openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Growth /CosetSubgroupGenerators.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Growth.CosetTransfer | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function | |
| open scoped Classical Pointwise | |
| namespace CriticalPercolation.RelativeGrowth | |
| variable {A B : Type*} [Group A] [Group B] | |
| lemma Bounded.image_trans {S : Set B} {T U : Set A} (f : A →* B) | |
| (hT : Bounded S (f '' T)) (hU : Bounded T U) : Bounded S (f '' U) := by | |
| obtain ⟨L,hL⟩ := hT | |
| obtain ⟨n,hn⟩ := hU | |
| refine ⟨L*n,?_⟩ | |
| rw [pow_mul] | |
| apply Set.Subset.trans (Set.image_mono hn) | |
| rw [Set.image_pow] | |
| exact Set.pow_subset_pow_left hL | |
| lemma finiteIndex_generators_containing_stabilizer (K : Subgroup A) (S : Set A) | |
| (h1 : (1:A)∈S) (hi : S⁻¹=S) (hgen : Subgroup.closure S=⊤) | |
| (hK : Bounded S (K:Set A)) (H : Subgroup A) [H.FiniteIndex] : | |
| ∃ T : Set H, (1:H)∈T ∧ T⁻¹=T ∧ Subgroup.closure T=⊤ ∧ | |
| Bounded S (H.subtype '' T) ∧ (K.subgroupOf H:Set H)⊆T := by | |
| obtain ⟨U,hU1,hUi,hUgen,hUb⟩ := finiteIndex_bounded_generators S h1 hi hgen H | |
| let T := (K.subgroupOf H:Set H)∪U | |
| refine ⟨T,Or.inr hU1,?_,?_,?_,Set.subset_union_left⟩ | |
| · simp only [T,Set.union_inv,inv_coe_set,hUi] | |
| · apply top_unique | |
| rw [←hUgen] | |
| exact Subgroup.closure_mono Set.subset_union_right | |
| · rw [Set.image_union] | |
| exact (hK.mono (by rintro _ ⟨k,hk,rfl⟩; exact hk)).union h1 hUb | |
| lemma normalizer_of_normal_subgroup (H : Subgroup A) (N : Subgroup H) [N.Normal] : | |
| H≤Subgroup.normalizer ((N.map H.subtype):Set A) := by | |
| have hh := N.le_normalizer_map H.subtype | |
| rwa [Subgroup.normalizer_eq_top,←MonoidHom.range_eq_map,Subgroup.range_subtype] at hh | |
| lemma relIndex_map_subgroupOf_nonzero (K H : Subgroup A) [H.FiniteIndex] | |
| (N : Subgroup H) (hN : N.relIndex (K.subgroupOf H)≠0) : | |
| (N.map H.subtype).relIndex K≠0 := by | |
| have he : (K.subgroupOf H).map H.subtype=K⊓H := by | |
| rw [Subgroup.subgroupOf,Subgroup.map_comap_eq,Subgroup.range_subtype,inf_comm] | |
| have hfirst : (N.map H.subtype).relIndex (K⊓H)≠0 := by | |
| rw [←he,Subgroup.relIndex_map_map_of_injective N (K.subgroupOf H) H.subtype_injective] | |
| exact hN | |
| apply Subgroup.relIndex_ne_zero_trans hfirst | |
| rw [Subgroup.inf_relIndex_left] | |
| exact fun h => (inferInstance : H.FiniteIndex).index_ne_zero (Subgroup.index_eq_zero_of_relIndex_eq_zero h) | |
| end CriticalPercolation.RelativeGrowth | |
| end | |
| end | |
| end OAI | |