@@ -113,47 +113,41 @@ theorem normal_iff_eq_cosets : normal_subgroup s ↔ ∀ g, g *l s = s *r g :=
113113
114114end coset_subgroup
115115
116- def left_cosets [group α] (s : set α) : set (set α) := range (λa, a *l s)
116+ instance left_rel [group α] (s : set α) [is_subgroup s] : setoid α :=
117+ ⟨λ x y, x⁻¹ * y ∈ s,
118+ assume x, by simp [is_submonoid.one_mem],
119+ assume x y hxy,
120+ have (x⁻¹ * y)⁻¹ ∈ s, from is_subgroup.inv_mem hxy,
121+ by simpa using this ,
122+ assume x y z hxy hyz,
123+ have x⁻¹ * y * (y⁻¹ * z) ∈ s, from is_submonoid.mul_mem hxy hyz,
124+ by simpa [mul_assoc] using this ⟩
125+
126+ def left_cosets [group α] (s : set α) [is_subgroup s] : Type * := quotient (left_rel s)
127+
128+ instance left_cosets.inhabited [group α] (s : set α) [is_subgroup s] : inhabited (left_cosets s) := ⟨⟦1 ⟧⟩
129+
130+ def left_cosets.left_coset [group α] (s : set α) [is_subgroup s] (g : α) : {x | ⟦x⟧ = ⟦g⟧} = left_coset g s :=
131+ set.ext $ λ z, by simp [eq_comm, mem_left_coset_iff]; refl
117132
118133namespace is_subgroup
119- open is_submonoid
120- variable [group α]
121-
122- lemma subgroup_mem_left_cosets (s : set α) [is_subgroup s] : s ∈ left_cosets s :=
123- ⟨1 , by simp⟩
124-
125- lemma left_cosets_disjoint {s : set α} [is_subgroup s] :
126- ∀{s₁ s₂ : set α}, s₁ ∈ left_cosets s → s₂ ∈ left_cosets s → ∀{a}, a ∈ s₁ → a ∈ s₂ → s₁ = s₂
127- | _ _ ⟨a₁, rfl⟩ ⟨a₂, rfl⟩ a h₁ h₂ :=
128- have h₁ : a₁⁻¹ * a ∈ s, by simpa [mem_left_coset_iff] using h₁,
129- have h₂ : a₂⁻¹ * a ∈ s, by simpa [mem_left_coset_iff] using h₂,
130- have a₁⁻¹ * a₂ ∈ s, by simpa [mul_assoc] using mul_mem h₁ (inv_mem h₂),
131- have a₁ *l s = a₁ *l ((a₁⁻¹ * a₂) *l s), by rw [left_coset_mem_left_coset _ this ],
132- by simpa
133-
134- lemma pairwise_left_cosets_disjoint {s : set α} (hs : is_subgroup s) :
135- pairwise_on (left_cosets s) disjoint :=
136- assume s₁ h₁ s₂ h₂ ne, eq_empty_iff_forall_not_mem.mpr $ assume a ⟨ha₁, ha₂⟩,
137- ne $ left_cosets_disjoint h₁ h₂ ha₁ ha₂
138-
139- lemma left_cosets_equiv_subgroup {s : set α} (hs : is_subgroup s) :
140- ∀{t : set α}, t ∈ left_cosets s → nonempty (t ≃ s)
141- | _ ⟨a, rfl⟩ := ⟨(equiv.set.image ((*) a) s injective_mul).symm⟩
142-
143- lemma Union_left_cosets_eq_univ {s : set α} (hs : is_subgroup s) : ⋃₀ left_cosets s = univ :=
144- eq_univ_of_forall $ assume a, ⟨(*) a '' s, mem_range_self _, ⟨1 , hs.one_mem, mul_one _⟩⟩
145-
146- lemma group_equiv_left_cosets_times_subgroup {s : set α} (hs : is_subgroup s) :
147- nonempty (α ≃ (left_cosets s × s)) :=
148- ⟨calc α ≃ (@set.univ α) :
149- (equiv.set.univ α).symm
150- ... ≃ (⋃t∈left_cosets s, t) :
151- by rw [←hs.Union_left_cosets_eq_univ]; simp
152- ... ≃ (Σt:left_cosets s, t) :
153- equiv.set.bUnion_eq_sigma_of_disjoint hs.pairwise_left_cosets_disjoint
154- ... ≃ (Σt:left_cosets s, s) :
155- equiv.sigma_congr_right $ λ⟨t, ht⟩, classical.choice $ hs.left_cosets_equiv_subgroup ht
156- ... ≃ (left_cosets s × s) :
157- equiv.sigma_equiv_prod _ _⟩
134+ variables [group α] {s : set α}
135+
136+ def left_coset_equiv_subgroup (g : α) : left_coset g s ≃ s :=
137+ ⟨λ x, ⟨g⁻¹ * x.1 , (mem_left_coset_iff _).1 x.2 ⟩,
138+ λ x, ⟨g * x.1 , x.1 , x.2 , rfl⟩,
139+ λ ⟨x, hx⟩, subtype.eq $ by simp,
140+ λ ⟨g, hg⟩, subtype.eq $ by simp⟩
141+
142+ noncomputable def group_equiv_left_cosets_times_subgroup (hs : is_subgroup s) :
143+ α ≃ (left_cosets s × s) :=
144+ calc α ≃ Σ L : left_cosets s, {x // ⟦x⟧ = L} :
145+ equiv.equiv_fib quotient.mk
146+ ... ≃ Σ L : left_cosets s, left_coset (quotient.out L) s :
147+ equiv.sigma_congr_right (λ L, by rw ← left_cosets.left_coset; simp)
148+ ... ≃ Σ L : left_cosets s, s :
149+ equiv.sigma_congr_right (λ L, left_coset_equiv_subgroup _)
150+ ... ≃ (left_cosets s × s) :
151+ equiv.sigma_equiv_prod _ _
158152
159153end is_subgroup
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