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

• H.index : the index of H : Subgroup G as a natural number, and returns 0 if the index is infinite.
• H.relindex K : the relative index of H : Subgroup G in K : Subgroup G as a natural number, and returns 0 if the relative index is infinite.

Main results

• card_mul_index : Nat.card H * H.index = Nat.card G
• index_mul_card : H.index * Fintype.card H = Fintype.card G
• index_dvd_card : H.index ∣ Fintype.card G
• relindex_mul_index : If H ≤ K, then H.relindex K * K.index = H.index
• index_dvd_of_le : If H ≤ K, then K.index ∣ H.index
• relindex_mul_relindex : relindex is multiplicative in towers
• MulAction.index_stabilizer: the index of the stabilizer is the cardinality of the orbit

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

The index of a subgroup as a natural number. Returns 0 if the index is infinite.

Equations

• H.index = Nat.card (G ⧸ H)

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

The index of an additive subgroup as a natural number. Returns 0 if the index is infinite.

Equations

• H.index = Nat.card (G ⧸ H)

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

If H and K are subgroups of a group G, then relindex H K : ℕ is the index of H ∩ K in K. The function returns 0 if the index is infinite.

Equations

• H.relindex K = (H.subgroupOf K).index

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

If H and K are subgroups of an additive group G, then relindex H K : ℕ is the index of H ∩ K in K. The function returns 0 if the index is infinite.

Equations

• H.relindex K = (H.addSubgroupOf K).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) : (comap f H).index = H.index

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) : (comap f H).index = H.index

theorem Subgroup.index_comap {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (H : Subgroup G) (f : G' →* G) : (comap f H).index = H.relindex f.range

theorem AddSubgroup.index_comap {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) (f : G' →+ G) : (comap f H).index = H.relindex f.range

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') : (comap f H).relindex K = H.relindex (map f K)

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') : (comap f H).relindex K = H.relindex (map f K)

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

theorem AddSubgroup.relindex_mul_index {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} (h : H ≤ K) : H.relindex K * K.index = H.index

theorem Subgroup.index_dvd_of_le {G : Type u_1} [Group G] {H K : Subgroup G} (h : H ≤ K) : K.index ∣ H.index

theorem AddSubgroup.index_dvd_of_le {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} (h : H ≤ K) : K.index ∣ H.index

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

theorem AddSubgroup.relindex_dvd_index_of_le {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} (h : H ≤ K) : H.relindex K ∣ H.index

theorem Subgroup.relindex_subgroupOf {G : Type u_1} [Group G] {H K L : Subgroup G} (hKL : K ≤ L) : (H.subgroupOf L).relindex (K.subgroupOf L) = H.relindex K

theorem AddSubgroup.relindex_addSubgroupOf {G : Type u_1} [AddGroup G] {H K L : AddSubgroup G} (hKL : K ≤ L) : (H.addSubgroupOf L).relindex (K.addSubgroupOf L) = H.relindex K

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

theorem AddSubgroup.relindex_mul_relindex {G : Type u_1} [AddGroup G] (H K L : AddSubgroup G) (hHK : H ≤ K) (hKL : K ≤ L) : H.relindex K * K.relindex L = H.relindex L

theorem Subgroup.inf_relindex_right {G : Type u_1} [Group G] (H K : Subgroup G) : (H ⊓ K).relindex K = H.relindex K

theorem AddSubgroup.inf_relindex_right {G : Type u_1} [AddGroup G] (H K : AddSubgroup G) : (H ⊓ K).relindex K = H.relindex K

theorem Subgroup.inf_relindex_left {G : Type u_1} [Group G] (H K : Subgroup G) : (H ⊓ K).relindex H = K.relindex H

theorem AddSubgroup.inf_relindex_left {G : Type u_1} [AddGroup G] (H K : AddSubgroup G) : (H ⊓ K).relindex H = K.relindex H

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

theorem AddSubgroup.relindex_inf_mul_relindex {G : Type u_1} [AddGroup G] (H K L : AddSubgroup G) : H.relindex (K ⊓ L) * K.relindex L = (H ⊓ K).relindex L

@[simp]

theorem Subgroup.relindex_sup_right {G : Type u_1} [Group G] (H K : Subgroup G) [K.Normal] : K.relindex (H ⊔ K) = K.relindex H

@[simp]

theorem AddSubgroup.relindex_sup_right {G : Type u_1} [AddGroup G] (H K : AddSubgroup G) [K.Normal] : K.relindex (H ⊔ K) = K.relindex H

@[simp]

theorem Subgroup.relindex_sup_left {G : Type u_1} [Group G] (H K : Subgroup G) [K.Normal] : K.relindex (K ⊔ H) = K.relindex H

@[simp]

theorem AddSubgroup.relindex_sup_left {G : Type u_1} [AddGroup G] (H K : AddSubgroup G) [K.Normal] : K.relindex (K ⊔ H) = K.relindex H

theorem Subgroup.relindex_dvd_index_of_normal {G : Type u_1} [Group G] (H K : Subgroup G) [H.Normal] : H.relindex K ∣ H.index

theorem AddSubgroup.relindex_dvd_index_of_normal {G : Type u_1} [AddGroup G] (H K : AddSubgroup G) [H.Normal] : H.relindex K ∣ H.index

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

theorem AddSubgroup.relindex_dvd_of_le_left {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} (L : AddSubgroup G) (hHK : H ≤ K) : K.relindex L ∣ H.relindex L

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.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.mul_mem_iff_of_index_two {G : Type u_1} [Group G] {H : Subgroup G} (h : H.index = 2) {a b : G} : a * b ∈ H ↔ (a ∈ H ↔ b ∈ H)

theorem AddSubgroup.add_mem_iff_of_index_two {G : Type u_1} [AddGroup G] {H : AddSubgroup G} (h : H.index = 2) {a b : G} : a + b ∈ H ↔ (a ∈ H ↔ b ∈ 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.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.sq_mem_of_index_two {G : Type u_1} [Group G] {H : Subgroup G} (h : H.index = 2) (a : G) : a ^ 2 ∈ 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

@[simp]

theorem Subgroup.index_top {G : Type u_1} [Group G] : ⊤.index = 1

@[simp]

theorem AddSubgroup.index_top {G : Type u_1} [AddGroup G] : ⊤.index = 1

@[simp]

theorem Subgroup.index_bot {G : Type u_1} [Group G] : ⊥.index = Nat.card G

@[simp]

theorem AddSubgroup.index_bot {G : Type u_1} [AddGroup G] : ⊥.index = Nat.card G

@[simp]

theorem Subgroup.relindex_top_left {G : Type u_1} [Group G] (H : Subgroup G) : ⊤.relindex H = 1

@[simp]

theorem AddSubgroup.relindex_top_left {G : Type u_1} [AddGroup G] (H : AddSubgroup G) : ⊤.relindex H = 1

@[simp]

theorem Subgroup.relindex_top_right {G : Type u_1} [Group G] (H : Subgroup G) : H.relindex ⊤ = H.index

@[simp]

theorem AddSubgroup.relindex_top_right {G : Type u_1} [AddGroup G] (H : AddSubgroup G) : H.relindex ⊤ = H.index

@[simp]

theorem Subgroup.relindex_bot_left {G : Type u_1} [Group G] (H : Subgroup G) : ⊥.relindex H = Nat.card ↥H

@[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_right {G : Type u_1} [Group G] (H : Subgroup G) : H.relindex ⊥ = 1

@[simp]

theorem AddSubgroup.relindex_bot_right {G : Type u_1} [AddGroup G] (H : AddSubgroup G) : H.relindex ⊥ = 1

@[simp]

theorem Subgroup.relindex_self {G : Type u_1} [Group G] (H : Subgroup G) : H.relindex H = 1

@[simp]

theorem AddSubgroup.relindex_self {G : Type u_1} [AddGroup G] (H : AddSubgroup G) : H.relindex H = 1

theorem Subgroup.index_ker {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (f : G →* G') : f.ker.index = Nat.card ↥f.range

theorem AddSubgroup.index_ker {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (f : G →+ G') : f.ker.index = Nat.card ↥f.range

theorem Subgroup.relindex_ker {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (K : Subgroup G) (f : G →* G') : f.ker.relindex K = Nat.card ↥(map f K)

theorem AddSubgroup.relindex_ker {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (K : AddSubgroup G) (f : G →+ G') : f.ker.relindex K = Nat.card ↥(map f K)

@[simp]

theorem Subgroup.card_mul_index {G : Type u_1} [Group G] (H : Subgroup G) : Nat.card ↥H * H.index = Nat.card G

@[simp]

theorem AddSubgroup.card_mul_index {G : Type u_1} [AddGroup G] (H : AddSubgroup G) : Nat.card ↥H * H.index = Nat.card G

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) : Nat.card G' ∣ Nat.card G

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) : Nat.card G' ∣ Nat.card G

theorem Subgroup.card_range_dvd {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (f : G →* G') : Nat.card ↥f.range ∣ Nat.card G

theorem AddSubgroup.card_range_dvd {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (f : G →+ G') : Nat.card ↥f.range ∣ Nat.card G

theorem Subgroup.card_map_dvd {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (H : Subgroup G) (f : G →* G') : Nat.card ↥(map f H) ∣ Nat.card ↥H

theorem AddSubgroup.card_map_dvd {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) (f : G →+ G') : Nat.card ↥(map f H) ∣ Nat.card ↥H

theorem Subgroup.index_map {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (H : Subgroup G) (f : G →* G') : (map f H).index = (H ⊔ f.ker).index * f.range.index

theorem AddSubgroup.index_map {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) (f : G →+ G') : (map f H).index = (H ⊔ f.ker).index * f.range.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) : (map f H).index ∣ H.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) : (map f H).index ∣ 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 ∣ (map f 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 ∣ (map f 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) : (map f H).index = 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) : (map f H).index = H.index

theorem Subgroup.index_map_of_injective {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (H : Subgroup G) {f : G →* G'} (hf : Function.Injective ⇑f) : (map f H).index = H.index * f.range.index

theorem AddSubgroup.index_map_of_injective {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) {f : G →+ G'} (hf : Function.Injective ⇑f) : (map f H).index = H.index * f.range.index

theorem Subgroup.index_map_subtype {G : Type u_1} [Group G] {H : Subgroup G} (K : Subgroup ↥H) : (map H.subtype K).index = K.index * H.index

theorem AddSubgroup.index_map_subtype {G : Type u_1} [AddGroup G] {H : AddSubgroup G} (K : AddSubgroup ↥H) : (map H.subtype K).index = K.index * H.index

theorem Subgroup.index_eq_card {G : Type u_1} [Group G] (H : Subgroup G) : H.index = Nat.card (G ⧸ H)

theorem AddSubgroup.index_eq_card {G : Type u_1} [AddGroup G] (H : AddSubgroup G) : H.index = Nat.card (G ⧸ H)

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_mul_card {G : Type u_1} [AddGroup G] (H : AddSubgroup G) : H.index * Nat.card ↥H = Nat.card G

theorem Subgroup.index_dvd_card {G : Type u_1} [Group G] (H : Subgroup G) : H.index ∣ Nat.card G

theorem AddSubgroup.index_dvd_card {G : Type u_1} [AddGroup G] (H : AddSubgroup G) : H.index ∣ Nat.card G

theorem Subgroup.relindex_dvd_card {G : Type u_1} [Group G] (H K : Subgroup G) : H.relindex K ∣ Nat.card ↥K

theorem AddSubgroup.relindex_dvd_card {G : Type u_1} [AddGroup G] (H K : AddSubgroup G) : H.relindex K ∣ Nat.card ↥K

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

theorem AddSubgroup.relindex_eq_zero_of_le_left {G : Type u_1} [AddGroup G] {H K L : AddSubgroup G} (hHK : H ≤ K) (hKL : K.relindex L = 0) : H.relindex L = 0

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

theorem AddSubgroup.relindex_eq_zero_of_le_right {G : Type u_1} [AddGroup G] {H K L : AddSubgroup G} (hKL : K ≤ L) (hHK : H.relindex K = 0) : H.relindex L = 0

theorem Subgroup.index_eq_zero_of_relindex_eq_zero {G : Type u_1} [Group G] {H K : Subgroup G} (h : H.relindex K = 0) : H.index = 0

theorem AddSubgroup.index_eq_zero_of_relindex_eq_zero {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} (h : H.relindex K = 0) : H.index = 0

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

theorem AddSubgroup.relindex_le_of_le_left {G : Type u_1} [AddGroup G] {H K L : AddSubgroup G} (hHK : H ≤ K) (hHL : H.relindex L ≠ 0) : K.relindex L ≤ H.relindex L

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

theorem AddSubgroup.relindex_le_of_le_right {G : Type u_1} [AddGroup G] {H K L : AddSubgroup G} (hKL : K ≤ L) (hHL : H.relindex L ≠ 0) : H.relindex K ≤ H.relindex L

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

theorem AddSubgroup.relindex_ne_zero_trans {G : Type u_1} [AddGroup G] {H K L : AddSubgroup G} (hHK : H.relindex K ≠ 0) (hKL : K.relindex L ≠ 0) : H.relindex L ≠ 0

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

theorem AddSubgroup.relindex_inf_ne_zero {G : Type u_1} [AddGroup G] {H K L : AddSubgroup G} (hH : H.relindex L ≠ 0) (hK : K.relindex L ≠ 0) : (H ⊓ K).relindex L ≠ 0

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

theorem AddSubgroup.index_inf_ne_zero {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} (hH : H.index ≠ 0) (hK : K.index ≠ 0) : (H ⊓ K).index ≠ 0

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

theorem AddSubgroup.relindex_inf_le {G : Type u_1} [AddGroup G] {H K L : AddSubgroup G} : (H ⊓ K).relindex L ≤ H.relindex L * K.relindex L

theorem Subgroup.index_inf_le {G : Type u_1} [Group G] {H K : Subgroup G} : (H ⊓ K).index ≤ H.index * K.index

theorem AddSubgroup.index_inf_le {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} : (H ⊓ K).index ≤ H.index * K.index

theorem Subgroup.relindex_iInf_ne_zero {G : Type u_1} [Group G] {L : Subgroup G} {ι : Type u_3} [_hι : Finite ι] {f : ι → Subgroup G} (hf : ∀ (i : ι), (f i).relindex L ≠ 0) : (⨅ (i : ι), f i).relindex L ≠ 0

theorem AddSubgroup.relindex_iInf_ne_zero {G : Type u_1} [AddGroup G] {L : AddSubgroup G} {ι : Type u_3} [_hι : Finite ι] {f : ι → AddSubgroup G} (hf : ∀ (i : ι), (f i).relindex L ≠ 0) : (⨅ (i : ι), f i).relindex L ≠ 0

theorem Subgroup.relindex_iInf_le {G : Type u_1} [Group G] {L : Subgroup G} {ι : Type u_3} [Fintype ι] (f : ι → Subgroup G) : (⨅ (i : ι), f i).relindex L ≤ ∏ i : ι, (f i).relindex L

theorem AddSubgroup.relindex_iInf_le {G : Type u_1} [AddGroup G] {L : AddSubgroup G} {ι : Type u_3} [Fintype ι] (f : ι → AddSubgroup G) : (⨅ (i : ι), f i).relindex L ≤ ∏ i : ι, (f i).relindex L

theorem Subgroup.index_iInf_ne_zero {G : Type u_1} [Group G] {ι : Type u_3} [Finite ι] {f : ι → Subgroup G} (hf : ∀ (i : ι), (f i).index ≠ 0) : (⨅ (i : ι), f i).index ≠ 0

theorem AddSubgroup.index_iInf_ne_zero {G : Type u_1} [AddGroup G] {ι : Type u_3} [Finite ι] {f : ι → AddSubgroup G} (hf : ∀ (i : ι), (f i).index ≠ 0) : (⨅ (i : ι), f i).index ≠ 0

theorem Subgroup.index_iInf_le {G : Type u_1} [Group G] {ι : Type u_3} [Fintype ι] (f : ι → Subgroup G) : (⨅ (i : ι), f i).index ≤ ∏ i : ι, (f i).index

theorem AddSubgroup.index_iInf_le {G : Type u_1} [AddGroup G] {ι : Type u_3} [Fintype ι] (f : ι → AddSubgroup G) : (⨅ (i : ι), f i).index ≤ ∏ i : ι, (f i).index

@[simp]

theorem Subgroup.index_eq_one {G : Type u_1} [Group G] {H : Subgroup G} : H.index = 1 ↔ H = ⊤

@[simp]

theorem AddSubgroup.index_eq_one {G : Type u_1} [AddGroup G] {H : AddSubgroup G} : H.index = 1 ↔ H = ⊤

@[simp]

theorem Subgroup.relindex_eq_one {G : Type u_1} [Group G] {H K : Subgroup G} : H.relindex K = 1 ↔ K ≤ H

@[simp]

theorem AddSubgroup.relindex_eq_one {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} : H.relindex K = 1 ↔ K ≤ H

@[simp]

theorem Subgroup.card_eq_one {G : Type u_1} [Group G] {H : Subgroup G} : Nat.card ↥H = 1 ↔ H = ⊥

@[simp]

theorem AddSubgroup.card_eq_one {G : Type u_1} [AddGroup G] {H : AddSubgroup G} : Nat.card ↥H = 1 ↔ H = ⊥

theorem Subgroup.inf_eq_bot_of_coprime {G : Type u_1} [Group G] {H K : Subgroup G} (h : (Nat.card ↥H).Coprime (Nat.card ↥K)) : H ⊓ K = ⊥

theorem AddSubgroup.inf_eq_bot_of_coprime {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} (h : (Nat.card ↥H).Coprime (Nat.card ↥K)) : H ⊓ K = ⊥

@[deprecated AddSubgroup.inf_eq_bot_of_coprime (since := "2024-12-18")]

theorem add_inf_eq_bot_of_coprime {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} (h : (Nat.card ↥H).Coprime (Nat.card ↥K)) : H ⊓ K = ⊥

Alias of AddSubgroup.inf_eq_bot_of_coprime.

theorem Subgroup.index_ne_zero_of_finite {G : Type u_1} [Group G] {H : Subgroup G} [hH : Finite (G ⧸ H)] : H.index ≠ 0

theorem AddSubgroup.index_ne_zero_of_finite {G : Type u_1} [AddGroup G] {H : AddSubgroup G} [hH : Finite (G ⧸ H)] : H.index ≠ 0

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

• Subgroup.fintypeOfIndexNeZero hH = Fintype.ofFinite (G ⧸ H)

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

• AddSubgroup.fintypeOfIndexNeZero hH = Fintype.ofFinite (G ⧸ H)

theorem Subgroup.index_eq_zero_iff_infinite {G : Type u_1} [Group G] {H : Subgroup G} : H.index = 0 ↔ Infinite (G ⧸ H)

theorem AddSubgroup.index_eq_zero_iff_infinite {G : Type u_1} [AddGroup G] {H : AddSubgroup G} : H.index = 0 ↔ Infinite (G ⧸ H)

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

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.finite_quotient_of_finite_quotient_of_index_ne_zero {G : Type u_1} [Group G] {H : Subgroup G} {X : Type u_3} [MulAction G X] [Finite (MulAction.orbitRel.Quotient G X)] (hi : H.index ≠ 0) : Finite (MulAction.orbitRel.Quotient (↥H) X)

theorem AddSubgroup.finite_quotient_of_finite_quotient_of_index_ne_zero {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {X : Type u_3} [AddAction G X] [Finite (AddAction.orbitRel.Quotient G X)] (hi : H.index ≠ 0) : Finite (AddAction.orbitRel.Quotient (↥H) X)

theorem Subgroup.finite_quotient_of_pretransitive_of_index_ne_zero {G : Type u_1} [Group G] {H : Subgroup G} {X : Type u_3} [MulAction G X] [MulAction.IsPretransitive G X] (hi : H.index ≠ 0) : Finite (MulAction.orbitRel.Quotient (↥H) X)

theorem AddSubgroup.finite_quotient_of_pretransitive_of_index_ne_zero {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {X : Type u_3} [AddAction G X] [AddAction.IsPretransitive G X] (hi : H.index ≠ 0) : Finite (AddAction.orbitRel.Quotient (↥H) X)

theorem Subgroup.exists_pow_mem_of_index_ne_zero {G : Type u_1} [Group G] {H : Subgroup G} (h : H.index ≠ 0) (a : G) : ∃ (n : ℕ), 0 < n ∧ n ≤ H.index ∧ a ^ n ∈ H

theorem AddSubgroup.exists_nsmul_mem_of_index_ne_zero {G : Type u_1} [AddGroup G] {H : AddSubgroup G} (h : H.index ≠ 0) (a : G) : ∃ (n : ℕ), 0 < n ∧ n ≤ H.index ∧ n • a ∈ H

theorem Subgroup.exists_pow_mem_of_relindex_ne_zero {G : Type u_1} [Group G] {H K : Subgroup G} (h : H.relindex K ≠ 0) {a : G} (ha : a ∈ K) : ∃ (n : ℕ), 0 < n ∧ n ≤ H.relindex K ∧ a ^ n ∈ H ⊓ K

theorem AddSubgroup.exists_nsmul_mem_of_relindex_ne_zero {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} (h : H.relindex K ≠ 0) {a : G} (ha : a ∈ K) : ∃ (n : ℕ), 0 < n ∧ n ≤ H.relindex K ∧ n • a ∈ H ⊓ K

theorem Subgroup.pow_mem_of_index_ne_zero_of_dvd {G : Type u_1} [Group G] {H : Subgroup G} (h : H.index ≠ 0) (a : G) {n : ℕ} (hn : ∀ (m : ℕ), 0 < m → m ≤ H.index → m ∣ n) : a ^ n ∈ H

theorem AddSubgroup.nsmul_mem_of_index_ne_zero_of_dvd {G : Type u_1} [AddGroup G] {H : AddSubgroup G} (h : H.index ≠ 0) (a : G) {n : ℕ} (hn : ∀ (m : ℕ), 0 < m → m ≤ H.index → m ∣ n) : n • a ∈ H

theorem Subgroup.pow_mem_of_relindex_ne_zero_of_dvd {G : Type u_1} [Group G] {H K : Subgroup G} (h : H.relindex K ≠ 0) {a : G} (ha : a ∈ K) {n : ℕ} (hn : ∀ (m : ℕ), 0 < m → m ≤ H.relindex K → m ∣ n) : a ^ n ∈ H ⊓ K

theorem AddSubgroup.nsmul_mem_of_relindex_ne_zero_of_dvd {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} (h : H.relindex K ≠ 0) {a : G} (ha : a ∈ K) {n : ℕ} (hn : ∀ (m : ℕ), 0 < m → m ≤ H.relindex K → m ∣ n) : n • a ∈ H ⊓ K

@[simp]

theorem Subgroup.index_prod {G : Type u_1} {G' : Type u_2} [Group G] [Group G'] (H : Subgroup G) (K : Subgroup G') : (H.prod K).index = H.index * K.index

@[simp]

theorem AddSubgroup.index_prod {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) (K : AddSubgroup G') : (H.prod K).index = H.index * K.index

@[deprecated AddSubgroup.index_prod (since := "2025-03-11")]

theorem AddSubgroup.index_sum {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) (K : AddSubgroup G') : (H.prod K).index = H.index * K.index

Alias of AddSubgroup.index_prod.

@[simp]

theorem Subgroup.index_pi {G : Type u_1} [Group G] {ι : Type u_3} [Fintype ι] (H : ι → Subgroup G) : (pi Set.univ H).index = ∏ i : ι, (H i).index

@[simp]

theorem AddSubgroup.index_pi {G : Type u_1} [AddGroup G] {ι : Type u_3} [Fintype ι] (H : ι → AddSubgroup G) : (pi Set.univ H).index = ∏ i : ι, (H i).index

@[simp]

theorem Subgroup.index_toAddSubgroup {G : Type u_1} [Group G] {H : Subgroup G} : (toAddSubgroup H).index = H.index

@[simp]

theorem AddSubgroup.index_toSubgroup {G : Type u_3} [AddGroup G] (H : AddSubgroup G) : (toSubgroup H).index = H.index

@[simp]

theorem Subgroup.relindex_toAddSubgroup {G : Type u_1} [Group G] {H K : Subgroup G} : (toAddSubgroup H).relindex (toAddSubgroup K) = H.relindex K

@[simp]

theorem AddSubgroup.relindex_toSubgroup {G : Type u_3} [AddGroup G] (H K : AddSubgroup G) : (toSubgroup H).relindex (toSubgroup K) = H.relindex K

class Subgroup.FiniteIndex {G : Type u_1} [Group G] (H : Subgroup G) : Prop

Typeclass for finite index subgroups.

• finiteIndex : H.index ≠ 0

  The subgroup has finite index

Instances

• Subgroup.finiteIndex_ker
• Subgroup.finiteIndex_normalCore
• Subgroup.finiteIndex_of_finite
• Subgroup.instFiniteIndexMin
• Subgroup.instFiniteIndexTop
• Subgroup.instFiniteIndex_subgroupOf

class AddSubgroup.FiniteIndex {G : Type u_3} [AddGroup G] (H : AddSubgroup G) : Prop

Typeclass for finite index subgroups.

• finiteIndex : H.index ≠ 0

  The additive subgroup has finite index

Instances

• AddSubgroup.finiteIndex_ker
• AddSubgroup.finiteIndex_of_finite
• AddSubgroup.instFiniteIndexMin
• AddSubgroup.instFiniteIndexTop
• AddSubgroup.instFiniteIndex_addSubgroupOf

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

• H.fintypeQuotientOfFiniteIndex = Subgroup.fintypeOfIndexNeZero ⋯

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

• H.fintypeQuotientOfFiniteIndex = AddSubgroup.fintypeOfIndexNeZero ⋯

instance Subgroup.finite_quotient_of_finiteIndex {G : Type u_1} [Group G] (H : Subgroup G) [H.FiniteIndex] : Finite (G ⧸ H)

instance AddSubgroup.finite_quotient_of_finiteIndex {G : Type u_1} [AddGroup G] (H : AddSubgroup G) [H.FiniteIndex] : Finite (G ⧸ H)

theorem Subgroup.finiteIndex_of_finite_quotient {G : Type u_1} [Group G] (H : Subgroup G) [Finite (G ⧸ H)] : H.FiniteIndex

theorem AddSubgroup.finiteIndex_of_finite_quotient {G : Type u_1} [AddGroup G] (H : AddSubgroup G) [Finite (G ⧸ H)] : H.FiniteIndex

@[instance 100]

instance Subgroup.finiteIndex_of_finite {G : Type u_1} [Group G] (H : Subgroup G) [Finite G] : H.FiniteIndex

@[instance 100]

instance AddSubgroup.finiteIndex_of_finite {G : Type u_1} [AddGroup G] (H : AddSubgroup G) [Finite G] : H.FiniteIndex

instance Subgroup.instFiniteIndexTop {G : Type u_1} [Group G] : ⊤.FiniteIndex

instance AddSubgroup.instFiniteIndexTop {G : Type u_1} [AddGroup G] : ⊤.FiniteIndex

instance Subgroup.instFiniteIndexMin {G : Type u_1} [Group G] (H K : Subgroup G) [H.FiniteIndex] [K.FiniteIndex] : (H ⊓ K).FiniteIndex

instance AddSubgroup.instFiniteIndexMin {G : Type u_1} [AddGroup G] (H K : AddSubgroup G) [H.FiniteIndex] [K.FiniteIndex] : (H ⊓ K).FiniteIndex

theorem Subgroup.finiteIndex_iInf {G : Type u_1} [Group G] {ι : Type u_3} [Finite ι] {f : ι → Subgroup 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} [Finite ι] {f : ι → AddSubgroup 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} {s : Finset ι} (f : ι → Subgroup G) (hs : ∀ i ∈ s, (f i).FiniteIndex) : (⨅ i ∈ s, f i).FiniteIndex

theorem AddSubgroup.finiteIndex_iInf' {G : Type u_1} [AddGroup G] {ι : Type u_3} {s : Finset ι} (f : ι → AddSubgroup G) (hs : ∀ i ∈ s, (f i).FiniteIndex) : (⨅ i ∈ s, f i).FiniteIndex

instance Subgroup.instFiniteIndex_subgroupOf {G : Type u_1} [Group G] (H K : Subgroup G) [H.FiniteIndex] : (H.subgroupOf K).FiniteIndex

instance AddSubgroup.instFiniteIndex_addSubgroupOf {G : Type u_1} [AddGroup G] (H K : AddSubgroup G) [H.FiniteIndex] : (H.addSubgroupOf K).FiniteIndex

theorem Subgroup.finiteIndex_of_le {G : Type u_1} [Group G] {H K : Subgroup G} [H.FiniteIndex] (h : H ≤ K) : K.FiniteIndex

theorem AddSubgroup.finiteIndex_of_le {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} [H.FiniteIndex] (h : H ≤ K) : K.FiniteIndex

theorem Subgroup.index_antitone {G : Type u_1} [Group G] {H K : Subgroup G} (h : H ≤ K) [H.FiniteIndex] : K.index ≤ H.index

theorem AddSubgroup.index_antitone {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} (h : H ≤ K) [H.FiniteIndex] : K.index ≤ H.index

theorem Subgroup.index_strictAnti {G : Type u_1} [Group G] {H K : Subgroup G} (h : H < K) [H.FiniteIndex] : K.index < H.index

theorem AddSubgroup.index_strictAnti {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} (h : H < K) [H.FiniteIndex] : K.index < H.index

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

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

instance Subgroup.finiteIndex_normalCore {G : Type u_1} [Group G] (H : Subgroup G) [H.FiniteIndex] : H.normalCore.FiniteIndex

theorem MulAction.index_stabilizer (G : Type u_1) {X : Type u_2} [Group G] [MulAction G X] (x : X) : (stabilizer G x).index = (orbit G x).ncard

theorem AddAction.index_stabilizer (G : Type u_1) {X : Type u_2} [AddGroup G] [AddAction G X] (x : X) : (stabilizer G x).index = (orbit G x).ncard

theorem MulAction.index_stabilizer_of_transitive (G : Type u_1) {X : Type u_2} [Group G] [MulAction G X] (x : X) [IsPretransitive G X] : (stabilizer G x).index = Nat.card X

theorem AddAction.index_stabilizer_of_transitive (G : Type u_1) {X : Type u_2} [AddGroup G] [AddAction G X] (x : X) [IsPretransitive G X] : (stabilizer G x).index = Nat.card X

theorem MonoidHom.surjective_of_card_ker_le_div {G : Type u_1} {M : Type u_2} [Group G] [Group M] [Finite G] [Finite M] (f : G →* M) (h : Nat.card ↥f.ker ≤ Nat.card G / Nat.card M) : Function.Surjective ⇑f

theorem AddMonoidHom.surjective_of_card_ker_le_div {G : Type u_1} {M : Type u_2} [AddGroup G] [AddGroup M] [Finite G] [Finite M] (f : G →+ M) (h : Nat.card ↥f.ker ≤ Nat.card G / Nat.card M) : Function.Surjective ⇑f

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 y : M} (hx : x ∈ Set.range ⇑f) (hy : y ∈ Set.range ⇑f) : {g : G | f g = x}.card = {g : G | f g = y}.card

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 y : M} (hx : x ∈ Set.range ⇑f) (hy : y ∈ Set.range ⇑f) : {g : G | f g = x}.card = {g : G | f g = y}.card