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

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} [] (H : ) :

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} [] (H : ) (K : ) :

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} [] (H : ) (K : ) :

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} [] (H : ) {G' : Type u_2} [AddGroup G'] {f : G' →+ G} (hf : ) :
().index = H.index
theorem Subgroup.index_comap_of_surjective {G : Type u_1} [] (H : ) {G' : Type u_2} [Group G'] {f : G' →* G} (hf : ) :
().index = H.index
theorem AddSubgroup.index_comap {G : Type u_1} [] (H : ) {G' : Type u_2} [AddGroup G'] (f : G' →+ G) :
().index = H.relindex f.range
theorem Subgroup.index_comap {G : Type u_1} [] (H : ) {G' : Type u_2} [Group G'] (f : G' →* G) :
().index = H.relindex f.range
theorem AddSubgroup.relindex_comap {G : Type u_1} [] (H : ) {G' : Type u_2} [AddGroup G'] (f : G' →+ G) (K : ) :
().relindex K = H.relindex ()
theorem Subgroup.relindex_comap {G : Type u_1} [] (H : ) {G' : Type u_2} [Group G'] (f : G' →* G) (K : Subgroup G') :
().relindex K = H.relindex ()
theorem AddSubgroup.relindex_mul_index {G : Type u_1} [] {H : } {K : } (h : H K) :
H.relindex K * K.index = H.index
theorem Subgroup.relindex_mul_index {G : Type u_1} [] {H : } {K : } (h : H K) :
H.relindex K * K.index = H.index
theorem AddSubgroup.index_dvd_of_le {G : Type u_1} [] {H : } {K : } (h : H K) :
K.index H.index
theorem Subgroup.index_dvd_of_le {G : Type u_1} [] {H : } {K : } (h : H K) :
K.index H.index
theorem AddSubgroup.relindex_dvd_index_of_le {G : Type u_1} [] {H : } {K : } (h : H K) :
H.relindex K H.index
theorem Subgroup.relindex_dvd_index_of_le {G : Type u_1} [] {H : } {K : } (h : H K) :
H.relindex K H.index
theorem AddSubgroup.relindex_addSubgroupOf {G : Type u_1} [] {H : } {K : } {L : } (hKL : K L) :
(H.addSubgroupOf L).relindex (K.addSubgroupOf L) = H.relindex K
theorem Subgroup.relindex_subgroupOf {G : Type u_1} [] {H : } {K : } {L : } (hKL : K L) :
(H.subgroupOf L).relindex (K.subgroupOf L) = H.relindex K
theorem AddSubgroup.relindex_mul_relindex {G : Type u_1} [] (H : ) (K : ) (L : ) (hHK : H K) (hKL : K L) :
H.relindex K * K.relindex L = H.relindex L
theorem Subgroup.relindex_mul_relindex {G : Type u_1} [] (H : ) (K : ) (L : ) (hHK : H K) (hKL : K L) :
H.relindex K * K.relindex L = H.relindex L
theorem AddSubgroup.inf_relindex_right {G : Type u_1} [] (H : ) (K : ) :
(H K).relindex K = H.relindex K
theorem Subgroup.inf_relindex_right {G : Type u_1} [] (H : ) (K : ) :
(H K).relindex K = H.relindex K
theorem AddSubgroup.inf_relindex_left {G : Type u_1} [] (H : ) (K : ) :
(H K).relindex H = K.relindex H
theorem Subgroup.inf_relindex_left {G : Type u_1} [] (H : ) (K : ) :
(H K).relindex H = K.relindex H
theorem AddSubgroup.relindex_inf_mul_relindex {G : Type u_1} [] (H : ) (K : ) (L : ) :
H.relindex (K L) * K.relindex L = (H K).relindex L
theorem Subgroup.relindex_inf_mul_relindex {G : Type u_1} [] (H : ) (K : ) (L : ) :
H.relindex (K L) * K.relindex L = (H K).relindex L
@[simp]
theorem AddSubgroup.relindex_sup_right {G : Type u_1} [] (H : ) (K : ) [K.Normal] :
K.relindex (H K) = K.relindex H
@[simp]
theorem Subgroup.relindex_sup_right {G : Type u_1} [] (H : ) (K : ) [K.Normal] :
K.relindex (H K) = K.relindex H
@[simp]
theorem AddSubgroup.relindex_sup_left {G : Type u_1} [] (H : ) (K : ) [K.Normal] :
K.relindex (K H) = K.relindex H
@[simp]
theorem Subgroup.relindex_sup_left {G : Type u_1} [] (H : ) (K : ) [K.Normal] :
K.relindex (K H) = K.relindex H
theorem AddSubgroup.relindex_dvd_index_of_normal {G : Type u_1} [] (H : ) (K : ) [H.Normal] :
H.relindex K H.index
theorem Subgroup.relindex_dvd_index_of_normal {G : Type u_1} [] (H : ) (K : ) [H.Normal] :
H.relindex K H.index
theorem AddSubgroup.relindex_dvd_of_le_left {G : Type u_1} [] {H : } {K : } (L : ) (hHK : H K) :
K.relindex L H.relindex L
theorem Subgroup.relindex_dvd_of_le_left {G : Type u_1} [] {H : } {K : } (L : ) (hHK : H K) :
K.relindex L H.relindex L
theorem AddSubgroup.index_eq_two_iff {G : Type u_1} [] {H : } :
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} [] {H : } :
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} [] {H : } (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} [] {H : } (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} [] {H : } (h : H.index = 2) (a : G) :
a + a H
theorem Subgroup.mul_self_mem_of_index_two {G : Type u_1} [] {H : } (h : H.index = 2) (a : G) :
a * a H
theorem AddSubgroup.two_smul_mem_of_index_two {G : Type u_1} [] {H : } (h : H.index = 2) (a : G) :
2 a H
theorem Subgroup.sq_mem_of_index_two {G : Type u_1} [] {H : } (h : H.index = 2) (a : G) :
a ^ 2 H
@[simp]
theorem AddSubgroup.index_top {G : Type u_1} [] :
.index = 1
@[simp]
theorem Subgroup.index_top {G : Type u_1} [] :
.index = 1
@[simp]
theorem AddSubgroup.index_bot {G : Type u_1} [] :
.index =
@[simp]
theorem Subgroup.index_bot {G : Type u_1} [] :
.index =
theorem AddSubgroup.index_bot_eq_card {G : Type u_1} [] [] :
.index =
theorem Subgroup.index_bot_eq_card {G : Type u_1} [] [] :
.index =
@[simp]
theorem AddSubgroup.relindex_top_left {G : Type u_1} [] (H : ) :
.relindex H = 1
@[simp]
theorem Subgroup.relindex_top_left {G : Type u_1} [] (H : ) :
.relindex H = 1
@[simp]
theorem AddSubgroup.relindex_top_right {G : Type u_1} [] (H : ) :
H.relindex = H.index
@[simp]
theorem Subgroup.relindex_top_right {G : Type u_1} [] (H : ) :
H.relindex = H.index
@[simp]
theorem AddSubgroup.relindex_bot_left {G : Type u_1} [] (H : ) :
.relindex H = Nat.card H
@[simp]
theorem Subgroup.relindex_bot_left {G : Type u_1} [] (H : ) :
.relindex H = Nat.card H
theorem AddSubgroup.relindex_bot_left_eq_card {G : Type u_1} [] (H : ) [Fintype H] :
.relindex H =
theorem Subgroup.relindex_bot_left_eq_card {G : Type u_1} [] (H : ) [Fintype H] :
.relindex H =
@[simp]
theorem AddSubgroup.relindex_bot_right {G : Type u_1} [] (H : ) :
H.relindex = 1
@[simp]
theorem Subgroup.relindex_bot_right {G : Type u_1} [] (H : ) :
H.relindex = 1
@[simp]
theorem AddSubgroup.relindex_self {G : Type u_1} [] (H : ) :
H.relindex H = 1
@[simp]
theorem Subgroup.relindex_self {G : Type u_1} [] (H : ) :
H.relindex H = 1
theorem AddSubgroup.index_ker {G : Type u_1} [] {H : Type u_2} [] (f : G →+ H) :
f.ker.index = Nat.card ()
theorem Subgroup.index_ker {G : Type u_1} [] {H : Type u_2} [] (f : G →* H) :
f.ker.index = Nat.card ()
theorem AddSubgroup.relindex_ker {G : Type u_1} [] {H : Type u_2} [] (f : G →+ H) (K : ) :
f.ker.relindex K = Nat.card (f '' K)
theorem Subgroup.relindex_ker {G : Type u_1} [] {H : Type u_2} [] (f : G →* H) (K : ) :
f.ker.relindex K = Nat.card (f '' K)
@[simp]
theorem AddSubgroup.card_mul_index {G : Type u_1} [] (H : ) :
Nat.card H * H.index =
@[simp]
theorem Subgroup.card_mul_index {G : Type u_1} [] (H : ) :
Nat.card H * H.index =
theorem AddSubgroup.nat_card_dvd_of_injective {G : Type u_2} {H : Type u_3} [] [] (f : G →+ H) (hf : ) :
theorem Subgroup.nat_card_dvd_of_injective {G : Type u_2} {H : Type u_3} [] [] (f : G →* H) (hf : ) :
theorem AddSubgroup.nat_card_dvd_of_le {G : Type u_1} [] (H : ) (K : ) (hHK : H K) :
theorem Subgroup.nat_card_dvd_of_le {G : Type u_1} [] (H : ) (K : ) (hHK : H K) :
theorem AddSubgroup.nat_card_dvd_of_surjective {G : Type u_2} {H : Type u_3} [] [] (f : G →+ H) (hf : ) :
theorem Subgroup.nat_card_dvd_of_surjective {G : Type u_2} {H : Type u_3} [] [] (f : G →* H) (hf : ) :
theorem AddSubgroup.card_dvd_of_surjective {G : Type u_2} {H : Type u_3} [] [] [] [] (f : G →+ H) (hf : ) :
theorem Subgroup.card_dvd_of_surjective {G : Type u_2} {H : Type u_3} [] [] [] [] (f : G →* H) (hf : ) :
theorem AddSubgroup.index_map {G : Type u_1} [] (H : ) {G' : Type u_2} [AddGroup G'] (f : G →+ G') :
().index = (H f.ker).index * f.range.index
theorem Subgroup.index_map {G : Type u_1} [] (H : ) {G' : Type u_2} [Group G'] (f : G →* G') :
().index = (H f.ker).index * f.range.index
theorem AddSubgroup.index_map_dvd {G : Type u_1} [] (H : ) {G' : Type u_2} [AddGroup G'] {f : G →+ G'} (hf : ) :
().index H.index
theorem Subgroup.index_map_dvd {G : Type u_1} [] (H : ) {G' : Type u_2} [Group G'] {f : G →* G'} (hf : ) :
().index H.index
theorem AddSubgroup.dvd_index_map {G : Type u_1} [] (H : ) {G' : Type u_2} [AddGroup G'] {f : G →+ G'} (hf : f.ker H) :
H.index ().index
theorem Subgroup.dvd_index_map {G : Type u_1} [] (H : ) {G' : Type u_2} [Group G'] {f : G →* G'} (hf : f.ker H) :
H.index ().index
theorem AddSubgroup.index_map_eq {G : Type u_1} [] (H : ) {G' : Type u_2} [AddGroup G'] {f : G →+ G'} (hf1 : ) (hf2 : f.ker H) :
().index = H.index
theorem Subgroup.index_map_eq {G : Type u_1} [] (H : ) {G' : Type u_2} [Group G'] {f : G →* G'} (hf1 : ) (hf2 : f.ker H) :
().index = H.index
theorem AddSubgroup.index_eq_card {G : Type u_1} [] (H : ) [Fintype (G H)] :
H.index = Fintype.card (G H)
theorem Subgroup.index_eq_card {G : Type u_1} [] (H : ) [Fintype (G H)] :
H.index = Fintype.card (G H)
theorem AddSubgroup.index_mul_card {G : Type u_1} [] (H : ) [] [hH : Fintype H] :
H.index * =
theorem Subgroup.index_mul_card {G : Type u_1} [] (H : ) [] [hH : Fintype H] :
H.index * =
theorem AddSubgroup.index_dvd_card {G : Type u_1} [] (H : ) [] :
H.index
theorem Subgroup.index_dvd_card {G : Type u_1} [] (H : ) [] :
H.index
theorem AddSubgroup.relindex_eq_zero_of_le_left {G : Type u_1} [] {H : } {K : } {L : } (hHK : H K) (hKL : K.relindex L = 0) :
H.relindex L = 0
theorem Subgroup.relindex_eq_zero_of_le_left {G : Type u_1} [] {H : } {K : } {L : } (hHK : H K) (hKL : K.relindex L = 0) :
H.relindex L = 0
theorem AddSubgroup.relindex_eq_zero_of_le_right {G : Type u_1} [] {H : } {K : } {L : } (hKL : K L) (hHK : H.relindex K = 0) :
H.relindex L = 0
theorem Subgroup.relindex_eq_zero_of_le_right {G : Type u_1} [] {H : } {K : } {L : } (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} [] {H : } {K : } (h : H.relindex K = 0) :
H.index = 0
theorem Subgroup.index_eq_zero_of_relindex_eq_zero {G : Type u_1} [] {H : } {K : } (h : H.relindex K = 0) :
H.index = 0
theorem AddSubgroup.relindex_le_of_le_left {G : Type u_1} [] {H : } {K : } {L : } (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} [] {H : } {K : } {L : } (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} [] {H : } {K : } {L : } (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} [] {H : } {K : } {L : } (hKL : K L) (hHL : H.relindex L 0) :
H.relindex K H.relindex L
theorem AddSubgroup.relindex_ne_zero_trans {G : Type u_1} [] {H : } {K : } {L : } (hHK : H.relindex K 0) (hKL : K.relindex L 0) :
H.relindex L 0
theorem Subgroup.relindex_ne_zero_trans {G : Type u_1} [] {H : } {K : } {L : } (hHK : H.relindex K 0) (hKL : K.relindex L 0) :
H.relindex L 0
theorem AddSubgroup.relindex_inf_ne_zero {G : Type u_1} [] {H : } {K : } {L : } (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} [] {H : } {K : } {L : } (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} [] {H : } {K : } (hH : H.index 0) (hK : K.index 0) :
(H K).index 0
theorem Subgroup.index_inf_ne_zero {G : Type u_1} [] {H : } {K : } (hH : H.index 0) (hK : K.index 0) :
(H K).index 0
theorem AddSubgroup.relindex_inf_le {G : Type u_1} [] {H : } {K : } {L : } :
(H K).relindex L H.relindex L * K.relindex L
theorem Subgroup.relindex_inf_le {G : Type u_1} [] {H : } {K : } {L : } :
(H K).relindex L H.relindex L * K.relindex L
theorem AddSubgroup.index_inf_le {G : Type u_1} [] {H : } {K : } :
(H K).index H.index * K.index
theorem Subgroup.index_inf_le {G : Type u_1} [] {H : } {K : } :
(H K).index H.index * K.index
theorem AddSubgroup.relindex_iInf_ne_zero {G : Type u_1} [] {L : } {ι : Type u_2} [_hι : ] {f : ι} (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} [] {L : } {ι : Type u_2} [_hι : ] {f : ι} (hf : ∀ (i : ι), (f i).relindex L 0) :
(⨅ (i : ι), f i).relindex L 0
theorem AddSubgroup.relindex_iInf_le {G : Type u_1} [] {L : } {ι : Type u_2} [] (f : ι) :
(⨅ (i : ι), f i).relindex L i : ι, (f i).relindex L
abbrev AddSubgroup.relindex_iInf_le.match_1 {G : Type u_2} [] {L : } {ι : Type u_1} [] (f : ι) (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 Subgroup.relindex_iInf_le {G : Type u_1} [] {L : } {ι : Type u_2} [] (f : ι) :
(⨅ (i : ι), f i).relindex L i : ι, (f i).relindex L
theorem AddSubgroup.index_iInf_ne_zero {G : Type u_1} [] {ι : Type u_2} [] {f : ι} (hf : ∀ (i : ι), (f i).index 0) :
(⨅ (i : ι), f i).index 0
theorem Subgroup.index_iInf_ne_zero {G : Type u_1} [] {ι : Type u_2} [] {f : ι} (hf : ∀ (i : ι), (f i).index 0) :
(⨅ (i : ι), f i).index 0
theorem AddSubgroup.index_iInf_le {G : Type u_1} [] {ι : Type u_2} [] (f : ι) :
(⨅ (i : ι), f i).index i : ι, (f i).index
theorem Subgroup.index_iInf_le {G : Type u_1} [] {ι : Type u_2} [] (f : ι) :
(⨅ (i : ι), f i).index i : ι, (f i).index
@[simp]
theorem AddSubgroup.index_eq_one {G : Type u_1} [] {H : } :
H.index = 1 H =
@[simp]
theorem Subgroup.index_eq_one {G : Type u_1} [] {H : } :
H.index = 1 H =
@[simp]
theorem AddSubgroup.relindex_eq_one {G : Type u_1} [] {H : } {K : } :
H.relindex K = 1 K H
@[simp]
theorem Subgroup.relindex_eq_one {G : Type u_1} [] {H : } {K : } :
H.relindex K = 1 K H
@[simp]
theorem AddSubgroup.card_eq_one {G : Type u_1} [] {H : } :
Nat.card H = 1 H =
@[simp]
theorem Subgroup.card_eq_one {G : Type u_1} [] {H : } :
Nat.card H = 1 H =
theorem AddSubgroup.index_ne_zero_of_finite {G : Type u_1} [] {H : } [hH : Finite (G H)] :
H.index 0
theorem Subgroup.index_ne_zero_of_finite {G : Type u_1} [] {H : } [hH : Finite (G H)] :
H.index 0
noncomputable def AddSubgroup.fintypeOfIndexNeZero {G : Type u_1} [] {H : } (hH : H.index 0) :
Fintype (G H)

Finite index implies finite quotient.

Equations
Instances For
theorem AddSubgroup.fintypeOfIndexNeZero.proof_1 {G : Type u_1} [] {H : } (hH : H.index 0) :
Finite (G H)
noncomputable def Subgroup.fintypeOfIndexNeZero {G : Type u_1} [] {H : } (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} [] {H : } [Finite (G H)] (hH : H ) :
1 < H.index
theorem Subgroup.one_lt_index_of_ne_top {G : Type u_1} [] {H : } [Finite (G H)] (hH : H ) :
1 < H.index
class Subgroup.FiniteIndex {G : Type u_1} [] (H : ) :

Typeclass for finite index subgroups.

• finiteIndex : H.index 0

The subgroup has finite index

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

The subgroup has finite index

class AddSubgroup.FiniteIndex {G : Type u_2} [] (H : ) :

Typeclass for finite index subgroups.

• finiteIndex : H.index 0

The additive subgroup has finite index

Instances
theorem AddSubgroup.FiniteIndex.finiteIndex {G : Type u_2} [] {H : } [self : H.FiniteIndex] :
H.index 0

The additive subgroup has finite index

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

A finite index subgroup has finite quotient

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

A finite index subgroup has finite quotient.

Equations
• H.fintypeQuotientOfFiniteIndex =
Instances For
instance AddSubgroup.finite_quotient_of_finiteIndex {G : Type u_1} [] (H : ) [H.FiniteIndex] :
Finite (G H)
Equations
• =
instance Subgroup.finite_quotient_of_finiteIndex {G : Type u_1} [] (H : ) [H.FiniteIndex] :
Finite (G H)
Equations
• =
theorem AddSubgroup.finiteIndex_of_finite_quotient {G : Type u_1} [] (H : ) [Finite (G H)] :
H.FiniteIndex
theorem Subgroup.finiteIndex_of_finite_quotient {G : Type u_1} [] (H : ) [Finite (G H)] :
H.FiniteIndex
@[instance 100]
instance AddSubgroup.finiteIndex_of_finite {G : Type u_1} [] (H : ) [] :
H.FiniteIndex
Equations
• =
@[instance 100]
instance Subgroup.finiteIndex_of_finite {G : Type u_1} [] (H : ) [] :
H.FiniteIndex
Equations
• =
instance AddSubgroup.instFiniteIndexTop {G : Type u_1} [] :
.FiniteIndex
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instance Subgroup.instFiniteIndexTop {G : Type u_1} [] :
.FiniteIndex
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instance AddSubgroup.instFiniteIndexInf {G : Type u_1} [] (H : ) (K : ) [H.FiniteIndex] [K.FiniteIndex] :
(H K).FiniteIndex
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instance Subgroup.instFiniteIndexInf {G : Type u_1} [] (H : ) (K : ) [H.FiniteIndex] [K.FiniteIndex] :
(H K).FiniteIndex
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theorem AddSubgroup.finiteIndex_of_le {G : Type u_1} [] {H : } {K : } [H.FiniteIndex] (h : H K) :
K.FiniteIndex
theorem Subgroup.finiteIndex_of_le {G : Type u_1} [] {H : } {K : } [H.FiniteIndex] (h : H K) :
K.FiniteIndex
instance AddSubgroup.finiteIndex_ker {G : Type u_1} [] {G' : Type u_2} [AddGroup G'] (f : G →+ G') [Finite f.range] :
f.ker.FiniteIndex
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instance Subgroup.finiteIndex_ker {G : Type u_1} [] {G' : Type u_2} [Group G'] (f : G →* G') [Finite f.range] :
f.ker.FiniteIndex
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instance Subgroup.finiteIndex_normalCore {G : Type u_1} [] (H : ) [H.FiniteIndex] :
H.normalCore.FiniteIndex
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instance Subgroup.finiteIndex_center (G : Type u_1) [] [Finite ()] [] :
().FiniteIndex
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theorem Subgroup.index_center_le_pow (G : Type u_1) [] [Finite ()] [] :
().index ^
theorem AddMonoidHom.card_fiber_eq_of_mem_range {G : Type u_1} {M : Type u_2} {F : Type u_3} [] [] [] [] [FunLike F G M] [] (f : F) {x : M} {y : M} (hx : x ) (hy : y ) :
(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} [] [] [] [] [FunLike F G M] [] (f : F) {x : M} {y : M} (hx : x ) (hy : y ) :
(Finset.filter (fun (g : G) => f g = x) Finset.univ).card = (Finset.filter (fun (g : G) => f g = y) Finset.univ).card