fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
@[to_additive]
Subgroup.toSubmonoid_zpowers {G : Type*} [Group G] (g : G) :
(Subgroup.zpowers g).toSubmonoid = Submonoid.powers g ⊔ Submonoid.powers g⁻¹ := by
rw [zpowers_eq_closure, closure_toSubmonoid, Submonoid.closure_union, Submonoid.powers_eq_closure,
Submonoid.powers_eq_closure, Set.inv_singleton]
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | Subgroup.toSubmonoid_zpowers | null |
Submonoid.powers_le_zpowers {G : Type*} [Group G] (g : G) :
Submonoid.powers g ≤ (Subgroup.zpowers g).toSubmonoid := by
rw [Subgroup.toSubmonoid_zpowers]
exact le_sup_left
open scoped Pointwise | lemma | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | Submonoid.powers_le_zpowers | null |
@[to_additive /-- The index of an additive subgroup as a natural number.
Returns 0 if the index is infinite. -/]
noncomputable index : ℕ :=
Nat.card (G ⧸ H) | def | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index | The index of a subgroup as a natural number. Returns `0` if the index is infinite. |
@[to_additive /-- 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. -/]
noncomputable relIndex : ℕ :=
(H.subgroupOf K).index
@[deprecated (since := "2025-08-12")] alias relindex := relIndex
@[to_additive] | def | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex | 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. |
index_comap_of_surjective {f : G' →* G} (hf : Function.Surjective f) :
(H.comap f).index = H.index := by
have key : ∀ x y : G',
QuotientGroup.leftRel (H.comap f) x y ↔ QuotientGroup.leftRel H (f x) (f y) := by
simp only [QuotientGroup.leftRel_apply]
exact fun x y => iff_of_eq (congr_arg (· ∈ H) (by rw [f.map_mul, f.map_inv]))
refine Cardinal.toNat_congr (Equiv.ofBijective (Quotient.map' f fun x y => (key x y).mp) ⟨?_, ?_⟩)
· simp_rw [← Quotient.eq''] at key
refine Quotient.ind' fun x => ?_
refine Quotient.ind' fun y => ?_
exact (key x y).mpr
· refine Quotient.ind' fun x => ?_
obtain ⟨y, hy⟩ := hf x
exact ⟨y, (Quotient.map'_mk'' f _ y).trans (congr_arg Quotient.mk'' hy)⟩
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_comap_of_surjective | null |
index_comap (f : G' →* G) :
(H.comap f).index = H.relIndex f.range :=
Eq.trans (congr_arg index (by rfl))
((H.subgroupOf f.range).index_comap_of_surjective f.rangeRestrict_surjective)
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_comap | null |
relIndex_comap (f : G' →* G) (K : Subgroup G') :
relIndex (comap f H) K = relIndex H (map f K) := by
rw [relIndex, subgroupOf, comap_comap, index_comap, ← f.map_range, K.range_subtype]
@[deprecated (since := "2025-08-12")] alias relindex_comap := relIndex_comap
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_comap | null |
relIndex_map_map_of_injective {f : G →* G'} (H K : Subgroup G) (hf : Function.Injective f) :
relIndex (map f H) (map f K) = relIndex H K := by
rw [← Subgroup.relIndex_comap, Subgroup.comap_map_eq_self_of_injective hf]
@[deprecated (since := "2025-08-12")]
alias relindex_map_map_of_injective := relIndex_map_map_of_injective
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_map_map_of_injective | null |
relIndex_map_map (f : G →* G') (H K : Subgroup G) :
(map f H).relIndex (map f K) = (H ⊔ f.ker).relIndex (K ⊔ f.ker) := by
rw [← comap_map_eq, ← comap_map_eq, relIndex_comap, (gc_map_comap f).l_u_l_eq_l]
variable {H K L}
@[to_additive relIndex_mul_index] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_map_map | null |
relIndex_mul_index (h : H ≤ K) : H.relIndex K * K.index = H.index :=
((mul_comm _ _).trans (Cardinal.toNat_mul _ _).symm).trans
(congr_arg Cardinal.toNat (Equiv.cardinal_eq (quotientEquivProdOfLE h))).symm
@[deprecated (since := "2025-08-12")] alias relindex_mul_index := relIndex_mul_index
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_mul_index | null |
index_dvd_of_le (h : H ≤ K) : K.index ∣ H.index :=
dvd_of_mul_left_eq (H.relIndex K) (relIndex_mul_index h)
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_dvd_of_le | null |
relIndex_dvd_index_of_le (h : H ≤ K) : H.relIndex K ∣ H.index :=
dvd_of_mul_right_eq K.index (relIndex_mul_index h)
@[deprecated (since := "2025-08-12")] alias relindex_dvd_index_of_le := relIndex_dvd_index_of_le
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_dvd_index_of_le | null |
relIndex_subgroupOf (hKL : K ≤ L) :
(H.subgroupOf L).relIndex (K.subgroupOf L) = H.relIndex K :=
((index_comap (H.subgroupOf L) (inclusion hKL)).trans (congr_arg _ (inclusion_range hKL))).symm
@[deprecated (since := "2025-08-12")] alias relindex_subgroupOf := relIndex_subgroupOf
variable (H K L)
@[to_additive relIndex_mul_relIndex] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_subgroupOf | null |
relIndex_mul_relIndex (hHK : H ≤ K) (hKL : K ≤ L) :
H.relIndex K * K.relIndex L = H.relIndex L := by
rw [← relIndex_subgroupOf hKL]
exact relIndex_mul_index fun x hx => hHK hx
@[deprecated (since := "2025-08-12")] alias relindex_mul_relindex := relIndex_mul_relIndex
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_mul_relIndex | null |
inf_relIndex_right : (H ⊓ K).relIndex K = H.relIndex K := by
rw [relIndex, relIndex, inf_subgroupOf_right]
@[deprecated (since := "2025-08-12")] alias inf_relindex_right := inf_relIndex_right
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | inf_relIndex_right | null |
inf_relIndex_left : (H ⊓ K).relIndex H = K.relIndex H := by
rw [inf_comm, inf_relIndex_right]
@[deprecated (since := "2025-08-12")] alias inf_relindex_left := inf_relIndex_left
@[to_additive relIndex_inf_mul_relIndex] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | inf_relIndex_left | null |
relIndex_inf_mul_relIndex : H.relIndex (K ⊓ L) * K.relIndex L = (H ⊓ K).relIndex L := by
rw [← inf_relIndex_right H (K ⊓ L), ← inf_relIndex_right K L, ← inf_relIndex_right (H ⊓ K) L,
inf_assoc, relIndex_mul_relIndex (H ⊓ (K ⊓ L)) (K ⊓ L) L inf_le_right inf_le_right]
@[deprecated (since := "2025-08-12")] alias relindex_inf_mul_relindex := relIndex_inf_mul_relIndex
@[to_additive (attr := simp)] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_inf_mul_relIndex | null |
relIndex_sup_right [K.Normal] : K.relIndex (H ⊔ K) = K.relIndex H :=
Nat.card_congr (QuotientGroup.quotientInfEquivProdNormalQuotient H K).toEquiv.symm
@[deprecated (since := "2025-08-12")] alias relindex_sup_right := relIndex_sup_right
@[to_additive (attr := simp)] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_sup_right | null |
relIndex_sup_left [K.Normal] : K.relIndex (K ⊔ H) = K.relIndex H := by
rw [sup_comm, relIndex_sup_right]
@[deprecated (since := "2025-08-12")] alias relindex_sup_left := relIndex_sup_left
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_sup_left | null |
relIndex_dvd_index_of_normal [H.Normal] : H.relIndex K ∣ H.index :=
relIndex_sup_right K H ▸ relIndex_dvd_index_of_le le_sup_right
@[deprecated (since := "2025-08-12")]
alias relindex_dvd_index_of_normal := relIndex_dvd_index_of_normal
variable {H K}
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_dvd_index_of_normal | null |
relIndex_dvd_of_le_left (hHK : H ≤ K) : K.relIndex L ∣ H.relIndex L :=
inf_of_le_left hHK ▸ dvd_of_mul_left_eq _ (relIndex_inf_mul_relIndex _ _ _)
@[deprecated (since := "2025-08-12")] alias relindex_dvd_of_le_left := relIndex_dvd_of_le_left | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_dvd_of_le_left | null |
@[to_additive /-- 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`. -/]
index_eq_two_iff : H.index = 2 ↔ ∃ a, ∀ b, Xor' (b * a ∈ H) (b ∈ H) := by
simp only [index, Nat.card_eq_two_iff' ((1 : G) : G ⧸ H), ExistsUnique, inv_mem_iff,
QuotientGroup.exists_mk, QuotientGroup.forall_mk, Ne, QuotientGroup.eq, mul_one,
xor_iff_iff_not]
refine exists_congr fun a =>
⟨fun ha b => ⟨fun hba hb => ?_, fun hb => ?_⟩, fun ha => ⟨?_, fun b hb => ?_⟩⟩
· exact ha.1 ((mul_mem_cancel_left hb).1 hba)
· exact inv_inv b ▸ ha.2 _ (mt (inv_mem_iff (x := b)).1 hb)
· rw [← inv_mem_iff (x := a), ← ha, inv_mul_cancel]
exact one_mem _
· rwa [ha, inv_mem_iff (x := b)]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_eq_two_iff | 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`. |
mul_mem_iff_of_index_two (h : H.index = 2) {a b : G} : a * b ∈ H ↔ (a ∈ H ↔ b ∈ H) := by
by_cases ha : a ∈ H; · simp only [ha, true_iff, mul_mem_cancel_left ha]
by_cases hb : b ∈ H; · simp only [hb, iff_true, mul_mem_cancel_right hb]
simp only [ha, hb, iff_true]
rcases index_eq_two_iff.1 h with ⟨c, hc⟩
refine (hc _).or.resolve_left ?_
rwa [mul_assoc, mul_mem_cancel_right ((hc _).or.resolve_right hb)]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | mul_mem_iff_of_index_two | null |
mul_self_mem_of_index_two (h : H.index = 2) (a : G) : a * a ∈ H := by
rw [mul_mem_iff_of_index_two h]
@[to_additive two_smul_mem_of_index_two] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | mul_self_mem_of_index_two | null |
sq_mem_of_index_two (h : H.index = 2) (a : G) : a ^ 2 ∈ H :=
(pow_two a).symm ▸ mul_self_mem_of_index_two h a
variable (H K) {f : G →* G'}
@[to_additive (attr := simp)] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | sq_mem_of_index_two | null |
index_top : (⊤ : Subgroup G).index = 1 :=
Nat.card_eq_one_iff_unique.mpr ⟨QuotientGroup.subsingleton_quotient_top, ⟨1⟩⟩
@[to_additive (attr := simp)] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_top | null |
index_bot : (⊥ : Subgroup G).index = Nat.card G :=
Cardinal.toNat_congr QuotientGroup.quotientBot.toEquiv
@[to_additive (attr := simp)] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_bot | null |
relIndex_top_left : (⊤ : Subgroup G).relIndex H = 1 :=
index_top
@[deprecated (since := "2025-08-12")] alias relindex_top_left := relIndex_top_left
@[to_additive (attr := simp)] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_top_left | null |
relIndex_top_right : H.relIndex ⊤ = H.index := by
rw [← relIndex_mul_index (show H ≤ ⊤ from le_top), index_top, mul_one]
@[deprecated (since := "2025-08-12")] alias relindex_top_right := relIndex_top_right
@[to_additive (attr := simp)] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_top_right | null |
relIndex_bot_left : (⊥ : Subgroup G).relIndex H = Nat.card H := by
rw [relIndex, bot_subgroupOf, index_bot]
@[deprecated (since := "2025-08-12")] alias relindex_bot_left := relIndex_bot_left
@[to_additive (attr := simp)] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_bot_left | null |
relIndex_bot_right : H.relIndex ⊥ = 1 := by rw [relIndex, subgroupOf_bot_eq_top, index_top]
@[deprecated (since := "2025-08-12")] alias relindex_bot_right := relIndex_bot_right
@[to_additive (attr := simp)] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_bot_right | null |
relIndex_self : H.relIndex H = 1 := by rw [relIndex, subgroupOf_self, index_top]
@[deprecated (since := "2025-08-12")] alias relindex_self := relIndex_self
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_self | null |
index_ker (f : G →* G') : f.ker.index = Nat.card f.range := by
rw [← MonoidHom.comap_bot, index_comap, relIndex_bot_left]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_ker | null |
relIndex_ker (f : G →* G') : f.ker.relIndex K = Nat.card (K.map f) := by
rw [← MonoidHom.comap_bot, relIndex_comap, relIndex_bot_left]
@[deprecated (since := "2025-08-12")] alias relindex_ker := relIndex_ker
@[to_additive (attr := simp) card_mul_index] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_ker | null |
card_mul_index : Nat.card H * H.index = Nat.card G := by
rw [← relIndex_bot_left, ← index_bot]
exact relIndex_mul_index bot_le
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | card_mul_index | null |
card_dvd_of_surjective (f : G →* G') (hf : Function.Surjective f) :
Nat.card G' ∣ Nat.card G := by
rw [← Nat.card_congr (QuotientGroup.quotientKerEquivOfSurjective f hf).toEquiv]
exact Dvd.intro_left (Nat.card f.ker) f.ker.card_mul_index
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | card_dvd_of_surjective | null |
card_range_dvd (f : G →* G') : Nat.card f.range ∣ Nat.card G :=
card_dvd_of_surjective f.rangeRestrict f.rangeRestrict_surjective
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | card_range_dvd | null |
card_map_dvd (f : G →* G') : Nat.card (H.map f) ∣ Nat.card H :=
card_dvd_of_surjective (f.subgroupMap H) (f.subgroupMap_surjective H)
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | card_map_dvd | null |
index_map (f : G →* G') :
(H.map f).index = (H ⊔ f.ker).index * f.range.index := by
rw [← comap_map_eq, index_comap, relIndex_mul_index (H.map_le_range f)]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_map | null |
index_map_dvd {f : G →* G'} (hf : Function.Surjective f) :
(H.map f).index ∣ H.index := by
rw [index_map, f.range_eq_top_of_surjective hf, index_top, mul_one]
exact index_dvd_of_le le_sup_left
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_map_dvd | null |
dvd_index_map {f : G →* G'} (hf : f.ker ≤ H) :
H.index ∣ (H.map f).index := by
rw [index_map, sup_of_le_left hf]
apply dvd_mul_right
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | dvd_index_map | null |
index_map_eq (hf1 : Surjective f) (hf2 : f.ker ≤ H) : (H.map f).index = H.index :=
Nat.dvd_antisymm (H.index_map_dvd hf1) (H.dvd_index_map hf2)
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_map_eq | null |
index_map_of_bijective (hf : Bijective f) (H : Subgroup G) : (H.map f).index = H.index :=
index_map_eq _ hf.2 (by rw [f.ker_eq_bot_iff.2 hf.1]; exact bot_le)
@[to_additive (attr := simp)] | lemma | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_map_of_bijective | null |
index_map_equiv (e : G ≃* G') : (map (e : G →* G') H).index = H.index :=
index_map_of_bijective e.bijective H
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_map_equiv | null |
index_map_of_injective {f : G →* G'} (hf : Function.Injective f) :
(H.map f).index = H.index * f.range.index := by
rw [H.index_map, f.ker_eq_bot_iff.mpr hf, sup_bot_eq]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_map_of_injective | null |
index_map_subtype {H : Subgroup G} (K : Subgroup H) :
(K.map H.subtype).index = K.index * H.index := by
rw [K.index_map_of_injective H.subtype_injective, H.range_subtype]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_map_subtype | null |
index_eq_card : H.index = Nat.card (G ⧸ H) :=
rfl
@[to_additive index_mul_card] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_eq_card | null |
index_mul_card : H.index * Nat.card H = Nat.card G := by
rw [mul_comm, card_mul_index]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_mul_card | null |
index_dvd_card : H.index ∣ Nat.card G :=
⟨Nat.card H, H.index_mul_card.symm⟩
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_dvd_card | null |
relIndex_dvd_card : H.relIndex K ∣ Nat.card K :=
(H.subgroupOf K).index_dvd_card
@[deprecated (since := "2025-08-12")] alias relindex_dvd_card := relIndex_dvd_card
variable {H K L}
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_dvd_card | null |
relIndex_eq_zero_of_le_left (hHK : H ≤ K) (hKL : K.relIndex L = 0) : H.relIndex L = 0 :=
eq_zero_of_zero_dvd (hKL ▸ relIndex_dvd_of_le_left L hHK)
@[deprecated (since := "2025-08-12")]
alias relindex_eq_zero_of_le_left := relIndex_eq_zero_of_le_left
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_eq_zero_of_le_left | null |
relIndex_eq_zero_of_le_right (hKL : K ≤ L) (hHK : H.relIndex K = 0) : H.relIndex L = 0 :=
Finite.card_eq_zero_of_embedding (quotientSubgroupOfEmbeddingOfLE H hKL) hHK
@[deprecated (since := "2025-08-12")]
alias relindex_eq_zero_of_le_right := relIndex_eq_zero_of_le_right | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_eq_zero_of_le_right | null |
@[to_additive /-- If `J` has finite index in `K`, then the same holds for their comaps under any
additive group hom. -/]
relIndex_comap_ne_zero (f : G →* G') {J K : Subgroup G'} (hJK : J.relIndex K ≠ 0) :
(J.comap f).relIndex (K.comap f) ≠ 0 := by
rw [relIndex_comap]
exact fun h ↦ hJK <| relIndex_eq_zero_of_le_right (map_comap_le _ _) h
@[deprecated (since := "2025-08-12")] alias relindex_comap_ne_zero := relIndex_comap_ne_zero
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_comap_ne_zero | If `J` has finite index in `K`, then the same holds for their comaps under any group hom. |
index_eq_zero_of_relIndex_eq_zero (h : H.relIndex K = 0) : H.index = 0 :=
H.relIndex_top_right.symm.trans (relIndex_eq_zero_of_le_right le_top h)
@[deprecated (since := "2025-08-12")]
alias index_eq_zero_of_relindex_eq_zero := index_eq_zero_of_relIndex_eq_zero
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_eq_zero_of_relIndex_eq_zero | null |
relIndex_le_of_le_left (hHK : H ≤ K) (hHL : H.relIndex L ≠ 0) :
K.relIndex L ≤ H.relIndex L :=
Nat.le_of_dvd (Nat.pos_of_ne_zero hHL) (relIndex_dvd_of_le_left L hHK)
@[deprecated (since := "2025-08-12")] alias relindex_le_of_le_left := relIndex_le_of_le_left
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_le_of_le_left | null |
relIndex_le_of_le_right (hKL : K ≤ L) (hHL : H.relIndex L ≠ 0) :
H.relIndex K ≤ H.relIndex L :=
Finite.card_le_of_embedding' (quotientSubgroupOfEmbeddingOfLE H hKL) fun h => (hHL h).elim
@[deprecated (since := "2025-08-12")] alias relindex_le_of_le_right := relIndex_le_of_le_right
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_le_of_le_right | null |
relIndex_ne_zero_trans (hHK : H.relIndex K ≠ 0) (hKL : K.relIndex L ≠ 0) :
H.relIndex L ≠ 0 := fun h =>
mul_ne_zero (mt (relIndex_eq_zero_of_le_right (show K ⊓ L ≤ K from inf_le_left)) hHK) hKL
((relIndex_inf_mul_relIndex H K L).trans (relIndex_eq_zero_of_le_left inf_le_left h))
@[deprecated (since := "2025-08-12")] alias relindex_ne_zero_trans := relIndex_ne_zero_trans
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_ne_zero_trans | null |
relIndex_inf_ne_zero (hH : H.relIndex L ≠ 0) (hK : K.relIndex L ≠ 0) :
(H ⊓ K).relIndex L ≠ 0 := by
replace hH : H.relIndex (K ⊓ L) ≠ 0 := mt (relIndex_eq_zero_of_le_right inf_le_right) hH
rw [← inf_relIndex_right] at hH hK ⊢
rw [inf_assoc]
exact relIndex_ne_zero_trans hH hK
@[deprecated (since := "2025-08-12")] alias relindex_inf_ne_zero := relIndex_inf_ne_zero
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_inf_ne_zero | null |
index_inf_ne_zero (hH : H.index ≠ 0) (hK : K.index ≠ 0) : (H ⊓ K).index ≠ 0 := by
rw [← relIndex_top_right] at hH hK ⊢
exact relIndex_inf_ne_zero hH hK | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_inf_ne_zero | null |
@[to_additive /-- If `J` has finite index in `K`, then `J ⊓ L` has finite index in `K ⊓ L` for any
`L`. -/]
relIndex_inter_ne_zero {J K : Subgroup G} (hJK : J.relIndex K ≠ 0) (L : Subgroup G) :
(J ⊓ L).relIndex (K ⊓ L) ≠ 0 := by
rw [← range_subtype L, inf_comm, ← map_comap_eq, inf_comm, ← map_comap_eq, ← relIndex_comap,
comap_map_eq_self_of_injective (subtype_injective L)]
exact relIndex_comap_ne_zero _ hJK
@[deprecated (since := "2025-08-12")] alias relindex_inter_ne_zero := relIndex_inter_ne_zero
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_inter_ne_zero | If `J` has finite index in `K`, then `J ⊓ L` has finite index in `K ⊓ L` for any `L`. |
relIndex_inf_le : (H ⊓ K).relIndex L ≤ H.relIndex L * K.relIndex L := by
by_cases h : H.relIndex L = 0
· exact (le_of_eq (relIndex_eq_zero_of_le_left inf_le_left h)).trans (zero_le _)
rw [← inf_relIndex_right, inf_assoc, ← relIndex_mul_relIndex _ _ L inf_le_right inf_le_right,
inf_relIndex_right, inf_relIndex_right]
exact mul_le_mul_right' (relIndex_le_of_le_right inf_le_right h) (K.relIndex L)
@[deprecated (since := "2025-08-12")] alias relindex_inf_le := relIndex_inf_le
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_inf_le | null |
index_inf_le : (H ⊓ K).index ≤ H.index * K.index := by
simp_rw [← relIndex_top_right, relIndex_inf_le]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_inf_le | null |
relIndex_iInf_ne_zero {ι : Type*} [_hι : Finite ι] {f : ι → Subgroup G}
(hf : ∀ i, (f i).relIndex L ≠ 0) : (⨅ i, f i).relIndex L ≠ 0 :=
haveI := Fintype.ofFinite ι
(Finset.prod_ne_zero_iff.mpr fun i _hi => hf i) ∘
Nat.card_pi.symm.trans ∘
Finite.card_eq_zero_of_embedding (quotientiInfSubgroupOfEmbedding f L)
@[deprecated (since := "2025-08-12")] alias relindex_iInf_ne_zero := relIndex_iInf_ne_zero
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_iInf_ne_zero | null |
relIndex_iInf_le {ι : Type*} [Fintype ι] (f : ι → Subgroup G) :
(⨅ i, f i).relIndex L ≤ ∏ i, (f i).relIndex L :=
le_of_le_of_eq
(Finite.card_le_of_embedding' (quotientiInfSubgroupOfEmbedding f L) fun h =>
let ⟨i, _hi, h⟩ := Finset.prod_eq_zero_iff.mp (Nat.card_pi.symm.trans h)
relIndex_eq_zero_of_le_left (iInf_le f i) h)
Nat.card_pi
@[deprecated (since := "2025-08-12")] alias relindex_iInf_le := relIndex_iInf_le
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_iInf_le | null |
index_iInf_ne_zero {ι : Type*} [Finite ι] {f : ι → Subgroup G}
(hf : ∀ i, (f i).index ≠ 0) : (⨅ i, f i).index ≠ 0 := by
simp_rw [← relIndex_top_right] at hf ⊢
exact relIndex_iInf_ne_zero hf
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_iInf_ne_zero | null |
index_iInf_le {ι : Type*} [Fintype ι] (f : ι → Subgroup G) :
(⨅ i, f i).index ≤ ∏ i, (f i).index := by simp_rw [← relIndex_top_right, relIndex_iInf_le]
@[to_additive (attr := simp) index_eq_one] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_iInf_le | null |
index_eq_one : H.index = 1 ↔ H = ⊤ :=
⟨fun h =>
QuotientGroup.subgroup_eq_top_of_subsingleton H (Nat.card_eq_one_iff_unique.mp h).1,
fun h => (congr_arg index h).trans index_top⟩
@[to_additive (attr := simp) relIndex_eq_one] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_eq_one | null |
relIndex_eq_one : H.relIndex K = 1 ↔ K ≤ H :=
index_eq_one.trans subgroupOf_eq_top
@[deprecated (since := "2025-08-12")] alias relindex_eq_one := relIndex_eq_one
@[to_additive (attr := simp) card_eq_one] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_eq_one | null |
card_eq_one : Nat.card H = 1 ↔ H = ⊥ :=
H.relIndex_bot_left ▸ relIndex_eq_one.trans le_bot_iff
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | card_eq_one | null |
inf_eq_bot_of_coprime (h : Nat.Coprime (Nat.card H) (Nat.card K)) : H ⊓ K = ⊥ :=
card_eq_one.1 <| Nat.eq_one_of_dvd_coprimes h
(card_dvd_of_le inf_le_left) (card_dvd_of_le inf_le_right)
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | inf_eq_bot_of_coprime | null |
index_ne_zero_of_finite [hH : Finite (G ⧸ H)] : H.index ≠ 0 := by
cases nonempty_fintype (G ⧸ H)
rw [index_eq_card]
exact Nat.card_pos.ne' | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_ne_zero_of_finite | null |
@[to_additive /-- Finite index implies finite quotient. -/]
noncomputable fintypeOfIndexNeZero (hH : H.index ≠ 0) : Fintype (G ⧸ H) :=
@Fintype.ofFinite _ (Nat.finite_of_card_ne_zero hH)
@[to_additive] | def | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | fintypeOfIndexNeZero | Finite index implies finite quotient. |
index_eq_zero_iff_infinite : H.index = 0 ↔ Infinite (G ⧸ H) := by
simp [index_eq_card, Nat.card_eq_zero]
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_eq_zero_iff_infinite | null |
index_ne_zero_iff_finite : H.index ≠ 0 ↔ Finite (G ⧸ H) := by
simp [index_eq_zero_iff_infinite]
@[to_additive one_lt_index_of_ne_top] | lemma | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_ne_zero_iff_finite | null |
one_lt_index_of_ne_top [Finite (G ⧸ H)] (hH : H ≠ ⊤) : 1 < H.index :=
Nat.one_lt_iff_ne_zero_and_ne_one.mpr ⟨index_ne_zero_of_finite, mt index_eq_one.mp hH⟩
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | one_lt_index_of_ne_top | null |
finite_quotient_of_finite_quotient_of_index_ne_zero {X : Type*} [MulAction G X]
[Finite <| MulAction.orbitRel.Quotient G X] (hi : H.index ≠ 0) :
Finite <| MulAction.orbitRel.Quotient H X := by
have := fintypeOfIndexNeZero hi
exact MulAction.finite_quotient_of_finite_quotient_of_finite_quotient
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | finite_quotient_of_finite_quotient_of_index_ne_zero | null |
finite_quotient_of_pretransitive_of_index_ne_zero {X : Type*} [MulAction G X]
[MulAction.IsPretransitive G X] (hi : H.index ≠ 0) :
Finite <| MulAction.orbitRel.Quotient H X := by
have := (MulAction.pretransitive_iff_subsingleton_quotient G X).1 inferInstance
exact finite_quotient_of_finite_quotient_of_index_ne_zero hi
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | finite_quotient_of_pretransitive_of_index_ne_zero | null |
exists_pow_mem_of_index_ne_zero (h : H.index ≠ 0) (a : G) :
∃ n, 0 < n ∧ n ≤ H.index ∧ a ^ n ∈ H := by
suffices ∃ n₁ n₂, n₁ < n₂ ∧ n₂ ≤ H.index ∧ ((a ^ n₂ : G) : G ⧸ H) = ((a ^ n₁ : G) : G ⧸ H) by
rcases this with ⟨n₁, n₂, hlt, hle, he⟩
refine ⟨n₂ - n₁, by cutsat, by cutsat, ?_⟩
rw [eq_comm, QuotientGroup.eq, ← zpow_natCast, ← zpow_natCast, ← zpow_neg, ← zpow_add,
add_comm] at he
rw [← zpow_natCast]
convert he
cutsat
suffices ∃ n₁ n₂, n₁ ≠ n₂ ∧ n₁ ≤ H.index ∧ n₂ ≤ H.index ∧
((a ^ n₂ : G) : G ⧸ H) = ((a ^ n₁ : G) : G ⧸ H) by
rcases this with ⟨n₁, n₂, hne, hle₁, hle₂, he⟩
rcases hne.lt_or_gt with hlt | hlt
· exact ⟨n₁, n₂, hlt, hle₂, he⟩
· exact ⟨n₂, n₁, hlt, hle₁, he.symm⟩
by_contra hc
simp_rw [not_exists] at hc
let f : (Set.Icc 0 H.index) → G ⧸ H := fun n ↦ (a ^ (n : ℕ) : G)
have hf : Function.Injective f := by
rintro ⟨n₁, h₁, hle₁⟩ ⟨n₂, h₂, hle₂⟩ he
have hc' := hc n₁ n₂
dsimp only [f] at he
simpa [hle₁, hle₂, he] using hc'
have := (fintypeOfIndexNeZero h).finite
have hcard := Nat.card_le_card_of_injective f hf
simp [← index_eq_card] at hcard
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | exists_pow_mem_of_index_ne_zero | null |
exists_pow_mem_of_relIndex_ne_zero (h : H.relIndex K ≠ 0) {a : G} (ha : a ∈ K) :
∃ n, 0 < n ∧ n ≤ H.relIndex K ∧ a ^ n ∈ H ⊓ K := by
rcases exists_pow_mem_of_index_ne_zero h ⟨a, ha⟩ with ⟨n, hlt, hle, he⟩
refine ⟨n, hlt, hle, ?_⟩
simpa [pow_mem ha, mem_subgroupOf] using he
@[deprecated (since := "2025-08-12")]
alias exists_pow_mem_of_relindex_ne_zero := exists_pow_mem_of_relIndex_ne_zero
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | exists_pow_mem_of_relIndex_ne_zero | null |
pow_mem_of_index_ne_zero_of_dvd (h : H.index ≠ 0) (a : G) {n : ℕ}
(hn : ∀ m, 0 < m → m ≤ H.index → m ∣ n) : a ^ n ∈ H := by
rcases exists_pow_mem_of_index_ne_zero h a with ⟨m, hlt, hle, he⟩
rcases hn m hlt hle with ⟨k, rfl⟩
rw [pow_mul]
exact pow_mem he _
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | pow_mem_of_index_ne_zero_of_dvd | null |
pow_mem_of_relIndex_ne_zero_of_dvd (h : H.relIndex K ≠ 0) {a : G} (ha : a ∈ K) {n : ℕ}
(hn : ∀ m, 0 < m → m ≤ H.relIndex K → m ∣ n) : a ^ n ∈ H ⊓ K := by
convert pow_mem_of_index_ne_zero_of_dvd h ⟨a, ha⟩ hn
simp [pow_mem ha, mem_subgroupOf]
@[deprecated (since := "2025-08-12")]
alias pow_mem_of_relindex_ne_zero_of_dvd := pow_mem_of_relIndex_ne_zero_of_dvd
@[to_additive (attr := simp) index_prod] | lemma | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | pow_mem_of_relIndex_ne_zero_of_dvd | null |
index_prod (H : Subgroup G) (K : Subgroup G') : (H.prod K).index = H.index * K.index := by
simp_rw [index, ← Nat.card_prod]
refine Nat.card_congr
((Quotient.congrRight (fun x y ↦ ?_)).trans (Setoid.prodQuotientEquiv _ _).symm)
rw [QuotientGroup.leftRel_prod]
@[deprecated (since := "2025-03-11")]
alias _root_.AddSubgroup.index_sum := AddSubgroup.index_prod
@[to_additive (attr := simp)] | lemma | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_prod | null |
index_pi {ι : Type*} [Fintype ι] (H : ι → Subgroup G) :
(Subgroup.pi Set.univ H).index = ∏ i, (H i).index := by
simp_rw [index, ← Nat.card_pi]
refine Nat.card_congr
((Quotient.congrRight (fun x y ↦ ?_)).trans (Setoid.piQuotientEquiv _).symm)
rw [QuotientGroup.leftRel_pi]
@[simp] | lemma | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_pi | null |
index_toAddSubgroup : (Subgroup.toAddSubgroup H).index = H.index :=
rfl
@[simp] | lemma | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | index_toAddSubgroup | null |
_root_.AddSubgroup.index_toSubgroup {G : Type*} [AddGroup G] (H : AddSubgroup G) :
(AddSubgroup.toSubgroup H).index = H.index :=
rfl
@[simp] | lemma | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | _root_.AddSubgroup.index_toSubgroup | null |
relIndex_toAddSubgroup :
(Subgroup.toAddSubgroup H).relIndex (Subgroup.toAddSubgroup K) = H.relIndex K :=
rfl
@[deprecated (since := "2025-08-12")] alias relindex_toAddSubgroup := relIndex_toAddSubgroup
@[simp] | lemma | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | relIndex_toAddSubgroup | null |
_root_.AddSubgroup.relIndex_toSubgroup {G : Type*} [AddGroup G] (H K : AddSubgroup G) :
(AddSubgroup.toSubgroup H).relIndex (AddSubgroup.toSubgroup K) = H.relIndex K :=
rfl
@[deprecated (since := "2025-08-12")]
alias _root_.AddSubgroup.relindex_toSubgroup := _root_.AddSubgroup.relIndex_toSubgroup | lemma | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | _root_.AddSubgroup.relIndex_toSubgroup | null |
_root_.AddSubgroup.FiniteIndex {G : Type*} [AddGroup G] (H : AddSubgroup G) : Prop where
/-- The additive subgroup has finite index;
recall that `AddSubgroup.index` returns 0 when the index is infinite. -/
index_ne_zero : H.index ≠ 0
@[deprecated (since := "2025-04-13")]
alias _root_AddSubgroup.FiniteIndex.finiteIndex := AddSubgroup.FiniteIndex.index_ne_zero
variable (H) in | class | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | _root_.AddSubgroup.FiniteIndex | Typeclass for finite index subgroups. |
@[to_additive] FiniteIndex : Prop where
/-- The subgroup has finite index;
recall that `Subgroup.index` returns 0 when the index is infinite. -/
index_ne_zero : H.index ≠ 0
@[deprecated (since := "2025-04-13")] alias FiniteIndex.finiteIndex := FiniteIndex.index_ne_zero | class | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | FiniteIndex | Typeclass for finite index subgroups. |
_root_.AddSubgroup.IsFiniteRelIndex {G : Type*} [AddGroup G] (H K : AddSubgroup G) :
Prop where
protected relIndex_ne_zero : H.relIndex K ≠ 0
variable (H K) in | class | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | _root_.AddSubgroup.IsFiniteRelIndex | Typeclass for a subgroup `H` to have finite index in a subgroup `K`. |
@[to_additive] IsFiniteRelIndex : Prop where
protected relIndex_ne_zero : H.relIndex K ≠ 0
@[to_additive] lemma relIndex_ne_zero [H.IsFiniteRelIndex K] : H.relIndex K ≠ 0 :=
IsFiniteRelIndex.relIndex_ne_zero
@[deprecated (since := "2025-08-12")] alias relindex_ne_zero := relIndex_ne_zero
@[to_additive] | class | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | IsFiniteRelIndex | Typeclass for a subgroup `H` to have finite index in a subgroup `K`. |
IsFiniteRelIndex.to_finiteIndex_subgroupOf [H.IsFiniteRelIndex K] :
(H.subgroupOf K).FiniteIndex where
index_ne_zero := relIndex_ne_zero
@[to_additive] | instance | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | IsFiniteRelIndex.to_finiteIndex_subgroupOf | null |
finiteIndex_iff : H.FiniteIndex ↔ H.index ≠ 0 :=
⟨fun h ↦ h.index_ne_zero, fun h ↦ ⟨h⟩⟩
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | finiteIndex_iff | null |
not_finiteIndex_iff {G : Type*} [Group G] {H : Subgroup G} :
¬ H.FiniteIndex ↔ H.index = 0 := by simp [finiteIndex_iff] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | not_finiteIndex_iff | null |
@[to_additive /-- A finite index subgroup has finite quotient -/]
noncomputable fintypeQuotientOfFiniteIndex [FiniteIndex H] : Fintype (G ⧸ H) :=
fintypeOfIndexNeZero FiniteIndex.index_ne_zero
@[to_additive] | def | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | fintypeQuotientOfFiniteIndex | A finite index subgroup has finite quotient. |
finite_quotient_of_finiteIndex [FiniteIndex H] : Finite (G ⧸ H) :=
fintypeQuotientOfFiniteIndex.finite
@[to_additive] | instance | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | finite_quotient_of_finiteIndex | null |
finiteIndex_of_finite_quotient [Finite (G ⧸ H)] : FiniteIndex H :=
⟨index_ne_zero_of_finite⟩
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | finiteIndex_of_finite_quotient | null |
finiteIndex_iff_finite_quotient : FiniteIndex H ↔ Finite (G ⧸ H) :=
⟨fun _ ↦ inferInstance, fun _ ↦ finiteIndex_of_finite_quotient⟩
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | finiteIndex_iff_finite_quotient | null |
@[to_additive]
finite_iff_finite_and_finiteIndex : Finite G ↔ Finite H ∧ H.FiniteIndex where
mp _ := ⟨inferInstance, inferInstance⟩
mpr := fun ⟨_, _⟩ ↦ Nat.finite_of_card_ne_zero <|
H.card_mul_index ▸ mul_ne_zero Nat.card_pos.ne' FiniteIndex.index_ne_zero
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | finite_iff_finite_and_finiteIndex | null |
_root_.MonoidHom.finite_iff_finite_ker_range (f : G →* G') :
Finite G ↔ Finite f.ker ∧ Finite f.range := by
rw [finite_iff_finite_and_finiteIndex f.ker, ← (QuotientGroup.quotientKerEquivRange f).finite_iff,
finiteIndex_iff_finite_quotient]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Finset",
"Mathlib.Algebra.GroupWithZero.Subgroup",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Finite.Prod",
"Mathlib.Data.Set.Card",
"Mathlib.GroupTheory.Coset.Card",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.QuotientGroup.Basic"
] | Mathlib/GroupTheory/Index.lean | _root_.MonoidHom.finite_iff_finite_ker_range | null |
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