| import OAI.Algebra.Universal.Congruence |
| import Mathlib.GroupTheory.GroupAction.Quotient |
| import Mathlib.Data.Fintype.EquivFin |
| import Mathlib.Data.Setoid.Basic |
|
|
| namespace OAI |
|
|
| /-! |
| # Congruences of coset actions |
|
|
| The full congruence lattice of the left action on a finite group's cosets is the |
| subgroup interval above the stabilizer. No normality assumption is needed. |
| -/ |
|
|
| namespace FiniteCongruence |
|
|
| noncomputable section |
| variable {G : Type} [Group G] [Fintype G] (H : Subgroup G) |
|
|
| /-- One unary operation for each element of G; no omitted operations. -/ |
| def cosetAlgebra : Algebra (G ⧸ H) where |
| numOps := Fintype.card G |
| arity := fun _ => 1 |
| op := fun i x => (Fintype.equivFin G).symm i • x 0 |
|
|
| lemma compatible_iff_invariant (r : Setoid (G ⧸ H)) : |
| Compatible (cosetAlgebra H) r ↔ |
| ∀ g : G, ∀ x y : G ⧸ H, r x y → r (g • x) (g • y) := by |
| constructor |
| · intro hr g x y hxy |
| have h := hr ((Fintype.equivFin G) g) (fun _ => x) (fun _ => y) (fun _ => hxy) |
| simpa [cosetAlgebra] using h |
| · intro hr i x y hxy |
| change Fin 1 → G ⧸ H at x y |
| change ∀ j : Fin 1, r (x j) (y j) at hxy |
| exact hr ((Fintype.equivFin G).symm i) (x 0) (y 0) (hxy 0) |
|
|
| abbrev base : G ⧸ H := (1 : G) |
|
|
| /-- The subgroup determined by the class of the identity coset. -/ |
| def subgroupOf (r : Congruence (cosetAlgebra H)) : Subgroup G where |
| carrier := {g | r.val (base H) (g • base H)} |
| one_mem' := by |
| change r.val (base H) ((1 : G) • base H) |
| rw [one_smul] |
| mul_mem' := by |
| intro a b ha hb |
| have hab := (compatible_iff_invariant H r.val).mp r.property a _ _ hb |
| simpa [mul_smul] using r.val.trans ha hab |
| inv_mem' := by |
| intro a ha |
| have h := (compatible_iff_invariant H r.val).mp r.property a⁻¹ _ _ ha |
| simpa using r.val.symm h |
|
|
| lemma le_subgroupOf (r : Congruence (cosetAlgebra H)) : H ≤ subgroupOf H r := by |
| intro g hg |
| have heq : (g : G ⧸ H) = (1 : G) := QuotientGroup.eq.mpr (by simpa using H.inv_mem hg) |
| change r.val (base H) (g • base H) |
| have hbase : g • base H = base H := by simpa [base] using heq |
| rw [hbase] |
|
|
| /-- The canonical quotient-set map exists without a normality assumption. -/ |
| def quotientMap (K : Subgroup G) (h : H ≤ K) : G ⧸ H → G ⧸ K := |
| Quotient.map' id (fun _ _ hxy => QuotientGroup.leftRel_apply.mpr |
| (h (QuotientGroup.leftRel_apply.mp hxy))) |
|
|
| omit [Fintype G] in |
| lemma quotientMap_mk (K : Subgroup G) (h : H ≤ K) (g : G) : |
| quotientMap H K h (g : G ⧸ H) = (g : G ⧸ K) := rfl |
|
|
| omit [Fintype G] in |
| lemma quotientMap_smul (K : Subgroup G) (h : H ≤ K) (g : G) (x : G ⧸ H) : |
| quotientMap H K h (g • x) = g • quotientMap H K h x := by |
| induction x using Quotient.inductionOn' with |
| | h a => rfl |
|
|
| /-- Pullback of equality on G/K, hence an actual equivalence relation on G/H. -/ |
| def congruenceOf (K : Subgroup G) (h : H ≤ K) : Congruence (cosetAlgebra H) := |
| ⟨Setoid.ker (quotientMap H K h), (compatible_iff_invariant H _).mpr (by |
| intro g x y hxy |
| change quotientMap H K h (g • x) = quotientMap H K h (g • y) |
| rw [quotientMap_smul, quotientMap_smul] |
| exact congrArg (fun q : G ⧸ K => g • q) hxy)⟩ |
|
|
| lemma congruenceOf_mk (K : Subgroup G) (h : H ≤ K) (a b : G) : |
| (congruenceOf H K h).val (a : G ⧸ H) (b : G ⧸ H) ↔ a⁻¹ * b ∈ K := by |
| change ((a : G ⧸ K) = (b : G ⧸ K)) ↔ _ |
| exact QuotientGroup.eq |
|
|
| lemma invariant_mk_iff (r : Congruence (cosetAlgebra H)) (a b : G) : |
| r.val (a : G ⧸ H) (b : G ⧸ H) ↔ a⁻¹ * b ∈ subgroupOf H r := by |
| have inv := (compatible_iff_invariant H r.val).mp r.property |
| constructor |
| · intro hab |
| have h := inv a⁻¹ _ _ hab |
| change r.val (base H) ((a⁻¹ * b) • base H) |
| simpa [base, MulAction.Quotient.smul_coe, smul_eq_mul] using h |
| · intro hab |
| change r.val (base H) ((a⁻¹ * b) • base H) at hab |
| have h := inv a _ _ hab |
| simpa [base, mul_smul, MulAction.Quotient.smul_coe, smul_eq_mul] using h |
|
|
| lemma subgroupOf_congruenceOf (K : Subgroup G) (h : H ≤ K) : |
| subgroupOf H (congruenceOf H K h) = K := by |
| ext g |
| change (congruenceOf H K h).val (base H) (g • base H) ↔ g ∈ K |
| simpa [base, MulAction.Quotient.smul_coe, smul_eq_mul] using congruenceOf_mk H K h 1 g |
|
|
| lemma congruenceOf_subgroupOf (r : Congruence (cosetAlgebra H)) : |
| congruenceOf H (subgroupOf H r) (le_subgroupOf H r) = r := by |
| apply Subtype.ext |
| ext x y |
| induction x using Quotient.inductionOn' with |
| | h a => |
| induction y using Quotient.inductionOn' with |
| | h b => |
| exact (congruenceOf_mk H _ _ a b).trans (invariant_mk_iff H r a b).symm |
|
|
| /-- The full congruence lattice of the coset algebra is order-isomorphic to the subgroup interval. -/ |
| def coset_action_correspondence : |
| Congruence (cosetAlgebra H) ≃o Set.Icc H (⊤ : Subgroup G) where |
| toFun r := ⟨subgroupOf H r, le_subgroupOf H r, le_top⟩ |
| invFun K := congruenceOf H K.val K.property.1 |
| left_inv r := congruenceOf_subgroupOf H r |
| right_inv K := by |
| apply Subtype.ext |
| exact subgroupOf_congruenceOf H K.val K.property.1 |
| map_rel_iff' := by |
| intro r s |
| change subgroupOf H r ≤ subgroupOf H s ↔ r.val ≤ s.val |
| constructor |
| · intro h x y hxy |
| induction x using Quotient.inductionOn' with |
| | h a => |
| induction y using Quotient.inductionOn' with |
| | h b => exact (invariant_mk_iff H s a b).mpr (h ((invariant_mk_iff H r a b).mp hxy)) |
| · intro h g hg |
| exact h hg |
|
|
| /-- The finite coset algebra realizes the full subgroup interval. -/ |
| theorem coset_action_correspondence_exists : |
| Nonempty (Congruence (cosetAlgebra H) ≃o Set.Icc H (⊤ : Subgroup G)) := |
| ⟨coset_action_correspondence H⟩ |
|
|
| end |
| end FiniteCongruence |
|
|
| end OAI |
|
|
