Documentation

Mathlib.GroupTheory.Index

Index of a Subgroup #

In this file we define the index of a subgroup, and prove several divisibility properties. Several theorems proved in this file are known as Lagrange's theorem.

Main definitions #

Main results #

noncomputable def AddSubgroup.index {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :

The index of a subgroup as a natural number, and returns 0 if the index is infinite.

Equations
Instances For
    noncomputable def Subgroup.index {G : Type u_1} [Group G] (H : Subgroup G) :

    The index of a subgroup as a natural number, and returns 0 if the index is infinite.

    Equations
    Instances For
      noncomputable def AddSubgroup.relindex {G : Type u_1} [AddGroup G] (H : AddSubgroup G) (K : AddSubgroup G) :

      The relative index of a subgroup as a natural number, and returns 0 if the relative index is infinite.

      Equations
      • H.relindex K = (H.addSubgroupOf K).index
      Instances For
        noncomputable def Subgroup.relindex {G : Type u_1} [Group G] (H : Subgroup G) (K : Subgroup G) :

        The relative index of a subgroup as a natural number, and returns 0 if the relative index is infinite.

        Equations
        • H.relindex K = (H.subgroupOf K).index
        Instances For
          theorem AddSubgroup.index_comap_of_surjective {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) {f : G' →+ G} (hf : Function.Surjective f) :
          (AddSubgroup.comap f H).index = H.index
          theorem Subgroup.index_comap_of_surjective {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (H : Subgroup G) {f : G' →* G} (hf : Function.Surjective f) :
          (Subgroup.comap f H).index = H.index
          theorem AddSubgroup.index_comap {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) (f : G' →+ G) :
          (AddSubgroup.comap f H).index = H.relindex f.range
          theorem Subgroup.index_comap {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (H : Subgroup G) (f : G' →* G) :
          (Subgroup.comap f H).index = H.relindex f.range
          theorem AddSubgroup.relindex_comap {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) (f : G' →+ G) (K : AddSubgroup G') :
          (AddSubgroup.comap f H).relindex K = H.relindex (AddSubgroup.map f K)
          theorem Subgroup.relindex_comap {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (H : Subgroup G) (f : G' →* G) (K : Subgroup G') :
          (Subgroup.comap f H).relindex K = H.relindex (Subgroup.map f K)
          theorem AddSubgroup.relindex_mul_index {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {K : AddSubgroup G} (h : H K) :
          H.relindex K * K.index = H.index
          theorem Subgroup.relindex_mul_index {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} (h : H K) :
          H.relindex K * K.index = H.index
          theorem AddSubgroup.index_dvd_of_le {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {K : AddSubgroup G} (h : H K) :
          K.index H.index
          theorem Subgroup.index_dvd_of_le {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} (h : H K) :
          K.index H.index
          theorem AddSubgroup.relindex_dvd_index_of_le {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {K : AddSubgroup G} (h : H K) :
          H.relindex K H.index
          theorem Subgroup.relindex_dvd_index_of_le {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} (h : H K) :
          H.relindex K H.index
          theorem AddSubgroup.relindex_addSubgroupOf {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {K : AddSubgroup G} {L : AddSubgroup G} (hKL : K L) :
          (H.addSubgroupOf L).relindex (K.addSubgroupOf L) = H.relindex K
          theorem Subgroup.relindex_subgroupOf {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} {L : Subgroup G} (hKL : K L) :
          (H.subgroupOf L).relindex (K.subgroupOf L) = H.relindex K
          theorem AddSubgroup.relindex_mul_relindex {G : Type u_1} [AddGroup G] (H : AddSubgroup G) (K : AddSubgroup G) (L : AddSubgroup G) (hHK : H K) (hKL : K L) :
          H.relindex K * K.relindex L = H.relindex L
          theorem Subgroup.relindex_mul_relindex {G : Type u_1} [Group G] (H : Subgroup G) (K : Subgroup G) (L : Subgroup G) (hHK : H K) (hKL : K L) :
          H.relindex K * K.relindex L = H.relindex L
          theorem AddSubgroup.inf_relindex_right {G : Type u_1} [AddGroup G] (H : AddSubgroup G) (K : AddSubgroup G) :
          (H K).relindex K = H.relindex K
          theorem Subgroup.inf_relindex_right {G : Type u_1} [Group G] (H : Subgroup G) (K : Subgroup G) :
          (H K).relindex K = H.relindex K
          theorem AddSubgroup.inf_relindex_left {G : Type u_1} [AddGroup G] (H : AddSubgroup G) (K : AddSubgroup G) :
          (H K).relindex H = K.relindex H
          theorem Subgroup.inf_relindex_left {G : Type u_1} [Group G] (H : Subgroup G) (K : Subgroup G) :
          (H K).relindex H = K.relindex H
          theorem AddSubgroup.relindex_inf_mul_relindex {G : Type u_1} [AddGroup G] (H : AddSubgroup G) (K : AddSubgroup G) (L : AddSubgroup G) :
          H.relindex (K L) * K.relindex L = (H K).relindex L
          theorem Subgroup.relindex_inf_mul_relindex {G : Type u_1} [Group G] (H : Subgroup G) (K : Subgroup G) (L : Subgroup G) :
          H.relindex (K L) * K.relindex L = (H K).relindex L
          @[simp]
          theorem AddSubgroup.relindex_sup_right {G : Type u_1} [AddGroup G] (H : AddSubgroup G) (K : AddSubgroup G) [K.Normal] :
          K.relindex (H K) = K.relindex H
          @[simp]
          theorem Subgroup.relindex_sup_right {G : Type u_1} [Group G] (H : Subgroup G) (K : Subgroup G) [K.Normal] :
          K.relindex (H K) = K.relindex H
          @[simp]
          theorem AddSubgroup.relindex_sup_left {G : Type u_1} [AddGroup G] (H : AddSubgroup G) (K : AddSubgroup G) [K.Normal] :
          K.relindex (K H) = K.relindex H
          @[simp]
          theorem Subgroup.relindex_sup_left {G : Type u_1} [Group G] (H : Subgroup G) (K : Subgroup G) [K.Normal] :
          K.relindex (K H) = K.relindex H
          theorem AddSubgroup.relindex_dvd_index_of_normal {G : Type u_1} [AddGroup G] (H : AddSubgroup G) (K : AddSubgroup G) [H.Normal] :
          H.relindex K H.index
          theorem Subgroup.relindex_dvd_index_of_normal {G : Type u_1} [Group G] (H : Subgroup G) (K : Subgroup G) [H.Normal] :
          H.relindex K H.index
          theorem AddSubgroup.relindex_dvd_of_le_left {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {K : AddSubgroup G} (L : AddSubgroup G) (hHK : H K) :
          K.relindex L H.relindex L
          theorem Subgroup.relindex_dvd_of_le_left {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} (L : Subgroup G) (hHK : H K) :
          K.relindex L H.relindex L
          theorem AddSubgroup.index_eq_two_iff {G : Type u_1} [AddGroup G] {H : AddSubgroup G} :
          H.index = 2 ∃ (a : G), ∀ (b : G), Xor' (b + a H) (b H)

          An additive subgroup has index two if and only if there exists a such that for all b, exactly one of b + a and b belong to H.

          theorem Subgroup.index_eq_two_iff {G : Type u_1} [Group G] {H : Subgroup G} :
          H.index = 2 ∃ (a : G), ∀ (b : G), Xor' (b * a H) (b H)

          A subgroup has index two if and only if there exists a such that for all b, exactly one of b * a and b belong to H.

          theorem AddSubgroup.add_mem_iff_of_index_two {G : Type u_1} [AddGroup G] {H : AddSubgroup G} (h : H.index = 2) {a : G} {b : G} :
          a + b H (a H b H)
          theorem Subgroup.mul_mem_iff_of_index_two {G : Type u_1} [Group G] {H : Subgroup G} (h : H.index = 2) {a : G} {b : G} :
          a * b H (a H b H)
          theorem AddSubgroup.add_self_mem_of_index_two {G : Type u_1} [AddGroup G] {H : AddSubgroup G} (h : H.index = 2) (a : G) :
          a + a H
          theorem Subgroup.mul_self_mem_of_index_two {G : Type u_1} [Group G] {H : Subgroup G} (h : H.index = 2) (a : G) :
          a * a H
          theorem AddSubgroup.two_smul_mem_of_index_two {G : Type u_1} [AddGroup G] {H : AddSubgroup G} (h : H.index = 2) (a : G) :
          2 a H
          theorem Subgroup.sq_mem_of_index_two {G : Type u_1} [Group G] {H : Subgroup G} (h : H.index = 2) (a : G) :
          a ^ 2 H
          @[simp]
          theorem AddSubgroup.index_top {G : Type u_1} [AddGroup G] :
          .index = 1
          @[simp]
          theorem Subgroup.index_top {G : Type u_1} [Group G] :
          .index = 1
          @[simp]
          theorem AddSubgroup.index_bot {G : Type u_1} [AddGroup G] :
          .index = Nat.card G
          @[simp]
          theorem Subgroup.index_bot {G : Type u_1} [Group G] :
          .index = Nat.card G
          @[deprecated Subgroup.index_bot]
          theorem Subgroup.index_bot_eq_card {G : Type u_1} [Group G] :
          .index = Nat.card G

          Alias of Subgroup.index_bot.

          @[simp]
          theorem AddSubgroup.relindex_top_left {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
          .relindex H = 1
          @[simp]
          theorem Subgroup.relindex_top_left {G : Type u_1} [Group G] (H : Subgroup G) :
          .relindex H = 1
          @[simp]
          theorem AddSubgroup.relindex_top_right {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
          H.relindex = H.index
          @[simp]
          theorem Subgroup.relindex_top_right {G : Type u_1} [Group G] (H : Subgroup G) :
          H.relindex = H.index
          @[simp]
          theorem AddSubgroup.relindex_bot_left {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
          .relindex H = Nat.card H
          @[simp]
          theorem Subgroup.relindex_bot_left {G : Type u_1} [Group G] (H : Subgroup G) :
          .relindex H = Nat.card H
          @[deprecated Subgroup.relindex_bot_left]
          theorem Subgroup.relindex_bot_left_eq_card {G : Type u_1} [Group G] (H : Subgroup G) :
          .relindex H = Nat.card H

          Alias of Subgroup.relindex_bot_left.

          @[simp]
          theorem AddSubgroup.relindex_bot_right {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
          H.relindex = 1
          @[simp]
          theorem Subgroup.relindex_bot_right {G : Type u_1} [Group G] (H : Subgroup G) :
          H.relindex = 1
          @[simp]
          theorem AddSubgroup.relindex_self {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
          H.relindex H = 1
          @[simp]
          theorem Subgroup.relindex_self {G : Type u_1} [Group G] (H : Subgroup G) :
          H.relindex H = 1
          theorem AddSubgroup.index_ker {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (f : G →+ G') :
          f.ker.index = Nat.card (Set.range f)
          theorem Subgroup.index_ker {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (f : G →* G') :
          f.ker.index = Nat.card (Set.range f)
          theorem AddSubgroup.relindex_ker {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (f : G →+ G') (K : AddSubgroup G) :
          f.ker.relindex K = Nat.card (f '' K)
          theorem Subgroup.relindex_ker {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (f : G →* G') (K : Subgroup G) :
          f.ker.relindex K = Nat.card (f '' K)
          @[simp]
          theorem AddSubgroup.card_mul_index {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
          Nat.card H * H.index = Nat.card G
          @[simp]
          theorem Subgroup.card_mul_index {G : Type u_1} [Group G] (H : Subgroup G) :
          Nat.card H * H.index = Nat.card G
          @[deprecated Subgroup.card_dvd_of_injective]
          theorem Subgroup.nat_card_dvd_of_injective {α : Type u_1} [Group α] {H : Type u_2} [Group H] (f : α →* H) (hf : Function.Injective f) :

          Alias of Subgroup.card_dvd_of_injective.

          @[deprecated Subgroup.card_dvd_of_le]
          theorem Subgroup.nat_card_dvd_of_le {α : Type u_1} [Group α] {H : Subgroup α} {K : Subgroup α} (hHK : H K) :

          Alias of Subgroup.card_dvd_of_le.

          theorem AddSubgroup.card_dvd_of_surjective {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (f : G →+ G') (hf : Function.Surjective f) :
          theorem Subgroup.card_dvd_of_surjective {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (f : G →* G') (hf : Function.Surjective f) :
          @[deprecated Subgroup.card_dvd_of_surjective]
          theorem Subgroup.nat_card_dvd_of_surjective {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (f : G →* G') (hf : Function.Surjective f) :

          Alias of Subgroup.card_dvd_of_surjective.

          theorem AddSubgroup.index_map {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) (f : G →+ G') :
          (AddSubgroup.map f H).index = (H f.ker).index * f.range.index
          theorem Subgroup.index_map {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (H : Subgroup G) (f : G →* G') :
          (Subgroup.map f H).index = (H f.ker).index * f.range.index
          theorem AddSubgroup.index_map_dvd {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) {f : G →+ G'} (hf : Function.Surjective f) :
          (AddSubgroup.map f H).index H.index
          theorem Subgroup.index_map_dvd {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (H : Subgroup G) {f : G →* G'} (hf : Function.Surjective f) :
          (Subgroup.map f H).index H.index
          theorem AddSubgroup.dvd_index_map {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) {f : G →+ G'} (hf : f.ker H) :
          H.index (AddSubgroup.map f H).index
          theorem Subgroup.dvd_index_map {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (H : Subgroup G) {f : G →* G'} (hf : f.ker H) :
          H.index (Subgroup.map f H).index
          theorem AddSubgroup.index_map_eq {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) {f : G →+ G'} (hf1 : Function.Surjective f) (hf2 : f.ker H) :
          (AddSubgroup.map f H).index = H.index
          theorem Subgroup.index_map_eq {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (H : Subgroup G) {f : G →* G'} (hf1 : Function.Surjective f) (hf2 : f.ker H) :
          (Subgroup.map f H).index = H.index
          theorem AddSubgroup.index_eq_card {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
          H.index = Nat.card (G H)
          theorem Subgroup.index_eq_card {G : Type u_1} [Group G] (H : Subgroup G) :
          H.index = Nat.card (G H)
          theorem AddSubgroup.index_mul_card {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
          H.index * Nat.card H = Nat.card G
          theorem Subgroup.index_mul_card {G : Type u_1} [Group G] (H : Subgroup G) :
          H.index * Nat.card H = Nat.card G
          theorem AddSubgroup.index_dvd_card {G : Type u_1} [AddGroup G] (H : AddSubgroup G) :
          H.index Nat.card G
          theorem Subgroup.index_dvd_card {G : Type u_1} [Group G] (H : Subgroup G) :
          H.index Nat.card G
          theorem AddSubgroup.relindex_eq_zero_of_le_left {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {K : AddSubgroup G} {L : AddSubgroup G} (hHK : H K) (hKL : K.relindex L = 0) :
          H.relindex L = 0
          theorem Subgroup.relindex_eq_zero_of_le_left {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} {L : Subgroup G} (hHK : H K) (hKL : K.relindex L = 0) :
          H.relindex L = 0
          theorem AddSubgroup.relindex_eq_zero_of_le_right {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {K : AddSubgroup G} {L : AddSubgroup G} (hKL : K L) (hHK : H.relindex K = 0) :
          H.relindex L = 0
          theorem Subgroup.relindex_eq_zero_of_le_right {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} {L : Subgroup G} (hKL : K L) (hHK : H.relindex K = 0) :
          H.relindex L = 0
          theorem AddSubgroup.index_eq_zero_of_relindex_eq_zero {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {K : AddSubgroup G} (h : H.relindex K = 0) :
          H.index = 0
          theorem Subgroup.index_eq_zero_of_relindex_eq_zero {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} (h : H.relindex K = 0) :
          H.index = 0
          theorem AddSubgroup.relindex_le_of_le_left {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {K : AddSubgroup G} {L : AddSubgroup G} (hHK : H K) (hHL : H.relindex L 0) :
          K.relindex L H.relindex L
          theorem Subgroup.relindex_le_of_le_left {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} {L : Subgroup G} (hHK : H K) (hHL : H.relindex L 0) :
          K.relindex L H.relindex L
          theorem AddSubgroup.relindex_le_of_le_right {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {K : AddSubgroup G} {L : AddSubgroup G} (hKL : K L) (hHL : H.relindex L 0) :
          H.relindex K H.relindex L
          theorem Subgroup.relindex_le_of_le_right {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} {L : Subgroup G} (hKL : K L) (hHL : H.relindex L 0) :
          H.relindex K H.relindex L
          theorem AddSubgroup.relindex_ne_zero_trans {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {K : AddSubgroup G} {L : AddSubgroup G} (hHK : H.relindex K 0) (hKL : K.relindex L 0) :
          H.relindex L 0
          theorem Subgroup.relindex_ne_zero_trans {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} {L : Subgroup G} (hHK : H.relindex K 0) (hKL : K.relindex L 0) :
          H.relindex L 0
          theorem AddSubgroup.relindex_inf_ne_zero {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {K : AddSubgroup G} {L : AddSubgroup G} (hH : H.relindex L 0) (hK : K.relindex L 0) :
          (H K).relindex L 0
          theorem Subgroup.relindex_inf_ne_zero {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} {L : Subgroup G} (hH : H.relindex L 0) (hK : K.relindex L 0) :
          (H K).relindex L 0
          theorem AddSubgroup.index_inf_ne_zero {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {K : AddSubgroup G} (hH : H.index 0) (hK : K.index 0) :
          (H K).index 0
          theorem Subgroup.index_inf_ne_zero {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} (hH : H.index 0) (hK : K.index 0) :
          (H K).index 0
          theorem AddSubgroup.relindex_inf_le {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {K : AddSubgroup G} {L : AddSubgroup G} :
          (H K).relindex L H.relindex L * K.relindex L
          theorem Subgroup.relindex_inf_le {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} {L : Subgroup G} :
          (H K).relindex L H.relindex L * K.relindex L
          theorem AddSubgroup.index_inf_le {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {K : AddSubgroup G} :
          (H K).index H.index * K.index
          theorem Subgroup.index_inf_le {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} :
          (H K).index H.index * K.index
          theorem AddSubgroup.relindex_iInf_ne_zero {G : Type u_1} [AddGroup G] {L : AddSubgroup G} {ι : Type u_3} [_hι : Finite ι] {f : ιAddSubgroup G} (hf : ∀ (i : ι), (f i).relindex L 0) :
          (⨅ (i : ι), f i).relindex L 0
          theorem Subgroup.relindex_iInf_ne_zero {G : Type u_1} [Group G] {L : Subgroup G} {ι : Type u_3} [_hι : Finite ι] {f : ιSubgroup G} (hf : ∀ (i : ι), (f i).relindex L 0) :
          (⨅ (i : ι), f i).relindex L 0
          abbrev AddSubgroup.relindex_iInf_le.match_1 {G : Type u_2} [AddGroup G] {L : AddSubgroup G} {ι : Type u_1} [Fintype ι] (f : ιAddSubgroup G) (motive : (aFinset.univ, Nat.card (L (f a).addSubgroupOf L) = 0)Prop) :
          ∀ (x : aFinset.univ, Nat.card (L (f a).addSubgroupOf L) = 0), (∀ (i : ι) (_hi : i Finset.univ) (h : Nat.card (L (f i).addSubgroupOf L) = 0), motive )motive x
          Equations
          • =
          Instances For
            theorem AddSubgroup.relindex_iInf_le {G : Type u_1} [AddGroup G] {L : AddSubgroup G} {ι : Type u_3} [Fintype ι] (f : ιAddSubgroup G) :
            (⨅ (i : ι), f i).relindex L i : ι, (f i).relindex L
            theorem Subgroup.relindex_iInf_le {G : Type u_1} [Group G] {L : Subgroup G} {ι : Type u_3} [Fintype ι] (f : ιSubgroup G) :
            (⨅ (i : ι), f i).relindex L i : ι, (f i).relindex L
            theorem AddSubgroup.index_iInf_ne_zero {G : Type u_1} [AddGroup G] {ι : Type u_3} [Finite ι] {f : ιAddSubgroup G} (hf : ∀ (i : ι), (f i).index 0) :
            (⨅ (i : ι), f i).index 0
            theorem Subgroup.index_iInf_ne_zero {G : Type u_1} [Group G] {ι : Type u_3} [Finite ι] {f : ιSubgroup G} (hf : ∀ (i : ι), (f i).index 0) :
            (⨅ (i : ι), f i).index 0
            theorem AddSubgroup.index_iInf_le {G : Type u_1} [AddGroup G] {ι : Type u_3} [Fintype ι] (f : ιAddSubgroup G) :
            (⨅ (i : ι), f i).index i : ι, (f i).index
            theorem Subgroup.index_iInf_le {G : Type u_1} [Group G] {ι : Type u_3} [Fintype ι] (f : ιSubgroup G) :
            (⨅ (i : ι), f i).index i : ι, (f i).index
            @[simp]
            theorem AddSubgroup.index_eq_one {G : Type u_1} [AddGroup G] {H : AddSubgroup G} :
            H.index = 1 H =
            @[simp]
            theorem Subgroup.index_eq_one {G : Type u_1} [Group G] {H : Subgroup G} :
            H.index = 1 H =
            @[simp]
            theorem AddSubgroup.relindex_eq_one {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {K : AddSubgroup G} :
            H.relindex K = 1 K H
            @[simp]
            theorem Subgroup.relindex_eq_one {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} :
            H.relindex K = 1 K H
            @[simp]
            theorem AddSubgroup.card_eq_one {G : Type u_1} [AddGroup G] {H : AddSubgroup G} :
            Nat.card H = 1 H =
            @[simp]
            theorem Subgroup.card_eq_one {G : Type u_1} [Group G] {H : Subgroup G} :
            Nat.card H = 1 H =
            theorem AddSubgroup.index_ne_zero_of_finite {G : Type u_1} [AddGroup G] {H : AddSubgroup G} [hH : Finite (G H)] :
            H.index 0
            theorem Subgroup.index_ne_zero_of_finite {G : Type u_1} [Group G] {H : Subgroup G} [hH : Finite (G H)] :
            H.index 0
            noncomputable def AddSubgroup.fintypeOfIndexNeZero {G : Type u_1} [AddGroup G] {H : AddSubgroup G} (hH : H.index 0) :
            Fintype (G H)

            Finite index implies finite quotient.

            Equations
            Instances For
              theorem AddSubgroup.fintypeOfIndexNeZero.proof_1 {G : Type u_1} [AddGroup G] {H : AddSubgroup G} (hH : H.index 0) :
              Finite (G H)
              noncomputable def Subgroup.fintypeOfIndexNeZero {G : Type u_1} [Group G] {H : Subgroup G} (hH : H.index 0) :
              Fintype (G H)

              Finite index implies finite quotient.

              Equations
              Instances For
                theorem AddSubgroup.one_lt_index_of_ne_top {G : Type u_1} [AddGroup G] {H : AddSubgroup G} [Finite (G H)] (hH : H ) :
                1 < H.index
                theorem Subgroup.one_lt_index_of_ne_top {G : Type u_1} [Group G] {H : Subgroup G} [Finite (G H)] (hH : H ) :
                1 < H.index
                class Subgroup.FiniteIndex {G : Type u_1} [Group G] (H : Subgroup G) :

                Typeclass for finite index subgroups.

                • finiteIndex : H.index 0

                  The subgroup has finite index

                Instances
                  theorem Subgroup.FiniteIndex.finiteIndex {G : Type u_1} [Group G] {H : Subgroup G} [self : H.FiniteIndex] :
                  H.index 0

                  The subgroup has finite index

                  Typeclass for finite index subgroups.

                  • finiteIndex : H.index 0

                    The additive subgroup has finite index

                  Instances
                    theorem AddSubgroup.FiniteIndex.finiteIndex {G : Type u_3} [AddGroup G] {H : AddSubgroup G} [self : H.FiniteIndex] :
                    H.index 0

                    The additive subgroup has finite index

                    noncomputable def AddSubgroup.fintypeQuotientOfFiniteIndex {G : Type u_1} [AddGroup G] (H : AddSubgroup G) [H.FiniteIndex] :
                    Fintype (G H)

                    A finite index subgroup has finite quotient

                    Equations
                    Instances For
                      noncomputable def Subgroup.fintypeQuotientOfFiniteIndex {G : Type u_1} [Group G] (H : Subgroup G) [H.FiniteIndex] :
                      Fintype (G H)

                      A finite index subgroup has finite quotient.

                      Equations
                      Instances For
                        instance AddSubgroup.finite_quotient_of_finiteIndex {G : Type u_1} [AddGroup G] (H : AddSubgroup G) [H.FiniteIndex] :
                        Finite (G H)
                        Equations
                        • =
                        instance Subgroup.finite_quotient_of_finiteIndex {G : Type u_1} [Group G] (H : Subgroup G) [H.FiniteIndex] :
                        Finite (G H)
                        Equations
                        • =
                        theorem AddSubgroup.finiteIndex_of_finite_quotient {G : Type u_1} [AddGroup G] (H : AddSubgroup G) [Finite (G H)] :
                        H.FiniteIndex
                        theorem Subgroup.finiteIndex_of_finite_quotient {G : Type u_1} [Group G] (H : Subgroup G) [Finite (G H)] :
                        H.FiniteIndex
                        @[instance 100]
                        instance AddSubgroup.finiteIndex_of_finite {G : Type u_1} [AddGroup G] (H : AddSubgroup G) [Finite G] :
                        H.FiniteIndex
                        Equations
                        • =
                        @[instance 100]
                        instance Subgroup.finiteIndex_of_finite {G : Type u_1} [Group G] (H : Subgroup G) [Finite G] :
                        H.FiniteIndex
                        Equations
                        • =
                        instance AddSubgroup.instFiniteIndexTop {G : Type u_1} [AddGroup G] :
                        .FiniteIndex
                        Equations
                        • =
                        instance Subgroup.instFiniteIndexTop {G : Type u_1} [Group G] :
                        .FiniteIndex
                        Equations
                        • =
                        instance AddSubgroup.instFiniteIndexInf {G : Type u_1} [AddGroup G] (H : AddSubgroup G) (K : AddSubgroup G) [H.FiniteIndex] [K.FiniteIndex] :
                        (H K).FiniteIndex
                        Equations
                        • =
                        instance Subgroup.instFiniteIndexInf {G : Type u_1} [Group G] (H : Subgroup G) (K : Subgroup G) [H.FiniteIndex] [K.FiniteIndex] :
                        (H K).FiniteIndex
                        Equations
                        • =
                        theorem AddSubgroup.finiteIndex_iInf {G : Type u_1} [AddGroup G] {ι : Type u_3} [Finite ι] {f : ιAddSubgroup G} (hf : ∀ (i : ι), (f i).FiniteIndex) :
                        (⨅ (i : ι), f i).FiniteIndex
                        theorem Subgroup.finiteIndex_iInf {G : Type u_1} [Group G] {ι : Type u_3} [Finite ι] {f : ιSubgroup G} (hf : ∀ (i : ι), (f i).FiniteIndex) :
                        (⨅ (i : ι), f i).FiniteIndex
                        theorem AddSubgroup.finiteIndex_iInf' {G : Type u_1} [AddGroup G] {ι : Type u_3} {s : Finset ι} (f : ιAddSubgroup G) (hs : is, (f i).FiniteIndex) :
                        (is, f i).FiniteIndex
                        abbrev AddSubgroup.finiteIndex_iInf'.match_1 {ι : Type u_1} {s : Finset ι} (motive : { i : ι // i s }Prop) :
                        ∀ (x : { i : ι // i s }), (∀ (i : ι) (hi : i s), motive i, hi)motive x
                        Equations
                        • =
                        Instances For
                          theorem Subgroup.finiteIndex_iInf' {G : Type u_1} [Group G] {ι : Type u_3} {s : Finset ι} (f : ιSubgroup G) (hs : is, (f i).FiniteIndex) :
                          (is, f i).FiniteIndex
                          instance AddSubgroup.instFiniteIndex_addSubgroupOf {G : Type u_1} [AddGroup G] (H : AddSubgroup G) (K : AddSubgroup G) [H.FiniteIndex] :
                          (H.addSubgroupOf K).FiniteIndex
                          Equations
                          • =
                          instance Subgroup.instFiniteIndex_subgroupOf {G : Type u_1} [Group G] (H : Subgroup G) (K : Subgroup G) [H.FiniteIndex] :
                          (H.subgroupOf K).FiniteIndex
                          Equations
                          • =
                          theorem AddSubgroup.finiteIndex_of_le {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {K : AddSubgroup G} [H.FiniteIndex] (h : H K) :
                          K.FiniteIndex
                          theorem Subgroup.finiteIndex_of_le {G : Type u_1} [Group G] {H : Subgroup G} {K : Subgroup G} [H.FiniteIndex] (h : H K) :
                          K.FiniteIndex
                          instance AddSubgroup.finiteIndex_ker {G : Type u_1} [AddGroup G] {G' : Type u_3} [AddGroup G'] (f : G →+ G') [Finite f.range] :
                          f.ker.FiniteIndex
                          Equations
                          • =
                          instance Subgroup.finiteIndex_ker {G : Type u_1} [Group G] {G' : Type u_3} [Group G'] (f : G →* G') [Finite f.range] :
                          f.ker.FiniteIndex
                          Equations
                          • =
                          instance Subgroup.finiteIndex_normalCore {G : Type u_1} [Group G] (H : Subgroup G) [H.FiniteIndex] :
                          H.normalCore.FiniteIndex
                          Equations
                          • =
                          instance Subgroup.finiteIndex_center (G : Type u_1) [Group G] [Finite (commutatorSet G)] [Group.FG G] :
                          (Subgroup.center G).FiniteIndex
                          Equations
                          • =
                          theorem AddMonoidHom.card_fiber_eq_of_mem_range {G : Type u_1} {M : Type u_2} {F : Type u_3} [AddGroup G] [Fintype G] [AddMonoid M] [DecidableEq M] [FunLike F G M] [AddMonoidHomClass F G M] (f : F) {x : M} {y : M} (hx : x Set.range f) (hy : y Set.range f) :
                          (Finset.filter (fun (g : G) => f g = x) Finset.univ).card = (Finset.filter (fun (g : G) => f g = y) Finset.univ).card
                          theorem MonoidHom.card_fiber_eq_of_mem_range {G : Type u_1} {M : Type u_2} {F : Type u_3} [Group G] [Fintype G] [Monoid M] [DecidableEq M] [FunLike F G M] [MonoidHomClass F G M] (f : F) {x : M} {y : M} (hx : x Set.range f) (hy : y Set.range f) :
                          (Finset.filter (fun (g : G) => f g = x) Finset.univ).card = (Finset.filter (fun (g : G) => f g = y) Finset.univ).card