509 lines
18 KiB
Text
509 lines
18 KiB
Text
/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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Basic group theory
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This file will be rewritten in the future, when we develop are more systematic notation for
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describing homomorphisms
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-/
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import algebra.category.category algebra.bundled
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open eq algebra pointed function is_trunc pi equiv is_equiv
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set_option class.force_new true
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namespace group
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definition pointed_Group [instance] [constructor] (G : Group) : pointed G :=
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pointed.mk 1
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definition Group.struct' [instance] [reducible] (G : Group) : group G :=
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Group.struct G
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definition comm_group_Group_of_CommGroup [instance] [constructor] [priority 900]
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(G : CommGroup) : comm_group (Group_of_CommGroup G) :=
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begin esimp, exact _ end
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definition comm_group_pSet_of_Group [instance] (G : CommGroup) : comm_group (pSet_of_Group G) :=
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CommGroup.struct G
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definition group_pSet_of_Group [instance] [priority 900] (G : Group) :
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group (pSet_of_Group G) :=
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Group.struct G
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/- group homomorphisms -/
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definition is_homomorphism [class] [reducible]
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{G₁ G₂ : Type} [has_mul G₁] [has_mul G₂] (φ : G₁ → G₂) : Type :=
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Π(g h : G₁), φ (g * h) = φ g * φ h
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section
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variables {G G₁ G₂ G₃ : Type} {g h : G₁} (ψ : G₂ → G₃) {φ₁ φ₂ : G₁ → G₂} (φ : G₁ → G₂)
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[group G] [group G₁] [group G₂] [group G₃]
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[is_homomorphism ψ] [is_homomorphism φ₁] [is_homomorphism φ₂] [is_homomorphism φ]
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definition respect_mul {G₁ G₂ : Type} [has_mul G₁] [has_mul G₂] (φ : G₁ → G₂)
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[is_homomorphism φ] : Π(g h : G₁), φ (g * h) = φ g * φ h :=
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by assumption
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theorem respect_one /- φ -/ : φ 1 = 1 :=
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mul.right_cancel
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(calc
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φ 1 * φ 1 = φ (1 * 1) : respect_mul φ
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... = φ 1 : ap φ !one_mul
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... = 1 * φ 1 : one_mul)
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theorem respect_inv /- φ -/ (g : G₁) : φ g⁻¹ = (φ g)⁻¹ :=
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eq_inv_of_mul_eq_one (!respect_mul⁻¹ ⬝ ap φ !mul.left_inv ⬝ !respect_one)
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definition is_embedding_homomorphism /- φ -/ (H : Π{g}, φ g = 1 → g = 1) : is_embedding φ :=
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begin
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apply function.is_embedding_of_is_injective,
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intro g g' p,
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apply eq_of_mul_inv_eq_one,
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apply H,
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refine !respect_mul ⬝ _,
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rewrite [respect_inv φ, p],
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apply mul.right_inv
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end
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definition is_homomorphism_compose {ψ : G₂ → G₃} {φ : G₁ → G₂}
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(H1 : is_homomorphism ψ) (H2 : is_homomorphism φ) : is_homomorphism (ψ ∘ φ) :=
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λg h, ap ψ !respect_mul ⬝ !respect_mul
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definition is_homomorphism_id (G : Type) [group G] : is_homomorphism (@id G) :=
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λg h, idp
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end
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section additive
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definition is_add_homomorphism [class] [reducible] {G₁ G₂ : Type} [has_add G₁] [has_add G₂]
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(φ : G₁ → G₂) : Type :=
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Π(g h : G₁), φ (g + h) = φ g + φ h
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variables {G₁ G₂ : Type} (φ : G₁ → G₂) [add_group G₁] [add_group G₂] [is_add_homomorphism φ]
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definition respect_add /- φ -/ : Π(g h : G₁), φ (g + h) = φ g + φ h :=
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by assumption
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theorem respect_zero /- φ -/ : φ 0 = 0 :=
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add.right_cancel
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(calc
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φ 0 + φ 0 = φ (0 + 0) : respect_add φ
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... = φ 0 : ap φ !zero_add
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... = 0 + φ 0 : zero_add)
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theorem respect_neg /- φ -/ (g : G₁) : φ (-g) = -(φ g) :=
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eq_neg_of_add_eq_zero (!respect_add⁻¹ ⬝ ap φ !add.left_inv ⬝ !respect_zero)
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end additive
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structure homomorphism (G₁ G₂ : Group) : Type :=
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(φ : G₁ → G₂)
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(p : is_homomorphism φ)
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infix ` →g `:55 := homomorphism
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definition group_fun [unfold 3] [coercion] := @homomorphism.φ
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definition homomorphism.struct [unfold 3] [instance] [priority 900] {G₁ G₂ : Group}
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(φ : G₁ →g G₂) : is_homomorphism φ :=
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homomorphism.p φ
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definition homomorphism.mulstruct [instance] [priority 2000] {G₁ G₂ : Group} (φ : G₁ →g G₂)
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: is_homomorphism φ :=
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homomorphism.p φ
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definition homomorphism.addstruct [instance] [priority 2000] {G₁ G₂ : AddGroup} (φ : G₁ →g G₂)
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: is_add_homomorphism φ :=
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homomorphism.p φ
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variables {G G₁ G₂ G₃ : Group} {g h : G₁} {ψ : G₂ →g G₃} {φ₁ φ₂ : G₁ →g G₂} (φ : G₁ →g G₂)
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definition to_respect_mul /- φ -/ (g h : G₁) : φ (g * h) = φ g * φ h :=
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respect_mul φ g h
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theorem to_respect_one /- φ -/ : φ 1 = 1 :=
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respect_one φ
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theorem to_respect_inv /- φ -/ (g : G₁) : φ g⁻¹ = (φ g)⁻¹ :=
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respect_inv φ g
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definition to_is_embedding_homomorphism /- φ -/ (H : Π{g}, φ g = 1 → g = 1) : is_embedding φ :=
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is_embedding_homomorphism φ @H
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variables (G₁ G₂)
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definition is_set_homomorphism [instance] : is_set (G₁ →g G₂) :=
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begin
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have H : G₁ →g G₂ ≃ Σ(f : G₁ → G₂), Π(g₁ g₂ : G₁), f (g₁ * g₂) = f g₁ * f g₂,
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begin
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fapply equiv.MK,
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{ intro φ, induction φ, constructor, assumption},
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{ intro v, induction v, constructor, assumption},
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{ intro v, induction v, reflexivity},
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{ intro φ, induction φ, reflexivity}
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end,
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apply is_trunc_equiv_closed_rev, exact H
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end
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variables {G₁ G₂}
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definition pmap_of_homomorphism [constructor] /- φ -/ : G₁ →* G₂ :=
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pmap.mk φ begin esimp, exact respect_one φ end
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definition homomorphism_change_fun [constructor] {G₁ G₂ : Group}
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(φ : G₁ →g G₂) (f : G₁ → G₂) (p : φ ~ f) : G₁ →g G₂ :=
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homomorphism.mk f (λg h, (p (g * h))⁻¹ ⬝ to_respect_mul φ g h ⬝ ap011 mul (p g) (p h))
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definition homomorphism_eq (p : group_fun φ₁ ~ group_fun φ₂) : φ₁ = φ₂ :=
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begin
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induction φ₁ with φ₁ q₁, induction φ₂ with φ₂ q₂, esimp at p, induction p,
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exact ap (homomorphism.mk φ₁) !is_prop.elim
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end
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section additive
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variables {H₁ H₂ : AddGroup} (χ : H₁ →g H₂)
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definition to_respect_add /- χ -/ (g h : H₁) : χ (g + h) = χ g + χ h :=
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respect_add χ g h
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theorem to_respect_zero /- χ -/ : χ 0 = 0 :=
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respect_zero χ
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theorem to_respect_neg /- χ -/ (g : H₁) : χ (-g) = -(χ g) :=
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respect_neg χ g
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end additive
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section add_mul
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variables {H₁ : AddGroup} {H₂ : Group} (χ : H₁ →g H₂)
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definition to_respect_add_mul /- χ -/ (g h : H₁) : χ (g + h) = χ g * χ h :=
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to_respect_mul χ g h
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theorem to_respect_zero_one /- χ -/ : χ 0 = 1 :=
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to_respect_one χ
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theorem to_respect_neg_inv /- χ -/ (g : H₁) : χ (-g) = (χ g)⁻¹ :=
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to_respect_inv χ g
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end add_mul
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section mul_add
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variables {H₁ : Group} {H₂ : AddGroup} (χ : H₁ →g H₂)
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definition to_respect_mul_add /- χ -/ (g h : H₁) : χ (g * h) = χ g + χ h :=
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to_respect_mul χ g h
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theorem to_respect_one_zero /- χ -/ : χ 1 = 0 :=
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to_respect_one χ
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theorem to_respect_inv_neg /- χ -/ (g : H₁) : χ g⁻¹ = -(χ g) :=
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to_respect_inv χ g
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end mul_add
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/- categorical structure of groups + homomorphisms -/
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definition homomorphism_compose [constructor] [trans] (ψ : G₂ →g G₃) (φ : G₁ →g G₂) : G₁ →g G₃ :=
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homomorphism.mk (ψ ∘ φ) (is_homomorphism_compose _ _)
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variable (G)
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definition homomorphism_id [constructor] [refl] : G →g G :=
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homomorphism.mk (@id G) (is_homomorphism_id G)
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variable {G}
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abbreviation gid [constructor] := @homomorphism_id
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infixr ` ∘g `:75 := homomorphism_compose
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notation 1 := homomorphism_id _
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structure isomorphism (A B : Group) :=
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(to_hom : A →g B)
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(is_equiv_to_hom : is_equiv to_hom)
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infix ` ≃g `:25 := isomorphism
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attribute isomorphism.to_hom [coercion]
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attribute isomorphism.is_equiv_to_hom [instance]
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attribute isomorphism._trans_of_to_hom [unfold 3]
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definition equiv_of_isomorphism [constructor] (φ : G₁ ≃g G₂) : G₁ ≃ G₂ :=
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equiv.mk φ _
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definition pequiv_of_isomorphism [constructor] (φ : G₁ ≃g G₂) :
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G₁ ≃* G₂ :=
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pequiv.mk φ begin esimp, exact _ end begin esimp, exact respect_one φ end
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definition isomorphism_of_equiv [constructor] (φ : G₁ ≃ G₂)
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(p : Πg₁ g₂, φ (g₁ * g₂) = φ g₁ * φ g₂) : G₁ ≃g G₂ :=
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isomorphism.mk (homomorphism.mk φ p) !to_is_equiv
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definition isomorphism_of_eq [constructor] {G₁ G₂ : Group} (φ : G₁ = G₂) : G₁ ≃g G₂ :=
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isomorphism_of_equiv (equiv_of_eq (ap Group.carrier φ))
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begin intros, induction φ, reflexivity end
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definition pequiv_of_isomorphism_of_eq {G₁ G₂ : Group} (p : G₁ = G₂) :
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pequiv_of_isomorphism (isomorphism_of_eq p) = pequiv_of_eq (ap pType_of_Group p) :=
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begin
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induction p,
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apply pequiv_eq,
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fapply pmap_eq,
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{ intro g, reflexivity},
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{ apply is_prop.elim}
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end
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definition to_ginv [constructor] (φ : G₁ ≃g G₂) : G₂ →g G₁ :=
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homomorphism.mk φ⁻¹
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abstract begin
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intro g₁ g₂, apply eq_of_fn_eq_fn' φ,
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rewrite [respect_mul φ, +right_inv φ]
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end end
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variable (G)
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definition isomorphism.refl [refl] [constructor] : G ≃g G :=
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isomorphism.mk 1 !is_equiv_id
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variable {G}
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definition isomorphism.symm [symm] [constructor] (φ : G₁ ≃g G₂) : G₂ ≃g G₁ :=
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isomorphism.mk (to_ginv φ) !is_equiv_inv
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definition isomorphism.trans [trans] [constructor] (φ : G₁ ≃g G₂) (ψ : G₂ ≃g G₃) : G₁ ≃g G₃ :=
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isomorphism.mk (ψ ∘g φ) !is_equiv_compose
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definition isomorphism.eq_trans [trans] [constructor]
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{G₁ G₂ : Group} {G₃ : Group} (φ : G₁ = G₂) (ψ : G₂ ≃g G₃) : G₁ ≃g G₃ :=
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proof isomorphism.trans (isomorphism_of_eq φ) ψ qed
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definition isomorphism.trans_eq [trans] [constructor]
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{G₁ : Group} {G₂ G₃ : Group} (φ : G₁ ≃g G₂) (ψ : G₂ = G₃) : G₁ ≃g G₃ :=
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isomorphism.trans φ (isomorphism_of_eq ψ)
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postfix `⁻¹ᵍ`:(max + 1) := isomorphism.symm
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infixl ` ⬝g `:75 := isomorphism.trans
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infixl ` ⬝gp `:75 := isomorphism.trans_eq
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infixl ` ⬝pg `:75 := isomorphism.eq_trans
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definition pmap_of_isomorphism [constructor] (φ : G₁ ≃g G₂) :
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G₁ →* G₂ :=
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pequiv_of_isomorphism φ
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/- category of groups -/
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section
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open category
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definition precategory_group [constructor] : precategory Group :=
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precategory.mk homomorphism
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@homomorphism_compose
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@homomorphism_id
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(λG₁ G₂ G₃ G₄ φ₃ φ₂ φ₁, homomorphism_eq (λg, idp))
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(λG₁ G₂ φ, homomorphism_eq (λg, idp))
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(λG₁ G₂ φ, homomorphism_eq (λg, idp))
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end
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-- TODO
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-- definition category_group : category Group :=
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-- category.mk precategory_group
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-- begin
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-- intro G₁ G₂,
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-- fapply adjointify,
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-- { intro φ, fapply Group_eq, },
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-- { },
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-- { }
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-- end
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/- given an equivalence A ≃ B we can transport a group structure on A to a group structure on B -/
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section
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parameters {A B : Type} (f : A ≃ B) [group A]
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definition group_equiv_mul (b b' : B) : B := f (f⁻¹ᶠ b * f⁻¹ᶠ b')
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definition group_equiv_one : B := f one
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definition group_equiv_inv (b : B) : B := f (f⁻¹ᶠ b)⁻¹
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local infix * := group_equiv_mul
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local postfix ^ := group_equiv_inv
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local notation 1 := group_equiv_one
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theorem group_equiv_mul_assoc (b₁ b₂ b₃ : B) : (b₁ * b₂) * b₃ = b₁ * (b₂ * b₃) :=
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by rewrite [↑group_equiv_mul, +left_inv f, mul.assoc]
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theorem group_equiv_one_mul (b : B) : 1 * b = b :=
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by rewrite [↑group_equiv_mul, ↑group_equiv_one, left_inv f, one_mul, right_inv f]
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theorem group_equiv_mul_one (b : B) : b * 1 = b :=
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by rewrite [↑group_equiv_mul, ↑group_equiv_one, left_inv f, mul_one, right_inv f]
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theorem group_equiv_mul_left_inv (b : B) : b^ * b = 1 :=
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by rewrite [↑group_equiv_mul, ↑group_equiv_one, ↑group_equiv_inv,
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+left_inv f, mul.left_inv]
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definition group_equiv_closed : group B :=
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⦃group,
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mul := group_equiv_mul,
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mul_assoc := group_equiv_mul_assoc,
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one := group_equiv_one,
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one_mul := group_equiv_one_mul,
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mul_one := group_equiv_mul_one,
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inv := group_equiv_inv,
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mul_left_inv := group_equiv_mul_left_inv,
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is_set_carrier := is_trunc_equiv_closed 0 f⦄
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end
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variable (G)
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/- the trivial group -/
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open unit
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definition trivial_group [constructor] : group unit :=
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group.mk (λx y, star) _ (λx y z, idp) star (unit.rec idp) (unit.rec idp) (λx, star) (λx, idp)
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definition Trivial_group [constructor] : Group :=
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Group.mk _ trivial_group
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abbreviation G0 := Trivial_group
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definition trivial_group_of_is_contr [H : is_contr G] : G ≃g G0 :=
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begin
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fapply isomorphism_of_equiv,
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{ apply equiv_unit_of_is_contr},
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{ intros, reflexivity}
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end
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variable {G}
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/-
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A group where the point in the pointed type corresponds with 1 in the group.
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We need this structure when we are given a pointed type, and want to say that there is a group
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structure on it which is compatible with the point. This is used in chain complexes.
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-/
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structure pgroup [class] (X : Type*) extends semigroup X, has_inv X :=
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(pt_mul : Πa, mul pt a = a)
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(mul_pt : Πa, mul a pt = a)
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(mul_left_inv_pt : Πa, mul (inv a) a = pt)
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definition group_of_pgroup [reducible] [instance] (X : Type*) [H : pgroup X]
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: group X :=
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⦃group, H,
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one := pt,
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one_mul := pgroup.pt_mul ,
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mul_one := pgroup.mul_pt,
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mul_left_inv := pgroup.mul_left_inv_pt⦄
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definition pgroup_of_group (X : Type*) [H : group X] (p : one = pt :> X) : pgroup X :=
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begin
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cases X with X x, esimp at *, induction p,
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exact ⦃pgroup, H,
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pt_mul := one_mul,
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mul_pt := mul_one,
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mul_left_inv_pt := mul.left_inv⦄
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end
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definition Group_of_pgroup (G : Type*) [pgroup G] : Group :=
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Group.mk G _
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definition pgroup_Group [instance] (G : Group) : pgroup G :=
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⦃ pgroup, Group.struct G,
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pt_mul := one_mul,
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mul_pt := mul_one,
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mul_left_inv_pt := mul.left_inv ⦄
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/- equality of groups and abelian groups -/
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definition group.to_has_mul {A : Type} (H : group A) : has_mul A := _
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definition group.to_has_inv {A : Type} (H : group A) : has_inv A := _
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definition group.to_has_one {A : Type} (H : group A) : has_one A := _
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local attribute group.to_has_mul group.to_has_inv [coercion]
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universe variable l
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variables {A B : Type.{l}}
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definition group_eq {G H : group A} (same_mul' : Π(g h : A), @mul A G g h = @mul A H g h)
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: G = H :=
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begin
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have foo : Π(g : A), @inv A G g = (@inv A G g * g) * @inv A H g,
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from λg, !mul_inv_cancel_right⁻¹,
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cases G with Gm Gs Gh1 G1 Gh2 Gh3 Gi Gh4,
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cases H with Hm Hs Hh1 H1 Hh2 Hh3 Hi Hh4,
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rewrite [↑[semigroup.to_has_mul,group.to_has_inv] at (same_mul',foo)],
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have same_mul : Gm = Hm, from eq_of_homotopy2 same_mul',
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cases same_mul,
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have same_one : G1 = H1, from calc
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G1 = Hm G1 H1 : Hh3
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... = H1 : Gh2,
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have same_inv : Gi = Hi, from eq_of_homotopy (take g, calc
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Gi g = Hm (Hm (Gi g) g) (Hi g) : foo
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... = Hm G1 (Hi g) : by rewrite Gh4
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... = Hi g : Gh2),
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cases same_one, cases same_inv,
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have ps : Gs = Hs, from !is_prop.elim,
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have ph1 : Gh1 = Hh1, from !is_prop.elim,
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have ph2 : Gh2 = Hh2, from !is_prop.elim,
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have ph3 : Gh3 = Hh3, from !is_prop.elim,
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have ph4 : Gh4 = Hh4, from !is_prop.elim,
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cases ps, cases ph1, cases ph2, cases ph3, cases ph4, reflexivity
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end
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definition group_pathover {G : group A} {H : group B} {p : A = B}
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(resp_mul : Π(g h : A), cast p (g * h) = cast p g * cast p h) : G =[p] H :=
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begin
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induction p,
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apply pathover_idp_of_eq, exact group_eq (resp_mul)
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end
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definition Group_eq_of_eq {G H : Group} (p : Group.carrier G = Group.carrier H)
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(resp_mul : Π(g h : G), cast p (g * h) = cast p g * cast p h) : G = H :=
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begin
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cases G with Gc G, cases H with Hc H,
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apply (apd011 Group.mk p),
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exact group_pathover resp_mul
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end
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definition Group_eq {G H : Group} (f : Group.carrier G ≃ Group.carrier H)
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(resp_mul : Π(g h : G), f (g * h) = f g * f h) : G = H :=
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Group_eq_of_eq (ua f) (λg h, !cast_ua ⬝ resp_mul g h ⬝ ap011 mul !cast_ua⁻¹ !cast_ua⁻¹)
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definition eq_of_isomorphism {G₁ G₂ : Group} (φ : G₁ ≃g G₂) : G₁ = G₂ :=
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Group_eq (equiv_of_isomorphism φ) (respect_mul φ)
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definition comm_group.to_has_mul {A : Type} (H : comm_group A) : has_mul A := _
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local attribute comm_group.to_has_mul [coercion]
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definition comm_group_eq {A : Type} {G H : comm_group A}
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(same_mul : Π(g h : A), @mul A G g h = @mul A H g h)
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: G = H :=
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begin
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have g_eq : @comm_group.to_group A G = @comm_group.to_group A H, from group_eq same_mul,
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cases G with Gm Gs Gh1 G1 Gh2 Gh3 Gi Gh4 Gh5,
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cases H with Hm Hs Hh1 H1 Hh2 Hh3 Hi Hh4 Hh5,
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have pm : Gm = Hm, from ap (@mul _ ∘ group.to_has_mul) g_eq,
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have pi : Gi = Hi, from ap (@inv _ ∘ group.to_has_inv) g_eq,
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have p1 : G1 = H1, from ap (@one _ ∘ group.to_has_one) g_eq,
|
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induction pm, induction pi, induction p1,
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have ps : Gs = Hs, from !is_prop.elim,
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have ph1 : Gh1 = Hh1, from !is_prop.elim,
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have ph2 : Gh2 = Hh2, from !is_prop.elim,
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have ph3 : Gh3 = Hh3, from !is_prop.elim,
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have ph4 : Gh4 = Hh4, from !is_prop.elim,
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have ph5 : Gh5 = Hh5, from !is_prop.elim,
|
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induction ps, induction ph1, induction ph2, induction ph3, induction ph4, induction ph5,
|
|
reflexivity
|
|
end
|
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|
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definition comm_group_pathover {A B : Type} {G : comm_group A} {H : comm_group B} {p : A = B}
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(resp_mul : Π(g h : A), cast p (g * h) = cast p g * cast p h) : G =[p] H :=
|
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begin
|
|
induction p,
|
|
apply pathover_idp_of_eq, exact comm_group_eq (resp_mul)
|
|
end
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|
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definition CommGroup_eq_of_isomorphism {G₁ G₂ : CommGroup} (φ : G₁ ≃g G₂) : G₁ = G₂ :=
|
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begin
|
|
induction G₁, induction G₂,
|
|
apply apd011 CommGroup.mk (ua (equiv_of_isomorphism φ)),
|
|
apply comm_group_pathover,
|
|
intro g h, exact !cast_ua ⬝ respect_mul φ g h ⬝ ap011 mul !cast_ua⁻¹ !cast_ua⁻¹
|
|
end
|
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definition trivial_group_of_is_contr' (G : Group) [H : is_contr G] : G = G0 :=
|
|
eq_of_isomorphism (trivial_group_of_is_contr G)
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end group
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