openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /LinearGroups /SolvableGrowth.lean
| 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 | |