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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 | 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
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