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Mathlib.Deprecated.Subgroup

Unbundled subgroups (deprecated) #

This file is deprecated, and is no longer imported by anything in mathlib other than other deprecated files, and test files. You should not need to import it.

This file defines unbundled multiplicative and additive subgroups. Instead of using this file, please use Subgroup G and AddSubgroup A, defined in GroupTheory.Subgroup.Basic.

Main definitions #

IsAddSubgroup (S : Set A) : the predicate that S is the underlying subset of an additive subgroup of A. The bundled variant AddSubgroup A should be used in preference to this.

IsSubgroup (S : Set G) : the predicate that S is the underlying subset of a subgroup of G. The bundled variant Subgroup G should be used in preference to this.

Tags #

subgroup, subgroups, IsSubgroup

structure IsAddSubgroup {A : Type u_1} [inst : AddGroup A] (s : Set A) extends IsAddSubmonoid :
  • The proposition that s is closed under negation.

    neg_mem : ∀ {a : A}, a s-a s

s is an additive subgroup: a set containing 0 and closed under addition and negation.

Instances For
    structure IsSubgroup {G : Type u_1} [inst : Group G] (s : Set G) extends IsSubmonoid :
    • The proposition that s is closed under inverse.

      inv_mem : ∀ {a : G}, a sa⁻¹ s

    s is a subgroup: a set containing 1 and closed under multiplication and inverse.

    Instances For
      theorem IsAddSubgroup.sub_mem {G : Type u_1} [inst : AddGroup G] {s : Set G} (hs : IsAddSubgroup s) {x : G} {y : G} (hx : x s) (hy : y s) :
      x - y s
      theorem IsSubgroup.div_mem {G : Type u_1} [inst : Group G] {s : Set G} (hs : IsSubgroup s) {x : G} {y : G} (hx : x s) (hy : y s) :
      x / y s
      theorem Additive.isAddSubgroup {G : Type u_1} [inst : Group G] {s : Set G} (hs : IsSubgroup s) :
      theorem Multiplicative.isSubgroup {A : Type u_1} [inst : AddGroup A] {s : Set A} (hs : IsAddSubgroup s) :
      theorem IsAddSubgroup.of_add_neg {G : Type u_1} [inst : AddGroup G] (s : Set G) (one_mem : 0 s) (div_mem : ∀ {a b : G}, a sb sa + -b s) :
      theorem IsSubgroup.of_div {G : Type u_1} [inst : Group G] (s : Set G) (one_mem : 1 s) (div_mem : ∀ {a b : G}, a sb sa * b⁻¹ s) :
      theorem IsAddSubgroup.of_sub {A : Type u_1} [inst : AddGroup A] (s : Set A) (zero_mem : 0 s) (sub_mem : ∀ {a b : A}, a sb sa - b s) :
      theorem IsAddSubgroup.inter {G : Type u_1} [inst : AddGroup G] {s₁ : Set G} {s₂ : Set G} (hs₁ : IsAddSubgroup s₁) (hs₂ : IsAddSubgroup s₂) :
      IsAddSubgroup (s₁ s₂)
      theorem IsSubgroup.inter {G : Type u_1} [inst : Group G] {s₁ : Set G} {s₂ : Set G} (hs₁ : IsSubgroup s₁) (hs₂ : IsSubgroup s₂) :
      IsSubgroup (s₁ s₂)
      theorem IsAddSubgroup.interᵢ {G : Type u_2} [inst : AddGroup G] {ι : Sort u_1} {s : ιSet G} (hs : ∀ (y : ι), IsAddSubgroup (s y)) :
      theorem IsSubgroup.interᵢ {G : Type u_2} [inst : Group G] {ι : Sort u_1} {s : ιSet G} (hs : ∀ (y : ι), IsSubgroup (s y)) :
      abbrev isAddSubgroup_unionᵢ_of_directed.match_1 {G : Type u_2} {ι : Type u_1} {s : ιSet G} :
      {a : G} → (motive : (i, a s i) → Prop) → ∀ (x : i, a s i), (∀ (i : ι) (hi : a s i), motive (_ : i, a s i)) → motive x
      Equations
      theorem isAddSubgroup_unionᵢ_of_directed {G : Type u_2} [inst : AddGroup G] {ι : Type u_1} [inst : Nonempty ι] {s : ιSet G} (hs : ∀ (i : ι), IsAddSubgroup (s i)) (directed : ∀ (i j : ι), k, s i s k s j s k) :
      theorem isSubgroup_unionᵢ_of_directed {G : Type u_2} [inst : Group G] {ι : Type u_1} [inst : Nonempty ι] {s : ιSet G} (hs : ∀ (i : ι), IsSubgroup (s i)) (directed : ∀ (i j : ι), k, s i s k s j s k) :
      IsSubgroup (Set.unionᵢ fun i => s i)
      theorem IsAddSubgroup.neg_mem_iff {G : Type u_1} {a : G} [inst : AddGroup G] {s : Set G} (hs : IsAddSubgroup s) :
      -a s a s
      theorem IsSubgroup.inv_mem_iff {G : Type u_1} {a : G} [inst : Group G] {s : Set G} (hs : IsSubgroup s) :
      a⁻¹ s a s
      theorem IsAddSubgroup.add_mem_cancel_right {G : Type u_1} {a : G} {b : G} [inst : AddGroup G] {s : Set G} (hs : IsAddSubgroup s) (h : a s) :
      b + a s b s
      theorem IsSubgroup.mul_mem_cancel_right {G : Type u_1} {a : G} {b : G} [inst : Group G] {s : Set G} (hs : IsSubgroup s) (h : a s) :
      b * a s b s
      theorem IsAddSubgroup.add_mem_cancel_left {G : Type u_1} {a : G} {b : G} [inst : AddGroup G] {s : Set G} (hs : IsAddSubgroup s) (h : a s) :
      a + b s b s
      theorem IsSubgroup.mul_mem_cancel_left {G : Type u_1} {a : G} {b : G} [inst : Group G] {s : Set G} (hs : IsSubgroup s) (h : a s) :
      a * b s b s
      structure IsNormalAddSubgroup {A : Type u_1} [inst : AddGroup A] (s : Set A) extends IsAddSubgroup :
      • The proposition that s is closed under (additive) conjugation.

        normal : ∀ (n : A), n s∀ (g : A), g + n + -g s

      IsNormalAddSubgroup (s : Set A) expresses the fact that s is a normal additive subgroup of the additive group A. Important: the preferred way to say this in Lean is via bundled subgroups S : AddSubgroup A and hs : S.normal, and not via this structure.

      Instances For
        structure IsNormalSubgroup {G : Type u_1} [inst : Group G] (s : Set G) extends IsSubgroup :
        • The proposition that s is closed under conjugation.

          normal : ∀ (n : G), n s∀ (g : G), g * n * g⁻¹ s

        IsNormalSubgroup (s : Set G) expresses the fact that s is a normal subgroup of the group G. Important: the preferred way to say this in Lean is via bundled subgroups S : Subgroup G and not via this structure.

        Instances For
          theorem isNormalSubgroup_of_commGroup {G : Type u_1} [inst : CommGroup G] {s : Set G} (hs : IsSubgroup s) :
          theorem IsAddSubgroup.mem_norm_comm {G : Type u_1} [inst : AddGroup G] {s : Set G} (hs : IsNormalAddSubgroup s) {a : G} {b : G} (hab : a + b s) :
          b + a s
          theorem IsSubgroup.mem_norm_comm {G : Type u_1} [inst : Group G] {s : Set G} (hs : IsNormalSubgroup s) {a : G} {b : G} (hab : a * b s) :
          b * a s
          theorem IsAddSubgroup.mem_norm_comm_iff {G : Type u_1} [inst : AddGroup G] {s : Set G} (hs : IsNormalAddSubgroup s) {a : G} {b : G} :
          a + b s b + a s
          theorem IsSubgroup.mem_norm_comm_iff {G : Type u_1} [inst : Group G] {s : Set G} (hs : IsNormalSubgroup s) {a : G} {b : G} :
          a * b s b * a s
          def IsAddSubgroup.trivial (G : Type u_1) [inst : AddGroup G] :
          Set G

          the trivial additive subgroup

          Equations
          def IsSubgroup.trivial (G : Type u_1) [inst : Group G] :
          Set G

          The trivial subgroup

          Equations
          @[simp]
          theorem IsAddSubgroup.mem_trivial {G : Type u_1} [inst : AddGroup G] {g : G} :
          @[simp]
          theorem IsSubgroup.mem_trivial {G : Type u_1} [inst : Group G] {g : G} :
          theorem IsAddSubgroup.eq_trivial_iff {G : Type u_1} [inst : AddGroup G] {s : Set G} (hs : IsAddSubgroup s) :
          s = IsAddSubgroup.trivial G ∀ (x : G), x sx = 0
          theorem IsSubgroup.eq_trivial_iff {G : Type u_1} [inst : Group G] {s : Set G} (hs : IsSubgroup s) :
          s = IsSubgroup.trivial G ∀ (x : G), x sx = 1
          theorem IsSubgroup.univ_subgroup {G : Type u_1} [inst : Group G] :
          def IsAddSubgroup.addCenter (G : Type u_1) [inst : AddGroup G] :
          Set G

          The underlying set of the center of an additive group.

          Equations
          def IsSubgroup.center (G : Type u_1) [inst : Group G] :
          Set G

          The underlying set of the center of a group.

          Equations
          theorem IsAddSubgroup.mem_add_center {G : Type u_1} [inst : AddGroup G] {a : G} :
          a IsAddSubgroup.addCenter G ∀ (g : G), g + a = a + g
          theorem IsSubgroup.mem_center {G : Type u_1} [inst : Group G] {a : G} :
          a IsSubgroup.center G ∀ (g : G), g * a = a * g
          def IsAddSubgroup.addNormalizer {G : Type u_1} [inst : AddGroup G] (s : Set G) :
          Set G

          The underlying set of the normalizer of a subset S : Set A of an additive group A. That is, the elements a : A such that a + S - a = S.

          Equations
          def IsSubgroup.normalizer {G : Type u_1} [inst : Group G] (s : Set G) :
          Set G

          The underlying set of the normalizer of a subset S : Set G of a group G. That is, the elements g : G such that g * S * g⁻¹ = S.

          Equations
          theorem IsSubgroup.subset_normalizer {G : Type u_1} [inst : Group G] {s : Set G} (hs : IsSubgroup s) :
          def IsAddGroupHom.ker {G : Type u_1} {H : Type u_2} [inst : AddGroup H] (f : GH) :
          Set G

          ker f : set A is the underlying subset of the kernel of a map A → B

          Equations
          def IsGroupHom.ker {G : Type u_1} {H : Type u_2} [inst : Group H] (f : GH) :
          Set G

          ker f : set G is the underlying subset of the kernel of a map G → H.

          Equations
          theorem IsAddGroupHom.mem_ker {G : Type u_2} {H : Type u_1} [inst : AddGroup H] (f : GH) {x : G} :
          theorem IsGroupHom.mem_ker {G : Type u_2} {H : Type u_1} [inst : Group H] (f : GH) {x : G} :
          theorem IsAddGroupHom.zero_ker_neg {G : Type u_1} {H : Type u_2} [inst : AddGroup G] [inst : AddGroup H] {f : GH} (hf : IsAddGroupHom f) {a : G} {b : G} (h : f (a + -b) = 0) :
          f a = f b
          theorem IsGroupHom.one_ker_inv {G : Type u_1} {H : Type u_2} [inst : Group G] [inst : Group H] {f : GH} (hf : IsGroupHom f) {a : G} {b : G} (h : f (a * b⁻¹) = 1) :
          f a = f b
          theorem IsAddGroupHom.zero_ker_neg' {G : Type u_1} {H : Type u_2} [inst : AddGroup G] [inst : AddGroup H] {f : GH} (hf : IsAddGroupHom f) {a : G} {b : G} (h : f (-a + b) = 0) :
          f a = f b
          theorem IsGroupHom.one_ker_inv' {G : Type u_1} {H : Type u_2} [inst : Group G] [inst : Group H] {f : GH} (hf : IsGroupHom f) {a : G} {b : G} (h : f (a⁻¹ * b) = 1) :
          f a = f b
          theorem IsAddGroupHom.neg_ker_zero {G : Type u_1} {H : Type u_2} [inst : AddGroup G] [inst : AddGroup H] {f : GH} (hf : IsAddGroupHom f) {a : G} {b : G} (h : f a = f b) :
          f (a + -b) = 0
          theorem IsGroupHom.inv_ker_one {G : Type u_1} {H : Type u_2} [inst : Group G] [inst : Group H] {f : GH} (hf : IsGroupHom f) {a : G} {b : G} (h : f a = f b) :
          f (a * b⁻¹) = 1
          theorem IsAddGroupHom.neg_ker_zero' {G : Type u_1} {H : Type u_2} [inst : AddGroup G] [inst : AddGroup H] {f : GH} (hf : IsAddGroupHom f) {a : G} {b : G} (h : f a = f b) :
          f (-a + b) = 0
          theorem IsGroupHom.inv_ker_one' {G : Type u_1} {H : Type u_2} [inst : Group G] [inst : Group H] {f : GH} (hf : IsGroupHom f) {a : G} {b : G} (h : f a = f b) :
          f (a⁻¹ * b) = 1
          theorem IsAddGroupHom.zero_iff_ker_neg {G : Type u_1} {H : Type u_2} [inst : AddGroup G] [inst : AddGroup H] {f : GH} (hf : IsAddGroupHom f) (a : G) (b : G) :
          f a = f b f (a + -b) = 0
          theorem IsGroupHom.one_iff_ker_inv {G : Type u_1} {H : Type u_2} [inst : Group G] [inst : Group H] {f : GH} (hf : IsGroupHom f) (a : G) (b : G) :
          f a = f b f (a * b⁻¹) = 1
          theorem IsAddGroupHom.zero_iff_ker_neg' {G : Type u_1} {H : Type u_2} [inst : AddGroup G] [inst : AddGroup H] {f : GH} (hf : IsAddGroupHom f) (a : G) (b : G) :
          f a = f b f (-a + b) = 0
          theorem IsGroupHom.one_iff_ker_inv' {G : Type u_1} {H : Type u_2} [inst : Group G] [inst : Group H] {f : GH} (hf : IsGroupHom f) (a : G) (b : G) :
          f a = f b f (a⁻¹ * b) = 1
          theorem IsAddGroupHom.neg_iff_ker {G : Type u_1} {H : Type u_2} [inst : AddGroup G] [inst : AddGroup H] {f : GH} (hf : IsAddGroupHom f) (a : G) (b : G) :
          f a = f b a + -b IsAddGroupHom.ker f
          theorem IsGroupHom.inv_iff_ker {G : Type u_1} {H : Type u_2} [inst : Group G] [inst : Group H] {f : GH} (hf : IsGroupHom f) (a : G) (b : G) :
          f a = f b a * b⁻¹ IsGroupHom.ker f
          theorem IsAddGroupHom.neg_iff_ker' {G : Type u_1} {H : Type u_2} [inst : AddGroup G] [inst : AddGroup H] {f : GH} (hf : IsAddGroupHom f) (a : G) (b : G) :
          f a = f b -a + b IsAddGroupHom.ker f
          theorem IsGroupHom.inv_iff_ker' {G : Type u_1} {H : Type u_2} [inst : Group G] [inst : Group H] {f : GH} (hf : IsGroupHom f) (a : G) (b : G) :
          f a = f b a⁻¹ * b IsGroupHom.ker f
          abbrev IsAddGroupHom.image_addSubgroup.match_1 {G : Type u_2} {H : Type u_1} {f : GH} {s : Set G} {a₂ : H} (motive : a₂ f '' sProp) :
          (x : a₂ f '' s) → ((b₂ : G) → (hb₂ : b₂ s) → (eq₂ : f b₂ = a₂) → motive (_ : a, a s f a = a₂)) → motive x
          Equations
          theorem IsAddGroupHom.image_addSubgroup {G : Type u_1} {H : Type u_2} [inst : AddGroup G] [inst : AddGroup H] {f : GH} (hf : IsAddGroupHom f) {s : Set G} (hs : IsAddSubgroup s) :
          theorem IsGroupHom.image_subgroup {G : Type u_1} {H : Type u_2} [inst : Group G] [inst : Group H] {f : GH} (hf : IsGroupHom f) {s : Set G} (hs : IsSubgroup s) :
          theorem IsAddGroupHom.range_addSubgroup {G : Type u_1} {H : Type u_2} [inst : AddGroup G] [inst : AddGroup H] {f : GH} (hf : IsAddGroupHom f) :
          theorem IsGroupHom.range_subgroup {G : Type u_1} {H : Type u_2} [inst : Group G] [inst : Group H] {f : GH} (hf : IsGroupHom f) :
          theorem IsAddGroupHom.preimage {G : Type u_1} {H : Type u_2} [inst : AddGroup G] [inst : AddGroup H] {f : GH} (hf : IsAddGroupHom f) {s : Set H} (hs : IsAddSubgroup s) :
          theorem IsGroupHom.preimage {G : Type u_1} {H : Type u_2} [inst : Group G] [inst : Group H] {f : GH} (hf : IsGroupHom f) {s : Set H} (hs : IsSubgroup s) :
          theorem IsAddGroupHom.preimage_normal {G : Type u_1} {H : Type u_2} [inst : AddGroup G] [inst : AddGroup H] {f : GH} (hf : IsAddGroupHom f) {s : Set H} (hs : IsNormalAddSubgroup s) :
          theorem IsGroupHom.preimage_normal {G : Type u_1} {H : Type u_2} [inst : Group G] [inst : Group H] {f : GH} (hf : IsGroupHom f) {s : Set H} (hs : IsNormalSubgroup s) :
          theorem IsAddGroupHom.isNormalAddSubgroup_ker {G : Type u_1} {H : Type u_2} [inst : AddGroup G] [inst : AddGroup H] {f : GH} (hf : IsAddGroupHom f) :
          theorem IsGroupHom.isNormalSubgroup_ker {G : Type u_1} {H : Type u_2} [inst : Group G] [inst : Group H] {f : GH} (hf : IsGroupHom f) :
          theorem IsAddGroupHom.injective_of_trivial_ker {G : Type u_1} {H : Type u_2} [inst : AddGroup G] [inst : AddGroup H] {f : GH} (hf : IsAddGroupHom f) (h : IsAddGroupHom.ker f = IsAddSubgroup.trivial G) :
          theorem IsGroupHom.injective_of_trivial_ker {G : Type u_1} {H : Type u_2} [inst : Group G] [inst : Group H] {f : GH} (hf : IsGroupHom f) (h : IsGroupHom.ker f = IsSubgroup.trivial G) :
          theorem IsAddGroupHom.trivial_ker_of_injective {G : Type u_1} {H : Type u_2} [inst : AddGroup G] [inst : AddGroup H] {f : GH} (hf : IsAddGroupHom f) (h : Function.Injective f) :
          theorem IsGroupHom.trivial_ker_of_injective {G : Type u_1} {H : Type u_2} [inst : Group G] [inst : Group H] {f : GH} (hf : IsGroupHom f) (h : Function.Injective f) :
          theorem IsGroupHom.injective_iff_trivial_ker {G : Type u_1} {H : Type u_2} [inst : Group G] [inst : Group H] {f : GH} (hf : IsGroupHom f) :
          theorem IsAddGroupHom.trivial_ker_iff_eq_zero {G : Type u_1} {H : Type u_2} [inst : AddGroup G] [inst : AddGroup H] {f : GH} (hf : IsAddGroupHom f) :
          IsAddGroupHom.ker f = IsAddSubgroup.trivial G ∀ (x : G), f x = 0x = 0
          theorem IsGroupHom.trivial_ker_iff_eq_one {G : Type u_1} {H : Type u_2} [inst : Group G] [inst : Group H] {f : GH} (hf : IsGroupHom f) :
          IsGroupHom.ker f = IsSubgroup.trivial G ∀ (x : G), f x = 1x = 1
          inductive AddGroup.InClosure {A : Type u_1} [inst : AddGroup A] (s : Set A) :
          AProp

          If A is an additive group and s : Set A, then InClosure s : Set A is the underlying subset of the subgroup generated by s.

          Instances For
            inductive Group.InClosure {G : Type u_1} [inst : Group G] (s : Set G) :
            GProp

            If G is a group and s : Set G, then InClosure s : Set G is the underlying subset of the subgroup generated by s.

            Instances For
              def AddGroup.closure {G : Type u_1} [inst : AddGroup G] (s : Set G) :
              Set G

              AddGroup.closure s is the additive subgroup generated by s, i.e., the smallest additive subgroup containing s.

              Equations
              def Group.closure {G : Type u_1} [inst : Group G] (s : Set G) :
              Set G

              Group.closure s is the subgroup generated by s, i.e. the smallest subgroup containg s.

              Equations
              theorem AddGroup.mem_closure {G : Type u_1} [inst : AddGroup G] {s : Set G} {a : G} :
              theorem Group.mem_closure {G : Type u_1} [inst : Group G] {s : Set G} {a : G} :
              a sa Group.closure s
              theorem Group.closure.isSubgroup {G : Type u_1} [inst : Group G] (s : Set G) :
              theorem AddGroup.subset_closure {G : Type u_1} [inst : AddGroup G] {s : Set G} :
              theorem Group.subset_closure {G : Type u_1} [inst : Group G] {s : Set G} :
              theorem AddGroup.closure_subset {G : Type u_1} [inst : AddGroup G] {s : Set G} {t : Set G} (ht : IsAddSubgroup t) (h : s t) :
              theorem Group.closure_subset {G : Type u_1} [inst : Group G] {s : Set G} {t : Set G} (ht : IsSubgroup t) (h : s t) :
              theorem AddGroup.closure_subset_iff {G : Type u_1} [inst : AddGroup G] {s : Set G} {t : Set G} (ht : IsAddSubgroup t) :
              theorem Group.closure_subset_iff {G : Type u_1} [inst : Group G] {s : Set G} {t : Set G} (ht : IsSubgroup t) :
              theorem AddGroup.closure_mono {G : Type u_1} [inst : AddGroup G] {s : Set G} {t : Set G} (h : s t) :
              theorem Group.closure_mono {G : Type u_1} [inst : Group G] {s : Set G} {t : Set G} (h : s t) :
              @[simp]
              theorem AddGroup.closure_addSubgroup {G : Type u_1} [inst : AddGroup G] {s : Set G} (hs : IsAddSubgroup s) :
              @[simp]
              theorem Group.closure_subgroup {G : Type u_1} [inst : Group G] {s : Set G} (hs : IsSubgroup s) :
              theorem AddGroup.exists_list_of_mem_closure {G : Type u_1} [inst : AddGroup G] {s : Set G} {a : G} (h : a AddGroup.closure s) :
              l, (∀ (x : G), x lx s -x s) List.sum l = a
              abbrev AddGroup.exists_list_of_mem_closure.match_1 {G : Type u_1} [inst : AddGroup G] (L : List G) (x : G) (motive : (a, a List.reverse L -a = x) → Prop) :
              (x : a, a List.reverse L -a = x) → ((y : G) → (hy1 : y List.reverse L) → (hy2 : -y = x) → motive (_ : a, a List.reverse L -a = x)) → motive x
              Equations
              abbrev AddGroup.exists_list_of_mem_closure.match_2 {G : Type u_1} [inst : AddGroup G] {s : Set G} {x : G} (motive : (l, (∀ (x : G), x lx s -x s) List.sum l = x) → Prop) :
              (x : l, (∀ (x : G), x lx s -x s) List.sum l = x) → ((L : List G) → (HL1 : ∀ (x : G), x Lx s -x s) → (HL2 : List.sum L = x) → motive (_ : l, (∀ (x : G), x lx s -x s) List.sum l = x)) → motive x
              Equations
              theorem Group.exists_list_of_mem_closure {G : Type u_1} [inst : Group G] {s : Set G} {a : G} (h : a Group.closure s) :
              l, (∀ (x : G), x lx s x⁻¹ s) List.prod l = a
              theorem AddGroup.image_closure {G : Type u_2} {H : Type u_1} [inst : AddGroup G] [inst : AddGroup H] {f : GH} (hf : IsAddGroupHom f) (s : Set G) :
              theorem Group.image_closure {G : Type u_2} {H : Type u_1} [inst : Group G] [inst : Group H] {f : GH} (hf : IsGroupHom f) (s : Set G) :
              theorem Group.mclosure_subset {G : Type u_1} [inst : Group G] {s : Set G} :
              theorem Group.mclosure_inv_subset {G : Type u_1} [inst : Group G] {s : Set G} :
              theorem Group.closure_eq_mclosure {G : Type u_1} [inst : Group G] {s : Set G} :
              theorem AddGroup.mem_closure_union_iff {G : Type u_1} [inst : AddCommGroup G] {s : Set G} {t : Set G} {x : G} :
              theorem Group.mem_closure_union_iff {G : Type u_1} [inst : CommGroup G] {s : Set G} {t : Set G} {x : G} :
              x Group.closure (s t) y, y Group.closure s z, z Group.closure t y * z = x
              theorem Group.conjugatesOf_subset {G : Type u_1} [inst : Group G] {t : Set G} (ht : IsNormalSubgroup t) {a : G} (h : a t) :
              theorem Group.conjugatesOfSet_subset' {G : Type u_1} [inst : Group G] {s : Set G} {t : Set G} (ht : IsNormalSubgroup t) (h : s t) :
              def Group.normalClosure {G : Type u_1} [inst : Group G] (s : Set G) :
              Set G

              The normal closure of a set s is the subgroup closure of all the conjugates of elements of s. It is the smallest normal subgroup containing s.

              Equations
              theorem Group.subset_normalClosure {G : Type u_1} {s : Set G} [inst : Group G] :

              The normal closure of a set is a subgroup.

              The normal closure of s is a normal subgroup.

              theorem Group.normalClosure_subset {G : Type u_1} [inst : Group G] {s : Set G} {t : Set G} (ht : IsNormalSubgroup t) (h : s t) :

              The normal closure of s is the smallest normal subgroup containing s.

              theorem Group.normalClosure_subset_iff {G : Type u_1} [inst : Group G] {s : Set G} {t : Set G} (ht : IsNormalSubgroup t) :
              theorem Group.normalClosure_mono {G : Type u_1} [inst : Group G] {s : Set G} {t : Set G} :
              def AddSubgroup.of.proof_1 {G : Type u_1} [inst : AddGroup G] {s : Set G} (h : IsAddSubgroup s) :
              ∀ {a b : G}, a sb sa + b s
              Equations
              def AddSubgroup.of.proof_3 {G : Type u_1} [inst : AddGroup G] {s : Set G} (h : IsAddSubgroup s) :
              ∀ {x : G}, x s-x s
              Equations
              def AddSubgroup.of {G : Type u_1} [inst : AddGroup G] {s : Set G} (h : IsAddSubgroup s) :

              Create a bundled additive subgroup from a set s and [is_add_subgroup s].

              Equations
              • One or more equations did not get rendered due to their size.
              def AddSubgroup.of.proof_2 {G : Type u_1} [inst : AddGroup G] {s : Set G} (h : IsAddSubgroup s) :
              0 s
              Equations
              def Subgroup.of {G : Type u_1} [inst : Group G] {s : Set G} (h : IsSubgroup s) :

              Create a bundled subgroup from a set s and [IsSubgroup s].

              Equations
              • One or more equations did not get rendered due to their size.
              theorem AddSubgroup.isAddSubgroup {G : Type u_1} [inst : AddGroup G] (K : AddSubgroup G) :
              theorem Subgroup.isSubgroup {G : Type u_1} [inst : Group G] (K : Subgroup G) :
              theorem Subgroup.of_normal {G : Type u_1} [inst : Group G] (s : Set G) (h : IsSubgroup s) (n : IsNormalSubgroup s) :