openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /LinearGroups /GeneralPeriodicCenter.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Nilpotent.NilpotentSubgroupFG | |
| import OAI.Probability.CriticalPercolation.Nilpotent.AbelianConjugationGrowth | |
| import OAI.Probability.CriticalPercolation.Nilpotent.NilpotentInfiniteCenter | |
| import OAI.Probability.CriticalPercolation.Nilpotent.FiniteCyclicExtensionIndex | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function | |
| open scoped Classical IsMulCommutative | |
| namespace CriticalPercolation.SparseGrowth | |
| variable {A : Type*} [Group A] [DecidableEq A] | |
| lemma HasPolynomialDegree.periodic_center_of_fg_nilpotent_kernel {d : ℕ} | |
| (hg : HasPolynomialDegree A d) | |
| (f : A →* Multiplicative ℤ) (hf : Surjective f) | |
| (L : Subgroup A) [L.Normal] [Group.IsNilpotent L] [Group.FG L] [Infinite L] | |
| (hLK : L ≤ f.ker) [(L.subgroupOf f.ker).FiniteIndex] : | |
| ∃ H : Subgroup A, H.FiniteIndex ∧ ∃ x : H, | |
| x∈Subgroup.center H ∧ ¬IsOfFinOrder x := by | |
| let Z := Subgroup.center L | |
| let _ : Infinite Z := infinite_center_of_nilpotent | |
| let C := Z.map L.subtype | |
| let e : Z ≃* C := Z.equivMapOfInjective L.subtype Subtype.val_injective | |
| let _ : C.Normal := characteristic_map_normal L Z | |
| let _ : IsMulCommutative C := ⟨⟨fun x y => e.symm.injective (by | |
| simp only [map_mul]; exact mul_comm _ _)⟩⟩ | |
| let _ : Infinite C := Infinite.of_injective e e.injective | |
| let _ : Group.FG Z := nilpotent_subgroup_fg L Z | |
| let _ : Group.FG C := Group.fg_of_surjective (f := e.toMonoidHom) e.surjective | |
| obtain ⟨a,ha⟩ := hf (Multiplicative.ofAdd 1) | |
| obtain ⟨m,hm,x,hx,hfix⟩ := hg.abelian_normal_periodic_direction C a | |
| let H := Subgroup.centralizer ({x.val} : Set A) | |
| have hLH : L≤H := by | |
| intro l hl | |
| obtain ⟨z,hz,hzx⟩ := x.property | |
| rw [Subgroup.mem_centralizer_singleton_iff,←hzx] | |
| exact congrArg Subtype.val ((Subgroup.mem_center_iff.mp hz) ⟨l,hl⟩) | |
| have haH : a^m∈H := by | |
| rw [Subgroup.mem_centralizer_singleton_iff] | |
| have he := congrArg Subtype.val hfix | |
| change a^m*x.val*(a^m)⁻¹=x.val at he | |
| exact mul_inv_eq_iff_eq_mul.mp he | |
| have hHi : H.FiniteIndex := finiteIndex_of_normal_kernel_subgroup f L hLK a ha H hLH m hm haH | |
| let y : H := ⟨x.val,by rw [Subgroup.mem_centralizer_singleton_iff]⟩ | |
| refine ⟨H,hHi,y,?_,?_⟩ | |
| · rw [Subgroup.mem_center_iff] | |
| intro w | |
| apply Subtype.ext | |
| exact Subgroup.mem_centralizer_singleton_iff.mp w.property | |
| · intro hy | |
| obtain ⟨k,hk,he⟩ := hy.exists_pow_eq_one | |
| apply hx | |
| apply isOfFinOrder_iff_pow_eq_one.mpr | |
| exact ⟨k,hk,Subtype.ext (congrArg (fun z : H => z.val) he)⟩ | |
| end CriticalPercolation.SparseGrowth | |
| end | |
| end | |
| end OAI | |