openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /LinearGroups /FiniteSubspaceOrbit.lean
| 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 | |