mathlib documentation

group_theory.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 #

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noncomputable def subgroup.index {G : Type u_1} [group G] (H : subgroup G) :
ℕ

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

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noncomputable def add_subgroup.index {G : Type u_1} [add_group G] (H : add_subgroup G) :
ℕ

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

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noncomputable def subgroup.relindex {G : Type u_1} [group G] (H K : subgroup G) :
ℕ

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

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noncomputable def add_subgroup.relindex {G : Type u_1} [add_group G] (H K : add_subgroup G) :
ℕ

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

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theorem subgroup.index_comap_of_surjective {G : Type u_1} [group G] (H : subgroup G) {G' : Type u_2} [group G'] {f : G' →* G} (hf : function.surjective ⇑f) :
(subgroup.comap f H).index = H.index
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theorem add_subgroup.index_comap_of_surjective {G : Type u_1} [add_group G] (H : add_subgroup G) {G' : Type u_2} [add_group G'] {f : G' →+ G} (hf : function.surjective ⇑f) :
(add_subgroup.comap f H).index = H.index
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theorem subgroup.index_comap {G : Type u_1} [group G] (H : subgroup G) {G' : Type u_2} [group G'] (f : G' →* G) :
(subgroup.comap f H).index = H.relindex f.range
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theorem add_subgroup.index_comap {G : Type u_1} [add_group G] (H : add_subgroup G) {G' : Type u_2} [add_group G'] (f : G' →+ G) :
(add_subgroup.comap f H).index = H.relindex f.range
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theorem subgroup.relindex_mul_index {G : Type u_1} [group G] {H K : subgroup G} (h : H ≤ K) :
H.relindex K * K.index = H.index
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theorem add_subgroup.relindex_mul_index {G : Type u_1} [add_group G] {H K : add_subgroup G} (h : H ≤ K) :
H.relindex K * K.index = H.index
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theorem subgroup.index_dvd_of_le {G : Type u_1} [group G] {H K : subgroup G} (h : H ≤ K) :
K.index ∣ H.index
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theorem add_subgroup.index_dvd_of_le {G : Type u_1} [add_group G] {H K : add_subgroup G} (h : H ≤ K) :
K.index ∣ H.index
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theorem add_subgroup.relindex_dvd_index_of_le {G : Type u_1} [add_group G] {H K : add_subgroup G} (h : H ≤ K) :
H.relindex K ∣ H.index
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theorem subgroup.relindex_dvd_index_of_le {G : Type u_1} [group G] {H K : subgroup G} (h : H ≤ K) :
H.relindex K ∣ H.index
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theorem add_subgroup.relindex_add_subgroup_of {G : Type u_1} [add_group G] {H K L : add_subgroup G} (hKL : K ≤ L) :
(H.add_subgroup_of L).relindex (K.add_subgroup_of L) = H.relindex K
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theorem subgroup.relindex_subgroup_of {G : Type u_1} [group G] {H K L : subgroup G} (hKL : K ≤ L) :
(H.subgroup_of L).relindex (K.subgroup_of L) = H.relindex K
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theorem subgroup.relindex_mul_relindex {G : Type u_1} [group G] (H K L : subgroup G) (hHK : H ≤ K) (hKL : K ≤ L) :
H.relindex K * K.relindex L = H.relindex L
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theorem add_subgroup.relindex_mul_relindex {G : Type u_1} [add_group G] (H K L : add_subgroup G) (hHK : H ≤ K) (hKL : K ≤ L) :
H.relindex K * K.relindex L = H.relindex L
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theorem add_subgroup.inf_relindex_right {G : Type u_1} [add_group G] (H K : add_subgroup G) :
(H ⊓ K).relindex K = H.relindex K
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theorem subgroup.inf_relindex_right {G : Type u_1} [group G] (H K : subgroup G) :
(H ⊓ K).relindex K = H.relindex K
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theorem subgroup.inf_relindex_left {G : Type u_1} [group G] (H K : subgroup G) :
(H ⊓ K).relindex H = K.relindex H
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theorem add_subgroup.inf_relindex_left {G : Type u_1} [add_group G] (H K : add_subgroup G) :
(H ⊓ K).relindex H = K.relindex H
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theorem add_subgroup.relindex_inf_mul_relindex {G : Type u_1} [add_group G] (H K L : add_subgroup G) :
H.relindex (K ⊓ L) * K.relindex L = (H ⊓ K).relindex L
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theorem subgroup.relindex_inf_mul_relindex {G : Type u_1} [group G] (H K L : subgroup G) :
H.relindex (K ⊓ L) * K.relindex L = (H ⊓ K).relindex L
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theorem add_subgroup.inf_relindex_eq_relindex_sup {G : Type u_1} [add_group G] (H K : add_subgroup G) [K.normal] :
(H ⊓ K).relindex H = K.relindex (H ⊔ K)
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theorem subgroup.inf_relindex_eq_relindex_sup {G : Type u_1} [group G] (H K : subgroup G) [K.normal] :
(H ⊓ K).relindex H = K.relindex (H ⊔ K)
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theorem add_subgroup.relindex_eq_relindex_sup {G : Type u_1} [add_group G] (H K : add_subgroup G) [K.normal] :
K.relindex H = K.relindex (H ⊔ K)
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theorem subgroup.relindex_eq_relindex_sup {G : Type u_1} [group G] (H K : subgroup G) [K.normal] :
K.relindex H = K.relindex (H ⊔ K)
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theorem add_subgroup.relindex_dvd_index_of_normal {G : Type u_1} [add_group G] (H K : add_subgroup G) [H.normal] :
H.relindex K ∣ H.index
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theorem subgroup.relindex_dvd_index_of_normal {G : Type u_1} [group G] (H K : subgroup G) [H.normal] :
H.relindex K ∣ H.index
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theorem add_subgroup.relindex_dvd_of_le_left {G : Type u_1} [add_group G] {H K : add_subgroup G} (L : add_subgroup G) (hHK : H ≤ K) :
K.relindex L ∣ H.relindex L
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theorem subgroup.relindex_dvd_of_le_left {G : Type u_1} [group G] {H K : subgroup G} (L : subgroup G) (hHK : H ≤ K) :
K.relindex L ∣ H.relindex L
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@[simp]
theorem add_subgroup.index_top {G : Type u_1} [add_group G] :
⊤.index = 1
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@[simp]
theorem subgroup.index_top {G : Type u_1} [group G] :
⊤.index = 1
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@[simp]
theorem subgroup.index_bot {G : Type u_1} [group G] :
⊥.index = nat.card G
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@[simp]
theorem add_subgroup.index_bot {G : Type u_1} [add_group G] :
⊥.index = nat.card G
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theorem add_subgroup.index_bot_eq_card {G : Type u_1} [add_group G] [fintype G] :
⊥.index = fintype.card G
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theorem subgroup.index_bot_eq_card {G : Type u_1} [group G] [fintype G] :
⊥.index = fintype.card G
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@[simp]
theorem add_subgroup.relindex_top_left {G : Type u_1} [add_group G] (H : add_subgroup G) :
⊤.relindex H = 1
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@[simp]
theorem subgroup.relindex_top_left {G : Type u_1} [group G] (H : subgroup G) :
⊤.relindex H = 1
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@[simp]
theorem subgroup.relindex_top_right {G : Type u_1} [group G] (H : subgroup G) :
H.relindex ⊤ = H.index
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@[simp]
theorem add_subgroup.relindex_top_right {G : Type u_1} [add_group G] (H : add_subgroup G) :
H.relindex ⊤ = H.index
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@[simp]
theorem add_subgroup.relindex_bot_left {G : Type u_1} [add_group G] (H : add_subgroup G) :
⊥.relindex H = nat.card ↥H
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@[simp]
theorem subgroup.relindex_bot_left {G : Type u_1} [group G] (H : subgroup G) :
⊥.relindex H = nat.card ↥H
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theorem subgroup.relindex_bot_left_eq_card {G : Type u_1} [group G] (H : subgroup G) [fintype ↥H] :
⊥.relindex H = fintype.card ↥H
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theorem add_subgroup.relindex_bot_left_eq_card {G : Type u_1} [add_group G] (H : add_subgroup G) [fintype ↥H] :
⊥.relindex H = fintype.card ↥H
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@[simp]
theorem subgroup.relindex_bot_right {G : Type u_1} [group G] (H : subgroup G) :
H.relindex ⊥ = 1
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@[simp]
theorem add_subgroup.relindex_bot_right {G : Type u_1} [add_group G] (H : add_subgroup G) :
H.relindex ⊥ = 1
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@[simp]
theorem add_subgroup.relindex_self {G : Type u_1} [add_group G] (H : add_subgroup G) :
H.relindex H = 1
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@[simp]
theorem subgroup.relindex_self {G : Type u_1} [group G] (H : subgroup G) :
H.relindex H = 1
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@[simp]
theorem add_subgroup.card_mul_index {G : Type u_1} [add_group G] (H : add_subgroup G) :
nat.card ↥H * H.index = nat.card G
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@[simp]
theorem subgroup.card_mul_index {G : Type u_1} [group G] (H : subgroup G) :
nat.card ↥H * H.index = nat.card G
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theorem add_subgroup.index_map {G : Type u_1} [add_group G] (H : add_subgroup G) {G' : Type u_2} [add_group G'] (f : G →+ G') :
(add_subgroup.map f H).index = (H ⊔ f.ker).index * f.range.index
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theorem subgroup.index_map {G : Type u_1} [group G] (H : subgroup G) {G' : Type u_2} [group G'] (f : G →* G') :
(subgroup.map f H).index = (H ⊔ f.ker).index * f.range.index
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theorem add_subgroup.index_map_dvd {G : Type u_1} [add_group G] (H : add_subgroup G) {G' : Type u_2} [add_group G'] {f : G →+ G'} (hf : function.surjective ⇑f) :
(add_subgroup.map f H).index ∣ H.index
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theorem subgroup.index_map_dvd {G : Type u_1} [group G] (H : subgroup G) {G' : Type u_2} [group G'] {f : G →* G'} (hf : function.surjective ⇑f) :
(subgroup.map f H).index ∣ H.index
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theorem add_subgroup.dvd_index_map {G : Type u_1} [add_group G] (H : add_subgroup G) {G' : Type u_2} [add_group G'] {f : G →+ G'} (hf : f.ker ≤ H) :
H.index ∣ (add_subgroup.map f H).index
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theorem subgroup.dvd_index_map {G : Type u_1} [group G] (H : subgroup G) {G' : Type u_2} [group G'] {f : G →* G'} (hf : f.ker ≤ H) :
H.index ∣ (subgroup.map f H).index
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theorem add_subgroup.index_map_eq {G : Type u_1} [add_group G] (H : add_subgroup G) {G' : Type u_2} [add_group G'] {f : G →+ G'} (hf1 : function.surjective ⇑f) (hf2 : f.ker ≤ H) :
(add_subgroup.map f H).index = H.index
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theorem subgroup.index_map_eq {G : Type u_1} [group G] (H : subgroup G) {G' : Type u_2} [group G'] {f : G →* G'} (hf1 : function.surjective ⇑f) (hf2 : f.ker ≤ H) :
(subgroup.map f H).index = H.index
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theorem subgroup.index_eq_card {G : Type u_1} [group G] (H : subgroup G) [fintype (G ⧸ H)] :
H.index = fintype.card (G ⧸ H)
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theorem add_subgroup.index_eq_card {G : Type u_1} [add_group G] (H : add_subgroup G) [fintype (G ⧸ H)] :
H.index = fintype.card (G ⧸ H)
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theorem subgroup.index_mul_card {G : Type u_1} [group G] (H : subgroup G) [fintype G] [hH : fintype ↥H] :
H.index * fintype.card ↥H = fintype.card G
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theorem add_subgroup.index_mul_card {G : Type u_1} [add_group G] (H : add_subgroup G) [fintype G] [hH : fintype ↥H] :
H.index * fintype.card ↥H = fintype.card G
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theorem subgroup.index_dvd_card {G : Type u_1} [group G] (H : subgroup G) [fintype G] :
H.index ∣ fintype.card G
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theorem add_subgroup.index_dvd_card {G : Type u_1} [add_group G] (H : add_subgroup G) [fintype G] :
H.index ∣ fintype.card G
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theorem add_subgroup.relindex_eq_zero_of_le_left {G : Type u_1} [add_group G] {H K L : add_subgroup G} (hHK : H ≤ K) (hKL : K.relindex L = 0) :
H.relindex L = 0
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theorem subgroup.relindex_eq_zero_of_le_left {G : Type u_1} [group G] {H K L : subgroup G} (hHK : H ≤ K) (hKL : K.relindex L = 0) :
H.relindex L = 0
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theorem subgroup.relindex_eq_zero_of_le_right {G : Type u_1} [group G] {H K L : subgroup G} (hKL : K ≤ L) (hHK : H.relindex K = 0) :
H.relindex L = 0
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theorem add_subgroup.relindex_eq_zero_of_le_right {G : Type u_1} [add_group G] {H K L : add_subgroup G} (hKL : K ≤ L) (hHK : H.relindex K = 0) :
H.relindex L = 0
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theorem subgroup.relindex_le_of_le_left {G : Type u_1} [group G] {H K L : subgroup G} (hHK : H ≤ K) (hHL : H.relindex L ≠ 0) :
K.relindex L ≤ H.relindex L
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theorem add_subgroup.relindex_le_of_le_left {G : Type u_1} [add_group G] {H K L : add_subgroup G} (hHK : H ≤ K) (hHL : H.relindex L ≠ 0) :
K.relindex L ≤ H.relindex L
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theorem add_subgroup.relindex_le_of_le_right {G : Type u_1} [add_group G] {H K L : add_subgroup G} (hKL : K ≤ L) (hHL : H.relindex L ≠ 0) :
H.relindex K ≤ H.relindex L
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theorem subgroup.relindex_le_of_le_right {G : Type u_1} [group G] {H K L : subgroup G} (hKL : K ≤ L) (hHL : H.relindex L ≠ 0) :
H.relindex K ≤ H.relindex L
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theorem add_subgroup.relindex_ne_zero_trans {G : Type u_1} [add_group G] {H K L : add_subgroup G} (hHK : H.relindex K ≠ 0) (hKL : K.relindex L ≠ 0) :
H.relindex L ≠ 0
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theorem subgroup.relindex_ne_zero_trans {G : Type u_1} [group G] {H K L : subgroup G} (hHK : H.relindex K ≠ 0) (hKL : K.relindex L ≠ 0) :
H.relindex L ≠ 0
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theorem subgroup.relindex_inf_ne_zero {G : Type u_1} [group G] {H K L : subgroup G} (hH : H.relindex L ≠ 0) (hK : K.relindex L ≠ 0) :
(H ⊓ K).relindex L ≠ 0
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theorem add_subgroup.relindex_inf_ne_zero {G : Type u_1} [add_group G] {H K L : add_subgroup G} (hH : H.relindex L ≠ 0) (hK : K.relindex L ≠ 0) :
(H ⊓ K).relindex L ≠ 0
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theorem subgroup.index_inf_ne_zero {G : Type u_1} [group G] {H K : subgroup G} (hH : H.index ≠ 0) (hK : K.index ≠ 0) :
(H ⊓ K).index ≠ 0
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theorem add_subgroup.index_inf_ne_zero {G : Type u_1} [add_group G] {H K : add_subgroup G} (hH : H.index ≠ 0) (hK : K.index ≠ 0) :
(H ⊓ K).index ≠ 0
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theorem subgroup.relindex_inf_le {G : Type u_1} [group G] {H K L : subgroup G} :
(H ⊓ K).relindex L ≤ H.relindex L * K.relindex L
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theorem add_subgroup.relindex_inf_le {G : Type u_1} [add_group G] {H K L : add_subgroup G} :
(H ⊓ K).relindex L ≤ H.relindex L * K.relindex L
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theorem add_subgroup.index_inf_le {G : Type u_1} [add_group G] {H K : add_subgroup G} :
(H ⊓ K).index ≤ H.index * K.index
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theorem subgroup.index_inf_le {G : Type u_1} [group G] {H K : subgroup G} :
(H ⊓ K).index ≤ H.index * K.index
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@[simp]
theorem add_subgroup.index_eq_one {G : Type u_1} [add_group G] {H : add_subgroup G} :
H.index = 1 ↔ H = ⊤
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@[simp]
theorem subgroup.index_eq_one {G : Type u_1} [group G] {H : subgroup G} :
H.index = 1 ↔ H = ⊤
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theorem subgroup.index_ne_zero_of_fintype {G : Type u_1} [group G] {H : subgroup G} [hH : fintype (G ⧸ H)] :
H.index ≠ 0
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theorem add_subgroup.index_ne_zero_of_fintype {G : Type u_1} [add_group G] {H : add_subgroup G} [hH : fintype (G ⧸ H)] :
H.index ≠ 0
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noncomputable def subgroup.fintype_of_index_ne_zero {G : Type u_1} [group G] {H : subgroup G} (hH : H.index ≠ 0) :
fintype (G ⧸ H)

Finite index implies finite quotient.

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noncomputable def add_subgroup.fintype_of_index_ne_zero {G : Type u_1} [add_group G] {H : add_subgroup G} (hH : H.index ≠ 0) :
fintype (G ⧸ H)

Finite index implies finite quotient.

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theorem add_subgroup.one_lt_index_of_ne_top {G : Type u_1} [add_group G] {H : add_subgroup G} [fintype (G ⧸ H)] (hH : H ≠ ⊤) :
1 < H.index
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theorem subgroup.one_lt_index_of_ne_top {G : Type u_1} [group G] {H : subgroup G} [fintype (G ⧸ H)] (hH : H ≠ ⊤) :
1 < H.index