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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 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.ComponentAction
namespace OAI
section
section
noncomputable section
open Set Function
open scoped Classical
namespace CriticalPercolation.BondGraph
variable {V Λ N T : Type*} [Group Λ] [Group N] [Group T] [MulAction Λ V]
lemma rangeRestrict_kernel_orbits (f : Λ →* N)
(hf : ∀ v : V, (MulAction.orbit f.ker v).Finite) :
∀ v : V, (MulAction.orbit f.rangeRestrict.ker v).Finite := by
intro v
apply (hf v).subset
rintro w ⟨g,hg⟩
have hgm : f g.val=1 := congrArg Subtype.val g.property
exact ⟨⟨g.val,hgm⟩,hg⟩
lemma rangeRestrict_stabilizer_images (f : Λ →* N)
(hf : ∀ v : V, ((MulAction.stabilizer Λ v).map f : Set N).Finite) :
∀ v : V, ((MulAction.stabilizer Λ v).map f.rangeRestrict : Set f.range).Finite := by
intro v
apply (Set.Finite.preimage Subtype.val_injective.injOn (hf v)).subset
rintro n ⟨g,hg,rfl⟩
exact ⟨g,hg,rfl⟩
lemma component_quotient_le_kernel (ρ : Λ →* N) (φ : N →* T) (C : Set V) :
(ρ.comp (componentKernel (φ.comp ρ) C).subtype).range≤φ.ker := by
rintro n ⟨g,rfl⟩
exact g.property.1
lemma infinite_index_of_le_coordinate_kernel [Infinite T]
(φ : N →* T) (hφ : Surjective φ) (J : Subgroup N) (hJ : J≤φ.ker) :
J.index=0 := by
have hr : Infinite φ.range := Infinite.of_surjective φ.range.subtype
(by intro t; obtain ⟨n,rfl⟩ := hφ t; exact ⟨⟨φ n,⟨n,rfl⟩⟩,rfl⟩)
have hk : φ.ker.index=0 := by rw [Subgroup.index_ker,Nat.card_eq_zero_of_infinite]
exact Nat.eq_zero_of_zero_dvd (hk ▸ Subgroup.index_dvd_of_le hJ)
end CriticalPercolation.BondGraph
end
end
end
end OAI
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