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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 | import Mathlib
namespace OAI
section
noncomputable section
open Set Function Module
open scoped Classical
namespace CriticalPercolation.LinearGrowth
variable {K V G : Type*} [Field K] [AddCommGroup V] [Module K V] [Group G]
@[instance_reducible] def subspaceAction (ρ : G →* (V ≃ₗ[K] V)) : MulAction G (Submodule K V) where
smul g W := W.map (ρ g).toLinearMap
one_smul W := by
change W.map (ρ 1).toLinearMap=W
rw [map_one]
exact W.map_id
mul_smul g h W := by
change W.map (ρ (g*h)).toLinearMap=(W.map (ρ h).toLinearMap).map (ρ g).toLinearMap
rw [map_mul]
exact (Submodule.map_comp _ _ W)
lemma finiteIndex_subspace_stabilizer (ρ : G →* (V ≃ₗ[K] V))
(W : Submodule K V)
(hf : (Set.range (fun g => W.map (ρ g).toLinearMap)).Finite) :
∃ H : Subgroup G, H.FiniteIndex ∧ ∀ h : H, W.map (ρ h).toLinearMap=W := by
let := subspaceAction ρ
let H := MulAction.stabilizer G W
have horb : (MulAction.orbit G W).Finite := hf
let : Finite (MulAction.orbit G W) := horb.to_subtype
let : Finite (G ⧸ H) := Finite.of_equiv _ (MulAction.orbitEquivQuotientStabilizer G W)
refine ⟨H,Subgroup.finiteIndex_of_finite_quotient,?_⟩
intro h
exact h.property
end CriticalPercolation.LinearGrowth
end
end
end OAI
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