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c8720e4 | 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 | import Mathlib
import OAI.RepresentationTheory.PartialPermutation.Model
namespace OAI
section
open scoped Classical
open scoped BigOperators ComplexConjugate MonoidAlgebra
open scoped BigOperators ComplexConjugate
open scoped MonoidAlgebra BigOperators
open scoped BigOperators MonoidAlgebra Classical
attribute [local instance] Classical.propDecidable
open scoped MonoidAlgebra BigOperators
open scoped BigOperators
namespace PartialPermutation
lemma blockSubgroup_commute {X : Type*} {M N : Set X} (h : Disjoint M N)
{g k : Equiv.Perm X} (hg : g ∈ blockSubgroup M) (hk : k ∈ blockSubgroup N) :
Commute g k := by
apply Equiv.Perm.Disjoint.commute
intro x
by_cases hx : x ∈ M
· right
exact hk x (fun hn => Set.disjoint_left.mp h hx hn)
· exact Or.inl (hg x hx)
lemma jointFactors_generate {G : Type*} [Group G] {b : ℕ} (H : Fin b → Subgroup G) :
⨆ i, (H i).subgroupOf (⨆ j, H j) = ⊤ := by
apply Subgroup.map_subtype_inj.mp
rw [Subgroup.map_iSup]
simp only [Subgroup.map_subgroupOf_eq_of_le (le_iSup H _),
← MonoidHom.range_eq_map, Subgroup.range_subtype]
lemma jointFactors_commute {G : Type*} [Group G] {b : ℕ} (H : Fin b → Subgroup G)
(h : ∀ i j, i ≠ j → ∀ g ∈ H i, ∀ k ∈ H j, Commute g k) :
∀ i j, i ≠ j → ∀ g ∈ (H i).subgroupOf (⨆ j, H j),
∀ k ∈ (H j).subgroupOf (⨆ j, H j), Commute g k := by
intro i j hij g hg k hk
apply Subtype.ext
exact h i j hij g hg k hk
end PartialPermutation
end
end OAI
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