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 ⌀ |
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@[to_additive]
finiteIndex_iInf {ι : Type*} [Finite ι] {f : ι → Subgroup G}
(hf : ∀ i, (f i).FiniteIndex) : (⨅ i, f i).FiniteIndex :=
⟨index_iInf_ne_zero fun i => (hf i).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 | finiteIndex_iInf | null |
finiteIndex_iInf' {ι : Type*} {s : Finset ι}
(f : ι → Subgroup G) (hs : ∀ i ∈ s, (f i).FiniteIndex) :
(⨅ i ∈ s, f i).FiniteIndex := by
rw [iInf_subtype']
exact finiteIndex_iInf fun ⟨i, hi⟩ => hs i hi
@[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_iInf' | null |
instFiniteIndex_subgroupOf (H K : Subgroup G) [H.FiniteIndex] :
(H.subgroupOf K).FiniteIndex :=
⟨fun h => H.index_ne_zero_of_finite <| H.index_eq_zero_of_relIndex_eq_zero h⟩
@[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 | instFiniteIndex_subgroupOf | null |
finiteIndex_of_le [FiniteIndex H] (h : H ≤ K) : FiniteIndex K :=
⟨ne_zero_of_dvd_ne_zero FiniteIndex.index_ne_zero (index_dvd_of_le h)⟩
@[to_additive (attr := gcongr)] | 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_le | null |
index_antitone (h : H ≤ K) [H.FiniteIndex] : K.index ≤ H.index :=
Nat.le_of_dvd (Nat.zero_lt_of_ne_zero FiniteIndex.index_ne_zero) (index_dvd_of_le h)
@[to_additive (attr := gcongr)] | 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_antitone | null |
index_strictAnti (h : H < K) [H.FiniteIndex] : K.index < H.index := by
have h0 : K.index ≠ 0 := (finiteIndex_of_le h.le).index_ne_zero
apply lt_of_le_of_ne (index_antitone h.le)
rw [← relIndex_mul_index h.le, Ne, eq_comm, mul_eq_right₀ h0, relIndex_eq_one]
exact h.not_ge
variable (H K)
@[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_strictAnti | null |
finiteIndex_ker {G' : Type*} [Group G'] (f : G →* G') [Finite f.range] :
f.ker.FiniteIndex :=
@finiteIndex_of_finite_quotient G _ f.ker
(Finite.of_equiv f.range (QuotientGroup.quotientKerEquivRange f).symm) | 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 | finiteIndex_ker | null |
finiteIndex_normalCore [H.FiniteIndex] : H.normalCore.FiniteIndex := by
rw [normalCore_eq_ker]
infer_instance
@[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 | finiteIndex_normalCore | null |
index_range {f : G →* G} [hf : f.ker.FiniteIndex] :
f.range.index = Nat.card f.ker := by
rw [← mul_left_inj' hf.index_ne_zero, card_mul_index, index_ker, 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_range | null |
Subgroup.relIndex_pointwise_smul [Group G] [MulDistribMulAction H G] (J K : Subgroup G) :
(h • J).relIndex (h • K) = J.relIndex K := by
rw [pointwise_smul_def K, ← relIndex_comap, pointwise_smul_def,
comap_map_eq_self_of_injective (by intro a b; simp)]
@[deprecated (since := "2025-08-12")]
alias Subgroup.relindex_pointwise_smul := Subgroup.relIndex_pointwise_smul | 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.relIndex_pointwise_smul | null |
AddSubgroup.relIndex_pointwise_smul [AddGroup G] [DistribMulAction H G]
(J K : AddSubgroup G) : (h • J).relIndex (h • K) = J.relIndex K := by
rw [pointwise_smul_def K, ← relIndex_comap, pointwise_smul_def,
comap_map_eq_self_of_injective (by intro a b; simp)]
@[deprecated (since := "2025-08-12")]
alias AddSubgroup.relindex_pointwise_smul := AddSubgroup.relIndex_pointwise_smul | 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 | AddSubgroup.relIndex_pointwise_smul | null |
@[to_additive] index_stabilizer :
(stabilizer G x).index = (orbit G x).ncard :=
(Nat.card_congr (MulAction.orbitEquivQuotientStabilizer G x)).symm.trans
(Nat.card_coe_set_eq (orbit G x))
@[to_additive] theorem index_stabilizer_of_transitive [IsPretransitive G X] :
(stabilizer G x).index = Nat.card X := by
rw [index_stabilizer, orbit_eq_univ, Set.ncard_univ] | 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_stabilizer | null |
@[to_additive AddMonoidHom.surjective_of_card_ker_le_div]
surjective_of_card_ker_le_div {G M : Type*} [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 := by
refine range_eq_top.1 <| SetLike.ext' <| Set.eq_of_subset_of_ncard_le (Set.subset_univ _) ?_
rw [Subgroup.coe_top, Set.ncard_univ, ← Nat.card_coe_set_eq, SetLike.coe_sort_coe,
← Nat.card_congr (QuotientGroup.quotientKerEquivRange f).toEquiv]
exact Nat.le_of_mul_le_mul_left (f.ker.card_mul_index ▸ Nat.mul_le_of_le_div _ _ _ h) Nat.card_pos
open Finset
variable {G M F : Type*} [Group G] [Fintype G] [Monoid M] [DecidableEq M]
[FunLike F G M] [MonoidHomClass F G M]
@[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 | surjective_of_card_ker_le_div | null |
card_fiber_eq_of_mem_range (f : F) {x y : M} (hx : x ∈ Set.range f) (hy : y ∈ Set.range f) :
#{g | f g = x} = #{g | f g = y} := by
rcases hx with ⟨x, rfl⟩
rcases hy with ⟨y, rfl⟩
rcases mul_left_surjective x y with ⟨y, rfl⟩
conv_lhs =>
rw [← map_univ_equiv (Equiv.mulRight y⁻¹), filter_map, card_map]
congr 2 with g
simp only [Function.comp, Equiv.toEmbedding_apply, Equiv.coe_mulRight, map_mul]
let f' := MonoidHomClass.toMonoidHom f
change f' g * f' y⁻¹ = f' x ↔ f' g = f' x * f' y
rw [← f'.coe_toHomUnits y⁻¹, map_inv, Units.mul_inv_eq_iff_eq_mul, f'.coe_toHomUnits] | 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 | card_fiber_eq_of_mem_range | null |
@[simp]
index_smul (a : G) (S : AddSubgroup A) : (a • S).index = S.index :=
index_map_of_bijective (MulAction.bijective _) _ | 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_smul | null |
normal_of_index_eq_one (hH : H.index = 1) : H.Normal := by
rw [index_eq_one] at hH
rw [hH]
infer_instance | theorem | GroupTheory | [
"Mathlib.Data.Finite.Perm",
"Mathlib.Data.Nat.Prime.Factorial",
"Mathlib.GroupTheory.Index"
] | Mathlib/GroupTheory/IndexNormal.lean | normal_of_index_eq_one | A subgroup of index 1 is normal (does not require finiteness of G) |
normal_of_index_eq_two (hH : H.index = 2) : H.Normal where
conj_mem x hxH g := by simp_rw [mul_mem_iff_of_index_two hH, hxH, iff_true, inv_mem_iff] | theorem | GroupTheory | [
"Mathlib.Data.Finite.Perm",
"Mathlib.Data.Nat.Prime.Factorial",
"Mathlib.GroupTheory.Index"
] | Mathlib/GroupTheory/IndexNormal.lean | normal_of_index_eq_two | A subgroup of index 2 is normal (does not require finiteness of G) |
normal_of_index_eq_minFac_card (hHp : H.index = (Nat.card G).minFac) :
H.Normal := by
by_cases hG0 : Nat.card G = 0
· rw [hG0, minFac_zero] at hHp
exact normal_of_index_eq_two hHp
by_cases hG1 : Nat.card G = 1
· rw [hG1, minFac_one] at hHp
exact normal_of_index_eq_one hHp
suffices H.normalCore.relIndex H = 1 by
convert H.normalCore_normal
exact le_antisymm (relIndex_eq_one.mp this) (normalCore_le H)
have : Finite G := finite_of_card_ne_zero hG0
have index_ne_zero : H.index ≠ 0 := index_ne_zero_of_finite
rw [← mul_left_inj' index_ne_zero, one_mul, relIndex_mul_index H.normalCore_le]
have hp : Nat.Prime H.index := hHp ▸ minFac_prime hG1
have h : H.normalCore.index ∣ H.index ! := by
rw [normalCore_eq_ker, index_ker, index_eq_card, ← Nat.card_perm]
exact card_subgroup_dvd_card (toPermHom G (G ⧸ H)).range
apply dvd_antisymm _ (index_dvd_of_le H.normalCore_le)
rwa [← Coprime.dvd_mul_right, mul_factorial_pred hp.ne_zero]
have hr1 : H.normalCore.index ≠ 1 := fun hr1 ↦ hp.ne_one <|
Nat.eq_one_of_dvd_one (hr1 ▸ H.normalCore.index_dvd_of_le H.normalCore_le)
rw [Nat.coprime_factorial_iff hr1]
exact lt_of_lt_of_le (Nat.sub_one_lt hp.ne_zero) <|
hHp ▸ minFac_le_of_dvd (Nat.minFac_prime hr1).two_le
(dvd_trans (minFac_dvd H.normalCore.index) (H.normalCore.index_dvd_card)) | theorem | GroupTheory | [
"Mathlib.Data.Finite.Perm",
"Mathlib.Data.Nat.Prime.Factorial",
"Mathlib.GroupTheory.Index"
] | Mathlib/GroupTheory/IndexNormal.lean | normal_of_index_eq_minFac_card | A subgroup of a finite group whose index is the smallest prime factor is normal.
Note : if `G` is infinite, then `Nat.card G = 0` and `(Nat.card G).minFac = 2` |
upperCentralSeriesStep : Subgroup G where
carrier := { x : G | ∀ y : G, x * y * x⁻¹ * y⁻¹ ∈ H }
one_mem' y := by simp
mul_mem' {a b} ha hb y := by
convert Subgroup.mul_mem _ (ha (b * y * b⁻¹)) (hb y) using 1
group
inv_mem' {x} hx y := by
specialize hx y⁻¹
rw [mul_assoc, inv_inv] at hx ⊢
exact Subgroup.Normal.mem_comm inferInstance hx | def | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | upperCentralSeriesStep | If `H` is a normal subgroup of `G`, then the set `{x : G | ∀ y : G, x*y*x⁻¹*y⁻¹ ∈ H}`
is a subgroup of `G` (because it is the preimage in `G` of the centre of the
quotient group `G/H`.) |
mem_upperCentralSeriesStep (x : G) :
x ∈ upperCentralSeriesStep H ↔ ∀ y, x * y * x⁻¹ * y⁻¹ ∈ H := Iff.rfl
open QuotientGroup | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | mem_upperCentralSeriesStep | null |
upperCentralSeriesStep_eq_comap_center :
upperCentralSeriesStep H = Subgroup.comap (mk' H) (center (G ⧸ H)) := by
ext
rw [mem_comap, mem_center_iff, forall_mk]
apply forall_congr'
intro y
rw [coe_mk', ← QuotientGroup.mk_mul, ← QuotientGroup.mk_mul, eq_comm, eq_iff_div_mem,
div_eq_mul_inv, mul_inv_rev, mul_assoc] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | upperCentralSeriesStep_eq_comap_center | The proof that `upperCentralSeriesStep H` is the preimage of the centre of `G/H` under
the canonical surjection. |
upperCentralSeriesAux : ℕ → Σ' H : Subgroup G, Normal H
| 0 => ⟨⊥, inferInstance⟩
| n + 1 =>
let un := upperCentralSeriesAux n
let _un_normal := un.2
⟨upperCentralSeriesStep un.1, inferInstance⟩ | def | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | upperCentralSeriesAux | An auxiliary type-theoretic definition defining both the upper central series of
a group, and a proof that it is normal, all in one go. |
upperCentralSeries (n : ℕ) : Subgroup G :=
(upperCentralSeriesAux G n).1 | def | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | upperCentralSeries | `upperCentralSeries G n` is the `n`th term in the upper central series of `G`.
This is the increasing chain of subgroups of `G` that starts with the trivial subgroup `⊥` of `G`
and then continues defining `upperCentralSeries G (n + 1)` to be all the elements of `G`
that, modulo `upperCentralSeries G n`, belong to the center of the quotient
`G ⧸ upperCentralSeries G n`.
In particular, the identities
* `upperCentralSeries G 0 = ⊥` (`upperCentralSeries_zero`);
* `upperCentralSeries G 1 = center G` (`upperCentralSeries_one`);
hold. |
upperCentralSeries_normal (n : ℕ) : Normal (upperCentralSeries G n) :=
(upperCentralSeriesAux G n).2
@[simp] | instance | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | upperCentralSeries_normal | null |
upperCentralSeries_zero : upperCentralSeries G 0 = ⊥ := rfl
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | upperCentralSeries_zero | null |
upperCentralSeries_one : upperCentralSeries G 1 = center G := by
ext
simp only [upperCentralSeries, upperCentralSeriesAux, upperCentralSeriesStep, mem_bot, mem_mk,
Submonoid.mem_mk, Subsemigroup.mem_mk, Set.mem_setOf_eq, mem_center_iff]
exact forall_congr' fun y => by rw [mul_inv_eq_one, mul_inv_eq_iff_eq_mul, eq_comm]
variable {G} | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | upperCentralSeries_one | null |
mem_upperCentralSeries_succ_iff {n : ℕ} {x : G} :
x ∈ upperCentralSeries G (n + 1) ↔ ∀ y : G, x * y * x⁻¹ * y⁻¹ ∈ upperCentralSeries G n :=
Iff.rfl
@[simp] lemma comap_upperCentralSeries {H : Type*} [Group H] (e : H ≃* G) :
∀ n, (upperCentralSeries G n).comap e = upperCentralSeries H n
| 0 => by simpa [MonoidHom.ker_eq_bot_iff] using e.injective
| n + 1 => by
ext
simp [mem_upperCentralSeries_succ_iff, ← comap_upperCentralSeries e n,
← e.toEquiv.forall_congr_right] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | mem_upperCentralSeries_succ_iff | The `n+1`st term of the upper central series `H i` has underlying set equal to the `x` such
that `⁅x,G⁆ ⊆ H n`. |
@[mk_iff]
IsNilpotent (G : Type*) [Group G] : Prop where
nilpotent' : ∃ n : ℕ, upperCentralSeries G n = ⊤ | class | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | IsNilpotent | A group `G` is nilpotent if its upper central series is eventually `G`. |
IsNilpotent.nilpotent (G : Type*) [Group G] [IsNilpotent G] :
∃ n : ℕ, upperCentralSeries G n = ⊤ := Group.IsNilpotent.nilpotent' | lemma | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | IsNilpotent.nilpotent | null |
isNilpotent_congr {H : Type*} [Group H] (e : G ≃* H) : IsNilpotent G ↔ IsNilpotent H := by
simp_rw [isNilpotent_iff]
refine exists_congr fun n ↦ ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· simp [← Subgroup.comap_top e.symm.toMonoidHom, ← h]
· simp [← Subgroup.comap_top e.toMonoidHom, ← h]
@[simp] lemma isNilpotent_top : IsNilpotent (⊤ : Subgroup G) ↔ IsNilpotent G :=
isNilpotent_congr Subgroup.topEquiv
variable (G) in | lemma | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | isNilpotent_congr | null |
IsVirtuallyNilpotent : Prop := ∃ N : Subgroup G, IsNilpotent N ∧ FiniteIndex N | def | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | IsVirtuallyNilpotent | A group `G` is virtually nilpotent if it has a nilpotent cofinite subgroup `N`. |
IsNilpotent.isVirtuallyNilpotent (hG : IsNilpotent G) : IsVirtuallyNilpotent G :=
⟨⊤, by simpa, inferInstance⟩ | lemma | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | IsNilpotent.isVirtuallyNilpotent | null |
IsAscendingCentralSeries (H : ℕ → Subgroup G) : Prop :=
H 0 = ⊥ ∧ ∀ (x : G) (n : ℕ), x ∈ H (n + 1) → ∀ g, x * g * x⁻¹ * g⁻¹ ∈ H n | def | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | IsAscendingCentralSeries | A sequence of subgroups of `G` is an ascending central series if `H 0` is trivial and
`⁅H (n + 1), G⁆ ⊆ H n` for all `n`. Note that we do not require that `H n = G` for some `n`. |
IsDescendingCentralSeries (H : ℕ → Subgroup G) :=
H 0 = ⊤ ∧ ∀ (x : G) (n : ℕ), x ∈ H n → ∀ g, x * g * x⁻¹ * g⁻¹ ∈ H (n + 1) | def | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | IsDescendingCentralSeries | A sequence of subgroups of `G` is a descending central series if `H 0` is `G` and
`⁅H n, G⁆ ⊆ H (n + 1)` for all `n`. Note that we do not require that `H n = {1}` for some `n`. |
ascending_central_series_le_upper (H : ℕ → Subgroup G) (hH : IsAscendingCentralSeries H) :
∀ n : ℕ, H n ≤ upperCentralSeries G n
| 0 => hH.1.symm ▸ le_refl ⊥
| n + 1 => by
intro x hx
rw [mem_upperCentralSeries_succ_iff]
exact fun y => ascending_central_series_le_upper H hH n (hH.2 x n hx y)
variable (G) | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | ascending_central_series_le_upper | Any ascending central series for a group is bounded above by the upper central series. |
upperCentralSeries_isAscendingCentralSeries :
IsAscendingCentralSeries (upperCentralSeries G) :=
⟨rfl, fun _x _n h => h⟩ | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | upperCentralSeries_isAscendingCentralSeries | The upper central series of a group is an ascending central series. |
upperCentralSeries_mono : Monotone (upperCentralSeries G) := by
refine monotone_nat_of_le_succ ?_
intro n x hx y
rw [mul_assoc, mul_assoc, ← mul_assoc y x⁻¹ y⁻¹]
exact mul_mem hx (Normal.conj_mem (upperCentralSeries_normal G n) x⁻¹ (inv_mem hx) y) | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | upperCentralSeries_mono | null |
nilpotent_iff_finite_ascending_central_series :
IsNilpotent G ↔ ∃ n : ℕ, ∃ H : ℕ → Subgroup G, IsAscendingCentralSeries H ∧ H n = ⊤ := by
constructor
· rintro ⟨n, nH⟩
exact ⟨_, _, upperCentralSeries_isAscendingCentralSeries G, nH⟩
· rintro ⟨n, H, hH, hn⟩
use n
rw [eq_top_iff, ← hn]
exact ascending_central_series_le_upper H hH n | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | nilpotent_iff_finite_ascending_central_series | A group `G` is nilpotent iff there exists an ascending central series which reaches `G` in
finitely many steps. |
is_descending_rev_series_of_is_ascending {H : ℕ → Subgroup G} {n : ℕ} (hn : H n = ⊤)
(hasc : IsAscendingCentralSeries H) : IsDescendingCentralSeries fun m : ℕ => H (n - m) := by
obtain ⟨h0, hH⟩ := hasc
refine ⟨hn, fun x m hx g => ?_⟩
dsimp at hx
by_cases hm : n ≤ m
· rw [tsub_eq_zero_of_le hm, h0, Subgroup.mem_bot] at hx
subst hx
rw [show (1 : G) * g * (1⁻¹ : G) * g⁻¹ = 1 by group]
exact Subgroup.one_mem _
· push_neg at hm
apply hH
convert hx using 1
rw [tsub_add_eq_add_tsub (Nat.succ_le_of_lt hm), Nat.succ_eq_add_one, Nat.add_sub_add_right] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | is_descending_rev_series_of_is_ascending | null |
is_ascending_rev_series_of_is_descending {H : ℕ → Subgroup G} {n : ℕ} (hn : H n = ⊥)
(hdesc : IsDescendingCentralSeries H) : IsAscendingCentralSeries fun m : ℕ => H (n - m) := by
obtain ⟨h0, hH⟩ := hdesc
refine ⟨hn, fun x m hx g => ?_⟩
dsimp only at hx ⊢
by_cases hm : n ≤ m
· have hnm : n - m = 0 := tsub_eq_zero_iff_le.mpr hm
rw [hnm, h0]
exact mem_top _
· push_neg at hm
convert hH x _ hx g using 1
rw [tsub_add_eq_add_tsub (Nat.succ_le_of_lt hm), Nat.succ_eq_add_one, Nat.add_sub_add_right] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | is_ascending_rev_series_of_is_descending | null |
nilpotent_iff_finite_descending_central_series :
IsNilpotent G ↔ ∃ n : ℕ, ∃ H : ℕ → Subgroup G, IsDescendingCentralSeries H ∧ H n = ⊥ := by
rw [nilpotent_iff_finite_ascending_central_series]
constructor
· rintro ⟨n, H, hH, hn⟩
refine ⟨n, fun m => H (n - m), is_descending_rev_series_of_is_ascending G hn hH, ?_⟩
dsimp only
rw [tsub_self]
exact hH.1
· rintro ⟨n, H, hH, hn⟩
refine ⟨n, fun m => H (n - m), is_ascending_rev_series_of_is_descending G hn hH, ?_⟩
dsimp only
rw [tsub_self]
exact hH.1 | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | nilpotent_iff_finite_descending_central_series | A group `G` is nilpotent iff there exists a descending central series which reaches the
trivial group in a finite time. |
lowerCentralSeries (G : Type*) [Group G] : ℕ → Subgroup G
| 0 => ⊤
| n + 1 => ⁅lowerCentralSeries G n, ⊤⁆
variable {G}
@[simp] | def | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | lowerCentralSeries | The lower central series of a group `G` is a sequence `H n` of subgroups of `G`, defined
by `H 0` is all of `G` and for `n≥1`, `H (n + 1) = ⁅H n, G⁆` |
lowerCentralSeries_zero : lowerCentralSeries G 0 = ⊤ := rfl
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | lowerCentralSeries_zero | null |
lowerCentralSeries_one : lowerCentralSeries G 1 = commutator G := rfl | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | lowerCentralSeries_one | null |
mem_lowerCentralSeries_succ_iff (n : ℕ) (q : G) :
q ∈ lowerCentralSeries G (n + 1) ↔
q ∈ closure { x | ∃ p ∈ lowerCentralSeries G n,
∃ q ∈ (⊤ : Subgroup G), p * q * p⁻¹ * q⁻¹ = x } := Iff.rfl | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | mem_lowerCentralSeries_succ_iff | null |
lowerCentralSeries_succ (n : ℕ) :
lowerCentralSeries G (n + 1) =
closure { x | ∃ p ∈ lowerCentralSeries G n, ∃ q ∈ (⊤ : Subgroup G), p * q * p⁻¹ * q⁻¹ = x } :=
rfl | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | lowerCentralSeries_succ | null |
lowerCentralSeries_normal (n : ℕ) : Normal (lowerCentralSeries G n) := by
induction n with
| zero => exact (⊤ : Subgroup G).normal_of_characteristic
| succ d hd => exact @Subgroup.commutator_normal _ _ (lowerCentralSeries G d) ⊤ hd _ | instance | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | lowerCentralSeries_normal | null |
lowerCentralSeries_antitone : Antitone (lowerCentralSeries G) := by
refine antitone_nat_of_succ_le fun n x hx => ?_
simp only [mem_lowerCentralSeries_succ_iff, mem_top,
true_and] at hx
refine
closure_induction ?_ (Subgroup.one_mem _) (fun _ _ _ _ ↦ mul_mem) (fun _ _ ↦ inv_mem) hx
rintro y ⟨z, hz, a, ha⟩
rw [← ha, mul_assoc, mul_assoc, ← mul_assoc a z⁻¹ a⁻¹]
exact mul_mem hz (Normal.conj_mem (lowerCentralSeries_normal n) z⁻¹ (inv_mem hz) a) | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | lowerCentralSeries_antitone | null |
lowerCentralSeries_isDescendingCentralSeries :
IsDescendingCentralSeries (lowerCentralSeries G) := by
constructor
· rfl
intro x n hxn g
exact commutator_mem_commutator hxn (mem_top g) | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | lowerCentralSeries_isDescendingCentralSeries | The lower central series of a group is a descending central series. |
descending_central_series_ge_lower (H : ℕ → Subgroup G) (hH : IsDescendingCentralSeries H) :
∀ n : ℕ, lowerCentralSeries G n ≤ H n
| 0 => hH.1.symm ▸ le_refl ⊤
| n + 1 => commutator_le.mpr fun x hx q _ =>
hH.2 x n (descending_central_series_ge_lower H hH n hx) q | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | descending_central_series_ge_lower | Any descending central series for a group is bounded below by the lower central series. |
nilpotent_iff_lowerCentralSeries : IsNilpotent G ↔ ∃ n, lowerCentralSeries G n = ⊥ := by
rw [nilpotent_iff_finite_descending_central_series]
constructor
· rintro ⟨n, H, ⟨h0, hs⟩, hn⟩
use n
rw [eq_bot_iff, ← hn]
exact descending_central_series_ge_lower H ⟨h0, hs⟩ n
· rintro ⟨n, hn⟩
exact ⟨n, lowerCentralSeries G, lowerCentralSeries_isDescendingCentralSeries, hn⟩ | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | nilpotent_iff_lowerCentralSeries | A group is nilpotent if and only if its lower central series eventually reaches
the trivial subgroup. |
noncomputable Group.nilpotencyClass : ℕ := Nat.find (IsNilpotent.nilpotent G)
open scoped Classical in
@[simp] | def | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | Group.nilpotencyClass | The nilpotency class of a nilpotent group is the smallest natural `n` such that
the `n`-th term of the upper central series is `G`. |
upperCentralSeries_nilpotencyClass : upperCentralSeries G (Group.nilpotencyClass G) = ⊤ :=
Nat.find_spec (IsNilpotent.nilpotent G) | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | upperCentralSeries_nilpotencyClass | null |
upperCentralSeries_eq_top_iff_nilpotencyClass_le {n : ℕ} :
upperCentralSeries G n = ⊤ ↔ Group.nilpotencyClass G ≤ n := by
classical
constructor
· intro h
exact Nat.find_le h
· intro h
rw [eq_top_iff, ← upperCentralSeries_nilpotencyClass]
exact upperCentralSeries_mono _ h
open scoped Classical in | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | upperCentralSeries_eq_top_iff_nilpotencyClass_le | null |
least_ascending_central_series_length_eq_nilpotencyClass :
Nat.find ((nilpotent_iff_finite_ascending_central_series G).mp hG) =
Group.nilpotencyClass G := by
refine le_antisymm (Nat.find_mono ?_) (Nat.find_mono ?_)
· intro n hn
exact ⟨upperCentralSeries G, upperCentralSeries_isAscendingCentralSeries G, hn⟩
· rintro n ⟨H, ⟨hH, hn⟩⟩
rw [← top_le_iff, ← hn]
exact ascending_central_series_le_upper H hH n
open scoped Classical in | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | least_ascending_central_series_length_eq_nilpotencyClass | The nilpotency class of a nilpotent `G` is equal to the smallest `n` for which an ascending
central series reaches `G` in its `n`-th term. |
least_descending_central_series_length_eq_nilpotencyClass :
Nat.find ((nilpotent_iff_finite_descending_central_series G).mp hG) =
Group.nilpotencyClass G := by
rw [← least_ascending_central_series_length_eq_nilpotencyClass]
refine le_antisymm (Nat.find_mono ?_) (Nat.find_mono ?_)
· rintro n ⟨H, ⟨hH, hn⟩⟩
refine ⟨fun m => H (n - m), is_descending_rev_series_of_is_ascending G hn hH, ?_⟩
dsimp only
rw [tsub_self]
exact hH.1
· rintro n ⟨H, ⟨hH, hn⟩⟩
refine ⟨fun m => H (n - m), is_ascending_rev_series_of_is_descending G hn hH, ?_⟩
dsimp only
rw [tsub_self]
exact hH.1
open scoped Classical in | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | least_descending_central_series_length_eq_nilpotencyClass | The nilpotency class of a nilpotent `G` is equal to the smallest `n` for which the descending
central series reaches `⊥` in its `n`-th term. |
lowerCentralSeries_length_eq_nilpotencyClass :
Nat.find (nilpotent_iff_lowerCentralSeries.mp hG) = Group.nilpotencyClass (G := G) := by
rw [← least_descending_central_series_length_eq_nilpotencyClass]
refine le_antisymm (Nat.find_mono ?_) (Nat.find_mono ?_)
· rintro n ⟨H, ⟨hH, hn⟩⟩
rw [← le_bot_iff, ← hn]
exact descending_central_series_ge_lower H hH n
· rintro n h
exact ⟨lowerCentralSeries G, ⟨lowerCentralSeries_isDescendingCentralSeries, h⟩⟩
@[simp] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | lowerCentralSeries_length_eq_nilpotencyClass | The nilpotency class of a nilpotent `G` is equal to the length of the lower central series. |
lowerCentralSeries_nilpotencyClass :
lowerCentralSeries G (Group.nilpotencyClass G) = ⊥ := by
classical
rw [← lowerCentralSeries_length_eq_nilpotencyClass]
exact Nat.find_spec (nilpotent_iff_lowerCentralSeries.mp hG) | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | lowerCentralSeries_nilpotencyClass | null |
lowerCentralSeries_eq_bot_iff_nilpotencyClass_le {n : ℕ} :
lowerCentralSeries G n = ⊥ ↔ Group.nilpotencyClass G ≤ n := by
classical
constructor
· intro h
rw [← lowerCentralSeries_length_eq_nilpotencyClass]
exact Nat.find_le h
· intro h
rw [eq_bot_iff, ← lowerCentralSeries_nilpotencyClass]
exact lowerCentralSeries_antitone h | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | lowerCentralSeries_eq_bot_iff_nilpotencyClass_le | null |
lowerCentralSeries_map_subtype_le (H : Subgroup G) (n : ℕ) :
(lowerCentralSeries H n).map H.subtype ≤ lowerCentralSeries G n := by
induction n with
| zero => simp
| succ d hd =>
rw [lowerCentralSeries_succ, lowerCentralSeries_succ, MonoidHom.map_closure]
apply Subgroup.closure_mono
rintro x1 ⟨x2, ⟨x3, hx3, x4, _hx4, rfl⟩, rfl⟩
exact ⟨x3, hd (mem_map.mpr ⟨x3, hx3, rfl⟩), x4, by simp⟩ | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | lowerCentralSeries_map_subtype_le | null |
Subgroup.isNilpotent (H : Subgroup G) [hG : IsNilpotent G] : IsNilpotent H := by
rw [nilpotent_iff_lowerCentralSeries] at *
rcases hG with ⟨n, hG⟩
use n
have := lowerCentralSeries_map_subtype_le H n
simp only [hG, SetLike.le_def, mem_map, exists_imp] at this
exact eq_bot_iff.mpr fun x hx => Subtype.ext (this x ⟨hx, rfl⟩) | instance | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | Subgroup.isNilpotent | A subgroup of a nilpotent group is nilpotent |
Subgroup.nilpotencyClass_le (H : Subgroup G) [hG : IsNilpotent G] :
Group.nilpotencyClass H ≤ Group.nilpotencyClass G := by
repeat rw [← lowerCentralSeries_length_eq_nilpotencyClass]
classical apply Nat.find_mono
intro n hG
have := lowerCentralSeries_map_subtype_le H n
simp only [hG, SetLike.le_def, mem_map, exists_imp] at this
exact eq_bot_iff.mpr fun x hx => Subtype.ext (this x ⟨hx, rfl⟩) | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | Subgroup.nilpotencyClass_le | The nilpotency class of a subgroup is less or equal to the nilpotency class of the group |
upperCentralSeries.map {H : Type*} [Group H] {f : G →* H} (h : Function.Surjective f)
(n : ℕ) : Subgroup.map f (upperCentralSeries G n) ≤ upperCentralSeries H n := by
induction n with
| zero => simp
| succ d hd =>
rintro _ ⟨x, hx : x ∈ upperCentralSeries G d.succ, rfl⟩ y'
rcases h y' with ⟨y, rfl⟩
simpa using hd (mem_map_of_mem f (hx y)) | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | upperCentralSeries.map | null |
lowerCentralSeries.map {H : Type*} [Group H] (f : G →* H) (n : ℕ) :
Subgroup.map f (lowerCentralSeries G n) ≤ lowerCentralSeries H n := by
induction n with
| zero => simp
| succ d hd =>
rintro a ⟨x, hx : x ∈ lowerCentralSeries G d.succ, rfl⟩
refine closure_induction (hx := hx) ?_ (by simp [f.map_one, Subgroup.one_mem _])
(fun y z _ _ hy hz => by simp [MonoidHom.map_mul, Subgroup.mul_mem _ hy hz]) (fun y _ hy => by
rw [f.map_inv]; exact Subgroup.inv_mem _ hy)
rintro a ⟨y, hy, z, ⟨-, rfl⟩⟩
apply mem_closure.mpr
exact fun K hK => hK ⟨f y, hd (mem_map_of_mem f hy), by simp [commutatorElement_def]⟩ | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | lowerCentralSeries.map | null |
lowerCentralSeries_succ_eq_bot {n : ℕ} (h : lowerCentralSeries G n ≤ center G) :
lowerCentralSeries G (n + 1) = ⊥ := by
rw [lowerCentralSeries_succ, closure_eq_bot_iff, Set.subset_singleton_iff]
rintro x ⟨y, hy1, z, ⟨⟩, rfl⟩
rw [mul_assoc, ← mul_inv_rev, mul_inv_eq_one, eq_comm]
exact mem_center_iff.mp (h hy1) z | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | lowerCentralSeries_succ_eq_bot | null |
isNilpotent_of_ker_le_center {H : Type*} [Group H] (f : G →* H) (hf1 : f.ker ≤ center G)
(hH : IsNilpotent H) : IsNilpotent G := by
rw [nilpotent_iff_lowerCentralSeries] at *
rcases hH with ⟨n, hn⟩
use n + 1
refine lowerCentralSeries_succ_eq_bot (le_trans ((Subgroup.map_eq_bot_iff _).mp ?_) hf1)
exact eq_bot_iff.mpr (hn ▸ lowerCentralSeries.map f n) | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | isNilpotent_of_ker_le_center | The preimage of a nilpotent group is nilpotent if the kernel of the homomorphism is contained
in the center |
nilpotencyClass_le_of_ker_le_center {H : Type*} [Group H] (f : G →* H)
(hf1 : f.ker ≤ center G) (hH : IsNilpotent H) :
Group.nilpotencyClass (hG := isNilpotent_of_ker_le_center f hf1 hH) ≤
Group.nilpotencyClass H + 1 := by
haveI : IsNilpotent G := isNilpotent_of_ker_le_center f hf1 hH
rw [← lowerCentralSeries_length_eq_nilpotencyClass]
classical apply Nat.find_min'
refine lowerCentralSeries_succ_eq_bot (le_trans ((Subgroup.map_eq_bot_iff _).mp ?_) hf1)
rw [eq_bot_iff]
apply le_trans (lowerCentralSeries.map f _)
simp only [lowerCentralSeries_nilpotencyClass, le_bot_iff] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | nilpotencyClass_le_of_ker_le_center | null |
nilpotent_of_surjective {G' : Type*} [Group G'] [h : IsNilpotent G] (f : G →* G')
(hf : Function.Surjective f) : IsNilpotent G' := by
rcases h with ⟨n, hn⟩
use n
apply eq_top_iff.mpr
calc
⊤ = f.range := symm (f.range_eq_top_of_surjective hf)
_ = Subgroup.map f ⊤ := MonoidHom.range_eq_map _
_ = Subgroup.map f (upperCentralSeries G n) := by rw [hn]
_ ≤ upperCentralSeries G' n := upperCentralSeries.map hf n | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | nilpotent_of_surjective | The range of a surjective homomorphism from a nilpotent group is nilpotent |
nilpotencyClass_le_of_surjective {G' : Type*} [Group G'] (f : G →* G')
(hf : Function.Surjective f) [h : IsNilpotent G] :
Group.nilpotencyClass (hG := nilpotent_of_surjective _ hf) ≤ Group.nilpotencyClass G := by
classical apply Nat.find_mono
intro n hn
rw [eq_top_iff]
calc
⊤ = f.range := symm (f.range_eq_top_of_surjective hf)
_ = Subgroup.map f ⊤ := MonoidHom.range_eq_map _
_ = Subgroup.map f (upperCentralSeries G n) := by rw [hn]
_ ≤ upperCentralSeries G' n := upperCentralSeries.map hf n | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | nilpotencyClass_le_of_surjective | The nilpotency class of the range of a surjective homomorphism from a
nilpotent group is less or equal the nilpotency class of the domain |
nilpotent_of_mulEquiv {G' : Type*} [Group G'] [_h : IsNilpotent G] (f : G ≃* G') :
IsNilpotent G' :=
nilpotent_of_surjective f.toMonoidHom (MulEquiv.surjective f) | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | nilpotent_of_mulEquiv | Nilpotency respects isomorphisms |
nilpotent_quotient_of_nilpotent (H : Subgroup G) [H.Normal] [_h : IsNilpotent G] :
IsNilpotent (G ⧸ H) :=
nilpotent_of_surjective (QuotientGroup.mk' H) QuotientGroup.mk_surjective | instance | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | nilpotent_quotient_of_nilpotent | A quotient of a nilpotent group is nilpotent |
nilpotencyClass_quotient_le (H : Subgroup G) [H.Normal] [_h : IsNilpotent G] :
Group.nilpotencyClass (G ⧸ H) ≤ Group.nilpotencyClass G :=
nilpotencyClass_le_of_surjective (QuotientGroup.mk' H) QuotientGroup.mk_surjective | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | nilpotencyClass_quotient_le | The nilpotency class of a quotient of `G` is less or equal the nilpotency class of `G` |
private comap_center_subst {H₁ H₂ : Subgroup G} [Normal H₁] [Normal H₂] (h : H₁ = H₂) :
comap (mk' H₁) (center (G ⧸ H₁)) = comap (mk' H₂) (center (G ⧸ H₂)) := by subst h; rfl | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | comap_center_subst | null |
comap_upperCentralSeries_quotient_center (n : ℕ) :
comap (mk' (center G)) (upperCentralSeries (G ⧸ center G) n) = upperCentralSeries G n.succ := by
induction n with
| zero =>
simp only [upperCentralSeries_zero, MonoidHom.comap_bot, ker_mk',
(upperCentralSeries_one G).symm]
| succ n ih =>
let Hn := upperCentralSeries (G ⧸ center G) n
calc
comap (mk' (center G)) (upperCentralSeriesStep Hn) =
comap (mk' (center G)) (comap (mk' Hn) (center ((G ⧸ center G) ⧸ Hn))) := by
rw [upperCentralSeriesStep_eq_comap_center]
_ = comap (mk' (comap (mk' (center G)) Hn)) (center (G ⧸ comap (mk' (center G)) Hn)) :=
QuotientGroup.comap_comap_center
_ = comap (mk' (upperCentralSeries G n.succ)) (center (G ⧸ upperCentralSeries G n.succ)) :=
(comap_center_subst ih)
_ = upperCentralSeriesStep (upperCentralSeries G n.succ) :=
symm (upperCentralSeriesStep_eq_comap_center _) | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | comap_upperCentralSeries_quotient_center | null |
nilpotencyClass_zero_iff_subsingleton [IsNilpotent G] :
Group.nilpotencyClass G = 0 ↔ Subsingleton G := by
classical
rw [Group.nilpotencyClass, Nat.find_eq_zero, upperCentralSeries_zero,
subsingleton_iff_bot_eq_top, Subgroup.subsingleton_iff] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | nilpotencyClass_zero_iff_subsingleton | null |
nilpotencyClass_quotient_center [hH : IsNilpotent G] :
Group.nilpotencyClass (G ⧸ center G) = Group.nilpotencyClass G - 1 := by
generalize hn : Group.nilpotencyClass G = n
rcases n with (rfl | n)
· simp only [nilpotencyClass_zero_iff_subsingleton, zero_tsub] at *
exact Quotient.instSubsingletonQuotient (leftRel (center G))
· suffices Group.nilpotencyClass (G ⧸ center G) = n by simpa
apply le_antisymm
· apply upperCentralSeries_eq_top_iff_nilpotencyClass_le.mp
apply comap_injective (f := (mk' (center G))) Quot.mk_surjective
rw [comap_upperCentralSeries_quotient_center, comap_top, Nat.succ_eq_add_one, ← hn]
exact upperCentralSeries_nilpotencyClass
· apply le_of_add_le_add_right
calc
n + 1 = Group.nilpotencyClass G := hn.symm
_ ≤ Group.nilpotencyClass (G ⧸ center G) + 1 :=
nilpotencyClass_le_of_ker_le_center _ (le_of_eq (ker_mk' _)) _ | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | nilpotencyClass_quotient_center | Quotienting the `center G` reduces the nilpotency class by 1 |
nilpotencyClass_eq_quotient_center_plus_one [hH : IsNilpotent G] [Nontrivial G] :
Group.nilpotencyClass G = Group.nilpotencyClass (G ⧸ center G) + 1 := by
rw [nilpotencyClass_quotient_center]
rcases h : Group.nilpotencyClass G with ⟨⟩
· exfalso
rw [nilpotencyClass_zero_iff_subsingleton] at h
apply false_of_nontrivial_of_subsingleton G
· simp | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | nilpotencyClass_eq_quotient_center_plus_one | The nilpotency class of a non-trivial group is one more than its quotient by the center |
of_quotient_center_nilpotent (h : IsNilpotent (G ⧸ center G)) : IsNilpotent G := by
obtain ⟨n, hn⟩ := h.nilpotent
use n.succ
simp [← comap_upperCentralSeries_quotient_center, hn] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | of_quotient_center_nilpotent | If the quotient by `center G` is nilpotent, then so is G. |
@[elab_as_elim]
nilpotent_center_quotient_ind {P : ∀ (G) [Group G] [IsNilpotent G], Prop}
(G : Type*) [Group G] [IsNilpotent G]
(hbase : ∀ (G) [Group G] [Subsingleton G], P G)
(hstep : ∀ (G) [Group G] [IsNilpotent G], P (G ⧸ center G) → P G) : P G := by
obtain ⟨n, h⟩ : ∃ n, Group.nilpotencyClass G = n := ⟨_, rfl⟩
induction n generalizing G with
| zero =>
haveI := nilpotencyClass_zero_iff_subsingleton.mp h
exact hbase _
| succ n ih =>
have hn : Group.nilpotencyClass (G ⧸ center G) = n := by
simp [nilpotencyClass_quotient_center, h]
exact hstep _ (ih _ hn) | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | nilpotent_center_quotient_ind | A custom induction principle for nilpotent groups. The base case is a trivial group
(`subsingleton G`), and in the induction step, one can assume the hypothesis for
the group quotiented by its center. |
derived_le_lower_central (n : ℕ) : derivedSeries G n ≤ lowerCentralSeries G n := by
induction n with
| zero => simp
| succ i ih => apply commutator_mono ih; simp | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | derived_le_lower_central | null |
CommGroup.nilpotencyClass_le_one {G : Type*} [CommGroup G] :
Group.nilpotencyClass G ≤ 1 := by
rw [← upperCentralSeries_eq_top_iff_nilpotencyClass_le, upperCentralSeries_one]
apply CommGroup.center_eq_top | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | CommGroup.nilpotencyClass_le_one | Abelian groups are nilpotent -/
instance (priority := 100) CommGroup.isNilpotent {G : Type*} [CommGroup G] : IsNilpotent G := by
use 1
rw [upperCentralSeries_one]
apply CommGroup.center_eq_top
/-- Abelian groups have nilpotency class at most one |
commGroupOfNilpotencyClass [IsNilpotent G] (h : Group.nilpotencyClass G ≤ 1) : CommGroup G :=
Group.commGroupOfCenterEqTop <| by
rw [← upperCentralSeries_one]
exact upperCentralSeries_eq_top_iff_nilpotencyClass_le.mpr h | def | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | commGroupOfNilpotencyClass | Groups with nilpotency class at most one are abelian |
lowerCentralSeries_prod (n : ℕ) :
lowerCentralSeries (G₁ × G₂) n = (lowerCentralSeries G₁ n).prod (lowerCentralSeries G₂ n) := by
induction n with
| zero => simp
| succ n ih =>
calc
lowerCentralSeries (G₁ × G₂) n.succ = ⁅lowerCentralSeries (G₁ × G₂) n, ⊤⁆ := rfl
_ = ⁅(lowerCentralSeries G₁ n).prod (lowerCentralSeries G₂ n), ⊤⁆ := by rw [ih]
_ = ⁅(lowerCentralSeries G₁ n).prod (lowerCentralSeries G₂ n), (⊤ : Subgroup G₁).prod ⊤⁆ := by
simp
_ = ⁅lowerCentralSeries G₁ n, (⊤ : Subgroup G₁)⁆.prod ⁅lowerCentralSeries G₂ n, ⊤⁆ :=
(commutator_prod_prod _ _ _ _)
_ = (lowerCentralSeries G₁ n.succ).prod (lowerCentralSeries G₂ n.succ) := rfl | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | lowerCentralSeries_prod | null |
isNilpotent_prod [IsNilpotent G₁] [IsNilpotent G₂] : IsNilpotent (G₁ × G₂) := by
rw [nilpotent_iff_lowerCentralSeries]
refine ⟨max (Group.nilpotencyClass G₁) (Group.nilpotencyClass G₂), ?_⟩
rw [lowerCentralSeries_prod,
lowerCentralSeries_eq_bot_iff_nilpotencyClass_le.mpr (le_max_left _ _),
lowerCentralSeries_eq_bot_iff_nilpotencyClass_le.mpr (le_max_right _ _), bot_prod_bot] | instance | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | isNilpotent_prod | Products of nilpotent groups are nilpotent |
nilpotencyClass_prod [IsNilpotent G₁] [IsNilpotent G₂] :
Group.nilpotencyClass (G₁ × G₂) =
max (Group.nilpotencyClass G₁) (Group.nilpotencyClass G₂) := by
refine eq_of_forall_ge_iff fun k => ?_
simp only [max_le_iff, ← lowerCentralSeries_eq_bot_iff_nilpotencyClass_le,
lowerCentralSeries_prod, prod_eq_bot_iff] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | nilpotencyClass_prod | The nilpotency class of a product is the max of the nilpotency classes of the factors |
lowerCentralSeries_pi_le (n : ℕ) :
lowerCentralSeries (∀ i, Gs i) n ≤ Subgroup.pi Set.univ
fun i => lowerCentralSeries (Gs i) n := by
let pi := fun f : ∀ i, Subgroup (Gs i) => Subgroup.pi Set.univ f
induction n with
| zero => simp [pi_top]
| succ n ih =>
calc
lowerCentralSeries (∀ i, Gs i) n.succ = ⁅lowerCentralSeries (∀ i, Gs i) n, ⊤⁆ := rfl
_ ≤ ⁅pi fun i => lowerCentralSeries (Gs i) n, ⊤⁆ := commutator_mono ih (le_refl _)
_ = ⁅pi fun i => lowerCentralSeries (Gs i) n, pi fun i => ⊤⁆ := by simp [pi, pi_top]
_ ≤ pi fun i => ⁅lowerCentralSeries (Gs i) n, ⊤⁆ := commutator_pi_pi_le _ _
_ = pi fun i => lowerCentralSeries (Gs i) n.succ := rfl | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | lowerCentralSeries_pi_le | null |
isNilpotent_pi_of_bounded_class [∀ i, IsNilpotent (Gs i)] (n : ℕ)
(h : ∀ i, Group.nilpotencyClass (Gs i) ≤ n) : IsNilpotent (∀ i, Gs i) := by
rw [nilpotent_iff_lowerCentralSeries]
refine ⟨n, ?_⟩
rw [eq_bot_iff]
apply le_trans (lowerCentralSeries_pi_le _)
rw [← eq_bot_iff, pi_eq_bot_iff]
intro i
apply lowerCentralSeries_eq_bot_iff_nilpotencyClass_le.mpr (h i) | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | isNilpotent_pi_of_bounded_class | products of nilpotent groups are nilpotent if their nilpotency class is bounded |
lowerCentralSeries_pi_of_finite [Finite η] (n : ℕ) :
lowerCentralSeries (∀ i, Gs i) n = Subgroup.pi Set.univ
fun i => lowerCentralSeries (Gs i) n := by
let pi := fun f : ∀ i, Subgroup (Gs i) => Subgroup.pi Set.univ f
induction n with
| zero => simp [pi_top]
| succ n ih =>
calc
lowerCentralSeries (∀ i, Gs i) n.succ = ⁅lowerCentralSeries (∀ i, Gs i) n, ⊤⁆ := rfl
_ = ⁅pi fun i => lowerCentralSeries (Gs i) n, ⊤⁆ := by rw [ih]
_ = ⁅pi fun i => lowerCentralSeries (Gs i) n, pi fun i => ⊤⁆ := by simp [pi, pi_top]
_ = pi fun i => ⁅lowerCentralSeries (Gs i) n, ⊤⁆ := commutator_pi_pi_of_finite _ _
_ = pi fun i => lowerCentralSeries (Gs i) n.succ := rfl | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | lowerCentralSeries_pi_of_finite | null |
isNilpotent_pi [Finite η] [∀ i, IsNilpotent (Gs i)] : IsNilpotent (∀ i, Gs i) := by
cases nonempty_fintype η
rw [nilpotent_iff_lowerCentralSeries]
refine ⟨Finset.univ.sup fun i => Group.nilpotencyClass (Gs i), ?_⟩
rw [lowerCentralSeries_pi_of_finite, pi_eq_bot_iff]
intro i
rw [lowerCentralSeries_eq_bot_iff_nilpotencyClass_le]
exact Finset.le_sup (f := fun i => Group.nilpotencyClass (Gs i)) (Finset.mem_univ i) | instance | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | isNilpotent_pi | n-ary products of nilpotent groups are nilpotent |
nilpotencyClass_pi [Fintype η] [∀ i, IsNilpotent (Gs i)] :
Group.nilpotencyClass (∀ i, Gs i) = Finset.univ.sup fun i => Group.nilpotencyClass (Gs i) := by
apply eq_of_forall_ge_iff
intro k
simp only [Finset.sup_le_iff, ← lowerCentralSeries_eq_bot_iff_nilpotencyClass_le,
lowerCentralSeries_pi_of_finite, pi_eq_bot_iff, Finset.mem_univ, true_imp_iff] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | nilpotencyClass_pi | The nilpotency class of an n-ary product is the sup of the nilpotency classes of the factors |
IsPGroup.isNilpotent [Finite G] {p : ℕ} [hp : Fact (Nat.Prime p)] (h : IsPGroup p G) :
IsNilpotent G := by
cases nonempty_fintype G
classical
revert hG
apply @Fintype.induction_subsingleton_or_nontrivial _ G _
· intro _ _ _ _
infer_instance
· intro G _ _ ih _ h
have hcq : Fintype.card (G ⧸ center G) < Fintype.card G := by
simp only [← Nat.card_eq_fintype_card]
rw [card_eq_card_quotient_mul_card_subgroup (center G)]
simp only [Nat.card_eq_fintype_card]
apply lt_mul_of_one_lt_right
· exact Fintype.card_pos_iff.mpr One.instNonempty
· simp only [← Nat.card_eq_fintype_card]
exact (Subgroup.one_lt_card_iff_ne_bot _).mpr (ne_of_gt h.bot_lt_center)
have hnq : IsNilpotent (G ⧸ center G) := ih _ hcq (h.to_quotient (center G))
exact of_quotient_center_nilpotent hnq
variable [Finite G] | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | IsPGroup.isNilpotent | A nilpotent subgroup is solvable -/
instance (priority := 100) IsNilpotent.to_isSolvable [h : IsNilpotent G] : IsSolvable G := by
obtain ⟨n, hn⟩ := nilpotent_iff_lowerCentralSeries.1 h
use n
rw [eq_bot_iff, ← hn]
exact derived_le_lower_central n
theorem normalizerCondition_of_isNilpotent [h : IsNilpotent G] : NormalizerCondition G := by
-- roughly based on https://groupprops.subwiki.org/wiki/Nilpotent_implies_normalizer_condition
rw [normalizerCondition_iff_only_full_group_self_normalizing]
apply @nilpotent_center_quotient_ind _ G _ _ <;> clear! G
· intro G _ _ H _
exact @Subsingleton.elim _ Unique.instSubsingleton _ _
· intro G _ _ ih H hH
have hch : center G ≤ H := Subgroup.center_le_normalizer.trans (le_of_eq hH)
have hkh : (mk' (center G)).ker ≤ H := by simpa using hch
have hsur : Function.Surjective (mk' (center G)) := Quot.mk_surjective
let H' := H.map (mk' (center G))
have hH' : H'.normalizer = H' := by
apply comap_injective hsur
rw [comap_normalizer_eq_of_surjective _ hsur, comap_map_eq_self hkh]
exact hH
apply map_injective_of_ker_le (mk' (center G)) hkh le_top
exact (ih H' hH').trans (symm (map_top_of_surjective _ hsur))
end WithGroup
section WithFiniteGroup
open Group Fintype
variable {G : Type*} [hG : Group G]
/-- A p-group is nilpotent |
isNilpotent_of_product_of_sylow_group
(e : (∀ p : (Nat.card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G) :
IsNilpotent G := by
classical
let ps := (Nat.card G).primeFactors
have : ∀ (p : ps) (P : Sylow p G), IsNilpotent (↑P : Subgroup G) := by
intro p P
haveI : Fact (Nat.Prime ↑p) := Fact.mk <| Nat.prime_of_mem_primeFactors p.2
exact P.isPGroup'.isNilpotent
exact nilpotent_of_mulEquiv e | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | isNilpotent_of_product_of_sylow_group | If a finite group is the direct product of its Sylow groups, it is nilpotent |
isNilpotent_of_finite_tfae :
List.TFAE
[IsNilpotent G, NormalizerCondition G, ∀ H : Subgroup G, IsCoatom H → H.Normal,
∀ (p : ℕ) (_hp : Fact p.Prime) (P : Sylow p G), (↑P : Subgroup G).Normal,
Nonempty
((∀ p : (Nat.card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G)] := by
tfae_have 1 → 2 := @normalizerCondition_of_isNilpotent _ _
tfae_have 2 → 3
| h, H => NormalizerCondition.normal_of_coatom H h
tfae_have 3 → 4
| h, p, _, P => Sylow.normal_of_all_max_subgroups_normal h _
tfae_have 4 → 5
| h => Nonempty.intro (Sylow.directProductOfNormal fun {p hp hP} => h p hp hP)
tfae_have 5 → 1
| ⟨e⟩ => isNilpotent_of_product_of_sylow_group e
tfae_finish | theorem | GroupTheory | [
"Mathlib.GroupTheory.Solvable",
"Mathlib.GroupTheory.Sylow",
"Mathlib.Algebra.Group.Subgroup.Order",
"Mathlib.GroupTheory.Commutator.Finite"
] | Mathlib/GroupTheory/Nilpotent.lean | isNilpotent_of_finite_tfae | A finite group is nilpotent iff the normalizer condition holds, and iff all maximal groups are
normal and iff all Sylow groups are normal and iff the group is the direct product of its Sylow
groups. |
@[to_additive (attr := simps)
/-- Coproduct of two `AddHom`s with the same codomain with `AddCommute` assumption:
`f.noncommCoprod g _ (p : M × N) = f p.1 + g p.2`.
(For the commutative case, use `AddHom.coprod`) -/]
noncommCoprod (comm : ∀ m n, Commute (f m) (g n)) : M × N →ₙ* P where
toFun mn := f mn.fst * g mn.snd
map_mul' mn mn' := by simpa using (comm _ _).mul_mul_mul_comm _ _ | def | GroupTheory | [
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Algebra.Group.Subgroup.Ker",
"Mathlib.Algebra.Group.Subgroup.Lattice",
"Mathlib.Order.Disjoint"
] | Mathlib/GroupTheory/NoncommCoprod.lean | noncommCoprod | Coproduct of two `MulHom`s with the same codomain with `Commute` assumption:
`f.noncommCoprod g _ (p : M × N) = f p.1 * g p.2`.
(For the commutative case, use `MulHom.coprod`) |
@[to_additive
/-- Variant of `AddHom.noncommCoprod_apply`, with the sum written in the other direction -/]
noncommCoprod_apply' (comm) (mn : M × N) :
(f.noncommCoprod g comm) mn = g mn.2 * f mn.1 := by
rw [← comm, noncommCoprod_apply]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Algebra.Group.Subgroup.Ker",
"Mathlib.Algebra.Group.Subgroup.Lattice",
"Mathlib.Order.Disjoint"
] | Mathlib/GroupTheory/NoncommCoprod.lean | noncommCoprod_apply' | Variant of `MulHom.noncommCoprod_apply` with the product written in the other direction` |
comp_noncommCoprod {Q : Type*} [Semigroup Q] (h : P →ₙ* Q)
(comm : ∀ m n, Commute (f m) (g n)) :
h.comp (f.noncommCoprod g comm) =
(h.comp f).noncommCoprod (h.comp g) (fun m n ↦ (comm m n).map h) :=
ext fun _ => map_mul h _ _ | theorem | GroupTheory | [
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Algebra.Group.Subgroup.Ker",
"Mathlib.Algebra.Group.Subgroup.Lattice",
"Mathlib.Order.Disjoint"
] | Mathlib/GroupTheory/NoncommCoprod.lean | comp_noncommCoprod | null |
@[to_additive (attr := simps)
/-- Coproduct of two `AddMonoidHom`s with the same codomain,
with a commutation assumption:
`f.noncommCoprod g (p : M × N) = f p.1 + g p.2`.
(Noncommutative case; in the commutative case, use `AddHom.coprod`.) -/]
noncommCoprod : M × N →* P where
toFun := fun mn ↦ (f mn.fst) * (g mn.snd)
map_one' := by simp only [Prod.fst_one, Prod.snd_one, map_one, mul_one]
__ := f.toMulHom.noncommCoprod g.toMulHom comm | def | GroupTheory | [
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Algebra.Group.Subgroup.Ker",
"Mathlib.Algebra.Group.Subgroup.Lattice",
"Mathlib.Order.Disjoint"
] | Mathlib/GroupTheory/NoncommCoprod.lean | noncommCoprod | Coproduct of two `MonoidHom`s with the same codomain,
with a commutation assumption:
`f.noncommCoprod g _ (p : M × N) = f p.1 * g p.2`.
(Noncommutative case; in the commutative case, use `MonoidHom.coprod`.) |
@[to_additive
/-- Variant of `AddMonoidHom.noncomCoprod_apply` with the sum written in the other direction -/]
noncommCoprod_apply' (comm) (mn : M × N) :
(f.noncommCoprod g comm) mn = g mn.2 * f mn.1 := by
rw [← comm, MonoidHom.noncommCoprod_apply]
@[to_additive (attr := simp)] | theorem | GroupTheory | [
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Algebra.Group.Subgroup.Ker",
"Mathlib.Algebra.Group.Subgroup.Lattice",
"Mathlib.Order.Disjoint"
] | Mathlib/GroupTheory/NoncommCoprod.lean | noncommCoprod_apply' | Variant of `MonoidHom.noncomCoprod_apply` with the product written in the other direction` |
noncommCoprod_comp_inl : (f.noncommCoprod g comm).comp (inl M N) = f :=
ext fun x => by simp
@[to_additive (attr := simp)] | theorem | GroupTheory | [
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Algebra.Group.Subgroup.Ker",
"Mathlib.Algebra.Group.Subgroup.Lattice",
"Mathlib.Order.Disjoint"
] | Mathlib/GroupTheory/NoncommCoprod.lean | noncommCoprod_comp_inl | null |
noncommCoprod_comp_inr : (f.noncommCoprod g comm).comp (inr M N) = g :=
ext fun x => by simp
@[to_additive (attr := simp)] | theorem | GroupTheory | [
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Algebra.Group.Subgroup.Ker",
"Mathlib.Algebra.Group.Subgroup.Lattice",
"Mathlib.Order.Disjoint"
] | Mathlib/GroupTheory/NoncommCoprod.lean | noncommCoprod_comp_inr | null |
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