74 lines
2.4 KiB
Text
74 lines
2.4 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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-/
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/-
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Groups are defined in the HoTT library in algebra/group, as part of the algebraic hierarchy.
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However, there is currently no group theory.
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The only relevant defintions are the trivial group (in types/unit) and some files in algebra/
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-/
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import algebra.group types.pointed types.pi
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open eq algebra pointed function is_trunc pi
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namespace group
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definition pointed_Group [instance] (G : Group) : pointed G := pointed.mk one
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definition Pointed_of_Group (G : Group) : Type* := pointed.mk' G
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definition Group_of_CommGroup [coercion] [constructor] (G : CommGroup) : Group :=
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Group.mk G _
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definition comm_group_Group_of_CommGroup [instance] [constructor] (G : CommGroup)
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: comm_group (Group_of_CommGroup G) :=
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begin esimp, exact _ end
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/- group homomorphisms -/
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structure homomorphism (G₁ G₂ : Group) : Type :=
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(φ : G₁ → G₂)
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(p : Π(g h : G₁), φ (g * h) = φ g * φ h)
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attribute homomorphism.φ [coercion]
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abbreviation group_fun [unfold 3] := @homomorphism.φ
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abbreviation respect_mul := @homomorphism.p
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infix ` →g `:55 := homomorphism
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variables {G₁ G₂ G₃ : Group} {g h : G₁} {ψ : G₂ →g G₃} {φ φ' : G₁ →g G₂}
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theorem respect_one (φ : G₁ →g G₂) : 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
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... = φ 1 : ap φ !one_mul
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... = 1 * φ 1 : one_mul)
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local attribute Pointed_of_Group [coercion]
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definition pmap_of_homomorphism [constructor] (φ : G₁ →g G₂) : G₁ →* G₂ :=
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pmap.mk φ !respect_one
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definition homomorphism_eq (p : group_fun φ = group_fun φ') : φ = φ' :=
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begin
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induction φ with φ q, induction φ' with φ' q', esimp at p, induction p,
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exact ap (homomorphism.mk φ) !is_hprop.elim
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end
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/- categorical structure of groups + homomorphisms -/
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definition homomorphism_compose [constructor] (ψ : G₂ →g G₃) (φ : G₁ →g G₂) : G₁ → G₃ :=
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homomorphism.mk (ψ ∘ φ) (λg h, ap ψ !respect_mul ⬝ !respect_mul)
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definition homomorphism_id [constructor] (G : Group) : G → G :=
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homomorphism.mk id (λg h, idp)
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-- TODO: maybe define this in more generality for pointed types?
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definition kernel [constructor] (φ : G₁ →g G₂) (g : G₁) : hprop := trunctype.mk (φ g = 1) _
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end group
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