| import OAI.GroupTheory.FiniteIndex.Extension |
|
|
| namespace OAI |
|
|
| /-! Realization of finite-index subgroups in extensions with free abelian quotient. -/ |
|
|
| namespace FiniteIndexRealization |
|
|
| lemma normal_map_of_characteristic {G : Type*} [Group G] (N : Subgroup G) |
| [N.Normal] (K : Subgroup N) [K.Characteristic] : (K.map N.subtype).Normal := by |
| classical |
| constructor |
| intro x hx g |
| obtain ⟨x, hx, rfl⟩ := hx |
| let e : N ≃* N := |
| { toFun := fun a => ⟨g * a * g⁻¹, (inferInstance : N.Normal).conj_mem a a.property g⟩ |
| invFun := fun a => ⟨g⁻¹ * a * (g⁻¹)⁻¹, (inferInstance : N.Normal).conj_mem a a.property g⁻¹⟩ |
| left_inv := by intro a; apply Subtype.ext; simp [mul_assoc] |
| right_inv := by intro a; apply Subtype.ext; simp [mul_assoc] |
| map_mul' := by intro a b; apply Subtype.ext; simp [mul_assoc] } |
| exact ⟨e x, Subgroup.characteristic_iff_le_comap.mp inferInstance e hx, rfl⟩ |
| |
| theorem realize_finite_index (G : Type*) [Group G] (N : Subgroup G) |
| [N.Normal] [Group.FG N] (r : Nat) |
| (e : G ⧸ N ≃* (Fin r → Multiplicative Int)) |
| (N0 : Subgroup N) [N0.FiniteIndex] : |
| ∃ G0 : Subgroup G, G0.FiniteIndex ∧ G0.comap N.subtype ≤ N0 := by |
| classical |
| obtain ⟨K, hKchar, hKfin, hKle⟩ := characteristic_finiteIndex_refinement N N0 |
| let := hKchar |
| let := hKfin |
| let L : Subgroup G := K.map N.subtype |
| let : L.Normal := normal_map_of_characteristic N K |
| have hLN : L ≤ N := by |
| rintro x ⟨a, ha, rfl⟩ |
| exact a.property |
| let π : G →* (Fin r → Multiplicative Int) := e.toMonoidHom.comp (QuotientGroup.mk' N) |
| have hπsurj : Function.Surjective π := e.surjective.comp (QuotientGroup.mk'_surjective N) |
| have hπker : π.ker = N := by |
| ext x |
| change e (QuotientGroup.mk' N x) = 1 ↔ x ∈ N |
| rw [← e.map_one, e.injective.eq_iff] |
| exact QuotientGroup.eq_one_iff x |
| have hπFG : Group.FG π.ker := by rw [hπker]; infer_instance |
| let := hπFG |
| let : Group.FG G := fg_of_surjective_with_fg_kernel π hπsurj |
| have hLπ : L ≤ π.ker := by rw [hπker]; exact hLN |
| let f : G ⧸ L →* (Fin r → Multiplicative Int) := QuotientGroup.lift L π hLπ |
| let t : N →* G ⧸ L := (QuotientGroup.mk' L).comp N.subtype |
| have htker : t.ker = K := by |
| ext x |
| change QuotientGroup.mk' L (x : G) = 1 ↔ x ∈ K |
| refine (QuotientGroup.eq_one_iff (N := L) (x : G)).trans ?_ |
| constructor |
| · rintro ⟨y, hy, heq⟩ |
| have : y = x := Subtype.ext heq |
| exact this ▸ hy |
| · intro hx |
| exact ⟨x, hx, rfl⟩ |
| have htrange : t.range = f.ker := by |
| rw [show f.ker = π.ker.map (QuotientGroup.mk' L) from QuotientGroup.ker_lift L π hLπ] |
| rw [hπker] |
| change ((QuotientGroup.mk' L).comp N.subtype).range = _ |
| rw [MonoidHom.range_comp, Subgroup.range_subtype] |
| have htfin : Finite (N ⧸ t.ker) := by rw [htker]; infer_instance |
| let := htfin |
| let : Finite t.range := Finite.of_surjective (QuotientGroup.quotientKerEquivRange t) |
| (QuotientGroup.quotientKerEquivRange t).surjective |
| have hffin : Finite f.ker := by rw [← htrange]; infer_instance |
| let := hffin |
| obtain ⟨J, hJ, hdisjoint⟩ := finite_kernel_free_abelian_disjoint r f |
| let := hJ |
| have hJcomap : (J.comap (QuotientGroup.mk' L)).FiniteIndex := by |
| constructor |
| rw [J.index_comap_of_surjective (QuotientGroup.mk'_surjective L)] |
| exact hJ.index_ne_zero |
| refine ⟨J.comap (QuotientGroup.mk' L), hJcomap, ?_⟩ |
| intro x hx |
| have hfx : f (QuotientGroup.mk' L (x : G)) = 1 := by |
| change π (x : G) = 1 |
| rw [← MonoidHom.mem_ker, hπker] |
| exact x.property |
| have hxJ : QuotientGroup.mk' L (x : G) ∈ J := hx |
| have hone : QuotientGroup.mk' L (x : G) = 1 := by |
| have hmem : QuotientGroup.mk' L (x : G) ∈ J ⊓ f.ker := ⟨hxJ, hfx⟩ |
| rw [hdisjoint] at hmem |
| exact hmem |
| have hxK : x ∈ K := by |
| rw [← htker] |
| exact hone |
| exact hKle hxK |
| end FiniteIndexRealization |
|
|
| end OAI |
|
|
