AddSubgroup α) :

                                      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.

                                      Equations
                                      • One or more equations did not get rendered due to their size.
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                                        noncomputable def MulAction.equivSubgroupOrbitsQuotientGroup {α : Type u} {β : Type v} [Group α] [MulAction α β] (x : β) [IsPretransitive α β] (free : ∀ (y : β), stabilizer α y = ) (H : Subgroup α) :
                                        orbitRel.Quotient (↥H) β α H

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

                                        Equations
                                        • One or more equations did not get rendered due to their size.
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                                          noncomputable def AddAction.equivAddSubgroupOrbitsQuotientAddGroup {α : Type u} {β : Type v} [AddGroup α] [AddAction α β] (x : β) [IsPretransitive α β] (free : ∀ (y : β), stabilizer α y = ) (H : AddSubgroup α) :
                                          orbitRel.Quotient (↥H) β α H

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

                                          Equations
                                          • One or more equations did not get rendered due to their size.
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                                            noncomputable def MulAction.selfEquivOrbitsQuotientProd' {α : Type u} {β : Type v} [Group α] [MulAction α β] {φ : Quotient (orbitRel α β)β} (hφ : Function.LeftInverse Quotient.mk'' φ) (h : ∀ (b : β), stabilizer α b = ) :
                                            β Quotient (orbitRel α β) × α

                                            If α acts on β with trivial stabilizers, β is equivalent to the product of the quotient of β by α and α. See MulAction.selfEquivOrbitsQuotientProd with φ = Quotient.out.

                                            Equations
                                            • One or more equations did not get rendered due to their size.
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                                              noncomputable def AddAction.selfEquivOrbitsQuotientSum' {α : Type u} {β : Type v} [AddGroup α] [AddAction α β] {φ : Quotient (orbitRel α β)β} (hφ : Function.LeftInverse Quotient.mk'' φ) (h : ∀ (b : β), stabilizer α b = ) :
                                              β Quotient (orbitRel α β) × α

                                              If α acts freely on β, β is equivalent to the product of the quotient of β by α and α. See AddAction.selfEquivOrbitsQuotientProd with φ = Quotient.out.

                                              Equations
                                              • One or more equations did not get rendered due to their size.
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                                                noncomputable def MulAction.selfEquivOrbitsQuotientProd {α : Type u} {β : Type v} [Group α] [MulAction α β] (h : ∀ (b : β), stabilizer α b = ) :
                                                β Quotient (orbitRel α β) × α

                                                If α acts freely on β, β is equivalent to the product of the quotient of β by α and α.

                                                Equations
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                                                  noncomputable def AddAction.selfEquivOrbitsQuotientSum {α : Type u} {β : Type v} [AddGroup α] [AddAction α β] (h : ∀ (b : β), stabilizer α b = ) :
                                                  β Quotient (orbitRel α β) × α

                                                  If α acts freely on β, β is equivalent to the product of the quotient of β by α and α.

                                                  Equations
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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
                                                    noncomputable def Subgroup.quotientCentralizerEmbedding {G : Type u_1} [Group G] (g : G) :

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

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                                                    • One or more equations did not get rendered due to their size.
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                                                      noncomputable def Subgroup.quotientCenterEmbedding {G : Type u_1} [Group G] {S : Set G} (hS : closure S = ) :
                                                      G 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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                                                      • One or more equations did not get rendered due to their size.
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                                                        theorem Subgroup.quotientCenterEmbedding_apply {G : Type u_1} [Group G] {S : Set G} (hS : closure S = ) (g : G) (s : S) :
                                                        (quotientCenterEmbedding hS) (↑g) s = g, s,