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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.NormalClosureGrowth
namespace OAI
section
noncomputable section
open Set Function
open scoped Classical Pointwise IsMulCommutative commutatorElement
namespace CriticalPercolation.SparseGrowth
variable {A B : Type*} [Group A] [Group B] [DecidableEq A]
omit [DecidableEq A] in
lemma isMulCommutative_of_generators (S : Set A) (hgen : Subgroup.closure S=⊤)
(hcomm : ∀ x∈S, ∀ y∈S, x*y=y*x) : IsMulCommutative A := by
have hc := Subgroup.isMulCommutative_closure
(fun left hleft right hright _ => hcomm left hleft right hright)
rw [hgen] at hc
let _ := hc
exact ⟨⟨fun x y => congrArg Subtype.val
(mul_comm' (⟨x,Subgroup.mem_top x⟩ : (⊤ : Subgroup A))
(⟨y,Subgroup.mem_top y⟩ : (⊤ : Subgroup A)))⟩⟩
lemma commutator_eq_normalClosure_generators (S : Finset A)
(hgen : Subgroup.closure (S : Set A)=⊤) :
commutator A = Subgroup.normalClosure
(((S.product S).image (fun p : A×A => ⁅p.1,p.2⁆)) : Set A) := by
let F := (S.product S).image (fun p : A×A => ⁅p.1,p.2⁆)
let N := Subgroup.normalClosure (F : Set A)
change commutator A=N
apply le_antisymm
· apply Subgroup.Normal.quotient_commutative_iff_commutator_le.mp
let q := QuotientGroup.mk' N
apply isMulCommutative_of_generators (q '' (S : Set A))
· rw [←MonoidHom.map_closure,hgen,←MonoidHom.range_eq_map]
exact MonoidHom.range_eq_top.mpr (QuotientGroup.mk'_surjective N)
· rintro _ ⟨x,hx,rfl⟩ _ ⟨y,hy,rfl⟩
apply commutatorElement_eq_one_iff_mul_comm.mp
rw [←map_commutatorElement]
apply (QuotientGroup.eq_one_iff ⁅x,y⁆).mpr
apply Subgroup.subset_normalClosure
exact Finset.mem_image.mpr ⟨(x,y),Finset.mem_product.mpr ⟨hx,hy⟩,rfl⟩
· apply Subgroup.normalClosure_le_normal
intro x hx
obtain ⟨⟨u,v⟩,huv,rfl⟩ := Finset.mem_image.mp hx
exact Subgroup.commutator_mem_commutator (Subgroup.mem_top u) (Subgroup.mem_top v)
theorem HasPolynomialDegree.commutator_fg {d : ℕ} (h : HasPolynomialDegree A d) :
(commutator A).FG := by
obtain ⟨S,h1,hsym,hgen,hg⟩ := (hasPolynomialDegree_iff d).mp h
rw [commutator_eq_normalClosure_generators S hgen]
exact normalClosure_fg_of_sparse_growth S _ h1 hsym hgen d hg
omit [DecidableEq A] in
lemma subgroup_fg_map {H : Subgroup A} (hH : H.FG) (f : A →* B) : (H.map f).FG := by
classical
obtain ⟨S,hS⟩ := hH
refine ⟨S.image f,?_⟩
rw [Finset.coe_image,←MonoidHom.map_closure,hS]
theorem HasPolynomialDegree.derivedSeries_fg {d : ℕ} (h : HasPolynomialDegree A d)
(n : ℕ) : (derivedSeries A n).FG := by
induction n with
| zero => exact h.fg.out
| succ n ih =>
let H := derivedSeries A n
let _ : Group.FG H := (Group.fg_iff_subgroup_fg H).mpr ih
have hH : HasPolynomialDegree H d := h.of_injective H.subtype Subtype.val_injective
have hc := subgroup_fg_map hH.commutator_fg H.subtype
rw [Subgroup.map_subtype_commutator] at hc
exact hc
omit [DecidableEq A] in
lemma finite_of_fg_torsion_finite_commutator [Group.FG A]
(ht : IsMulTorsion A) [Finite (commutator A)] : Finite A := by
let N := commutator A
let _ : IsMulCommutative (A ⧸ N) :=
Subgroup.Normal.quotient_commutative_iff_commutator_le.mpr le_rfl
let _ : Finite (A ⧸ N) := CommGroup.finite_of_fg_isMulTorsion _
(IsMulTorsion.of_surjective (QuotientGroup.mk'_surjective N) ht)
exact Finite.of_subgroup_quotient N
omit [DecidableEq A] in
lemma map_derivedSeries_commutator (n : ℕ) :
(derivedSeries (commutator A) n).map (commutator A).subtype =
derivedSeries A (n+1) := by
induction n with
| zero =>
change (⊤ : Subgroup (commutator A)).map (commutator A).subtype=commutator A
rw [←MonoidHom.range_eq_map,Subgroup.range_subtype]
| succ n ih =>
change (⁅derivedSeries (commutator A) n,derivedSeries (commutator A) n⁆).map _=_
rw [Subgroup.map_commutator,ih]
rfl
theorem finite_of_torsion_derivedSeries_eq_bot (n : ℕ) :
∀ (A : Type*) [Group A] [DecidableEq A], ∀ d : ℕ,
HasPolynomialDegree A d → IsMulTorsion A → derivedSeries A n=⊥ → Finite A := by
induction n with
| zero =>
intro A _ _ d hg ht hz
have hsub : Subsingleton A := by
constructor
intro x y
have hx : x=1 := by
have hm : x∈derivedSeries A 0 := Subgroup.mem_top x
simpa only [hz,Subgroup.mem_bot] using hm
have hy : y=1 := by
have hm : y∈derivedSeries A 0 := Subgroup.mem_top y
simpa only [hz,Subgroup.mem_bot] using hm
exact hx.trans hy.symm
exact Finite.of_subsingleton
| succ n ih =>
intro A _ _ d hg ht hz
let _ := hg.fg
let H := commutator A
let _ : Group.FG H := (Group.fg_iff_subgroup_fg H).mpr hg.commutator_fg
have hH : HasPolynomialDegree H d := hg.of_injective H.subtype Subtype.val_injective
have htH := ht.subgroup H
have hzH : derivedSeries H n=⊥ := by
apply Subgroup.map_injective (f := H.subtype) Subtype.val_injective
rw [Subgroup.map_bot,map_derivedSeries_commutator,hz]
let _ : Finite H := ih H d hH htH hzH
exact finite_of_fg_torsion_finite_commutator ht
omit [DecidableEq A] in
theorem finite_of_torsion_solvable_sparse_growth {d : ℕ} (hg : HasPolynomialDegree A d)
(ht : IsMulTorsion A) [Group.IsSolvable A] : Finite A := by
classical
obtain ⟨n,hn⟩ := Group.IsSolvable.solvable (G := A)
exact finite_of_torsion_derivedSeries_eq_bot n A d hg ht hn
end CriticalPercolation.SparseGrowth
end
end
end OAI
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