192 lines
6.9 KiB
Text
192 lines
6.9 KiB
Text
/-
|
|
Copyright (c) 2015 Haitao Zhang. All rights reserved.
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
|
|
|
Author : Haitao Zhang
|
|
-/
|
|
import algebra.group data.set .subgroup
|
|
|
|
namespace group_theory
|
|
|
|
-- ⁻¹ in eq.ops conflicts with group ⁻¹
|
|
-- open eq.ops
|
|
notation H1 ▸ H2 := eq.subst H1 H2
|
|
open set
|
|
open function
|
|
open group_theory.ops
|
|
open quot
|
|
local attribute set [reducible]
|
|
|
|
section defs
|
|
|
|
variables {A B : Type}
|
|
variable [s1 : group A]
|
|
variable [s2 : group B]
|
|
include s1
|
|
include s2
|
|
|
|
-- the Prop of being hom
|
|
definition homomorphic [reducible] (f : A → B) : Prop := ∀ a b, f (a*b) = (f a)*(f b)
|
|
-- type class for inference
|
|
structure is_hom_class [class] (f : A → B) : Type :=
|
|
(is_hom : homomorphic f)
|
|
-- the proof of hom_prop if the class can be inferred
|
|
definition is_hom (f : A → B) [h : is_hom_class f] : homomorphic f :=
|
|
@is_hom_class.is_hom A B s1 s2 f h
|
|
|
|
definition ker (f : A → B) [h : is_hom_class f] : set A := {a : A | f a = 1}
|
|
|
|
definition isomorphic (f : A → B) := injective f ∧ homomorphic f
|
|
structure is_iso_class [class] (f : A → B) extends is_hom_class f : Type :=
|
|
(inj : injective f)
|
|
lemma iso_is_inj (f : A → B) [h : is_iso_class f] : injective f:=
|
|
@is_iso_class.inj A B s1 s2 f h
|
|
lemma iso_is_iso (f : A → B) [h : is_iso_class f] : isomorphic f:=
|
|
and.intro (iso_is_inj f) (is_hom f)
|
|
|
|
end defs
|
|
|
|
section
|
|
variables {A B : Type}
|
|
variable [s1 : group A]
|
|
|
|
definition id_is_iso [instance] : @is_hom_class A A s1 s1 (@id A) :=
|
|
is_hom_class.mk (take a b, rfl)
|
|
|
|
variable [s2 : group B]
|
|
include s1
|
|
include s2
|
|
|
|
variable f : A → B
|
|
|
|
variable [h : is_hom_class f]
|
|
include h
|
|
|
|
theorem hom_map_one : f 1 = 1 :=
|
|
have P : f 1 = (f 1) * (f 1), from
|
|
calc f 1 = f (1*1) : mul_one
|
|
... = (f 1) * (f 1) : is_hom f,
|
|
eq.symm (mul.right_inv (f 1) ▸ (mul_inv_eq_of_eq_mul P))
|
|
|
|
theorem hom_map_inv (a : A) : f a⁻¹ = (f a)⁻¹ :=
|
|
assert P : f 1 = 1, from hom_map_one f,
|
|
assert P1 : f (a⁻¹ * a) = 1, from (eq.symm (mul.left_inv a)) ▸ P,
|
|
assert P2 : (f a⁻¹) * (f a) = 1, from (is_hom f a⁻¹ a) ▸ P1,
|
|
assert P3 : (f a⁻¹) * (f a) = (f a)⁻¹ * (f a), from eq.symm (mul.left_inv (f a)) ▸ P2,
|
|
mul_right_cancel P3
|
|
|
|
theorem hom_map_mul_closed (H : set A) : mul_closed_on H → mul_closed_on (f '[H]) :=
|
|
assume Pclosed, assume b1, assume b2,
|
|
assume Pb1 : b1 ∈ f '[ H], assume Pb2 : b2 ∈ f '[ H],
|
|
obtain a1 (Pa1 : a1 ∈ H ∧ f a1 = b1), from Pb1,
|
|
obtain a2 (Pa2 : a2 ∈ H ∧ f a2 = b2), from Pb2,
|
|
assert Pa1a2 : a1 * a2 ∈ H, from Pclosed a1 a2 (and.left Pa1) (and.left Pa2),
|
|
assert Pb1b2 : f (a1 * a2) = b1 * b2, from calc
|
|
f (a1 * a2) = f a1 * f a2 : is_hom f a1 a2
|
|
... = b1 * f a2 : {and.right Pa1}
|
|
... = b1 * b2 : {and.right Pa2},
|
|
mem_image Pa1a2 Pb1b2
|
|
|
|
lemma ker.has_one : 1 ∈ ker f := hom_map_one f
|
|
|
|
lemma ker.has_inv : subgroup.has_inv (ker f) :=
|
|
take a, assume Pa : f a = 1, calc
|
|
f a⁻¹ = (f a)⁻¹ : by rewrite (hom_map_inv f)
|
|
... = 1⁻¹ : by rewrite Pa
|
|
... = 1 : by rewrite one_inv
|
|
|
|
lemma ker.mul_closed : mul_closed_on (ker f) :=
|
|
take x y, assume (Px : f x = 1) (Py : f y = 1), calc
|
|
f (x*y) = (f x) * (f y) : by rewrite [is_hom f]
|
|
... = 1 : by rewrite [Px, Py, mul_one]
|
|
|
|
lemma ker.normal : same_left_right_coset (ker f) :=
|
|
take a, funext (assume x, begin
|
|
esimp [ker, set_of, glcoset, grcoset],
|
|
rewrite [*(is_hom f), mul_eq_one_iff_mul_eq_one (f a⁻¹) (f x)]
|
|
end)
|
|
|
|
definition ker_is_normal_subgroup : is_normal_subgroup (ker f) :=
|
|
is_normal_subgroup.mk (ker.has_one f) (ker.mul_closed f) (ker.has_inv f)
|
|
(ker.normal f)
|
|
|
|
-- additional subgroup variable
|
|
variable {H : set A}
|
|
variable [is_subgH : is_subgroup H]
|
|
include is_subgH
|
|
|
|
theorem hom_map_subgroup : is_subgroup (f '[H]) :=
|
|
have Pone : 1 ∈ f '[H], from mem_image (@subg_has_one _ _ H _) (hom_map_one f),
|
|
have Pclosed : mul_closed_on (f '[H]), from hom_map_mul_closed f H subg_mul_closed,
|
|
assert Pinv : ∀ b, b ∈ f '[H] → b⁻¹ ∈ f '[H], from
|
|
assume b, assume Pimg,
|
|
obtain a (Pa : a ∈ H ∧ f a = b), from Pimg,
|
|
assert Painv : a⁻¹ ∈ H, from subg_has_inv a (and.left Pa),
|
|
assert Pfainv : (f a)⁻¹ ∈ f '[H], from mem_image Painv (hom_map_inv f a),
|
|
and.right Pa ▸ Pfainv,
|
|
is_subgroup.mk Pone Pclosed Pinv
|
|
end
|
|
|
|
section hom_theorem
|
|
|
|
variables {A B : Type}
|
|
variable [s1 : group A]
|
|
variable [s2 : group B]
|
|
include s1
|
|
include s2
|
|
|
|
variable {f : A → B}
|
|
|
|
variable [h : is_hom_class f]
|
|
include h
|
|
|
|
definition ker_nsubg [instance] : is_normal_subgroup (ker f) :=
|
|
is_normal_subgroup.mk (ker.has_one f) (ker.mul_closed f)
|
|
(ker.has_inv f) (ker.normal f)
|
|
|
|
definition quot_over_ker [instance] : group (coset_of (ker f)) := mk_quotient_group (ker f)
|
|
-- under the wrap the tower of concepts collapse to a simple condition
|
|
example (a x : A) : (x ∈ a ∘> ker f) = (f (a⁻¹*x) = 1) := rfl
|
|
lemma ker_coset_same_val (a b : A): same_lcoset (ker f) a b → f a = f b :=
|
|
assume Psame,
|
|
assert Pin : f (b⁻¹*a) = 1, from subg_same_lcoset_in_lcoset a b Psame,
|
|
assert P : (f b)⁻¹ * (f a) = 1, from calc
|
|
(f b)⁻¹ * (f a) = (f b⁻¹) * (f a) : (hom_map_inv f)
|
|
... = f (b⁻¹*a) : by rewrite [is_hom f]
|
|
... = 1 : by rewrite Pin,
|
|
eq.symm (inv_inv (f b) ▸ inv_eq_of_mul_eq_one P)
|
|
|
|
definition ker_natural_map : (coset_of (ker f)) → B :=
|
|
quot.lift f ker_coset_same_val
|
|
|
|
example (a : A) : ker_natural_map ⟦a⟧ = f a := rfl
|
|
lemma ker_coset_hom (a b : A) :
|
|
ker_natural_map (⟦a⟧*⟦b⟧) = (ker_natural_map ⟦a⟧) * (ker_natural_map ⟦b⟧) := calc
|
|
ker_natural_map (⟦a⟧*⟦b⟧) = ker_natural_map ⟦a*b⟧ : rfl
|
|
... = f (a*b) : rfl
|
|
... = (f a) * (f b) : by rewrite [is_hom f]
|
|
... = (ker_natural_map ⟦a⟧) * (ker_natural_map ⟦b⟧) : rfl
|
|
|
|
lemma ker_map_is_hom : homomorphic (ker_natural_map : coset_of (ker f) → B) :=
|
|
take aK bK,
|
|
quot.ind (λ a, quot.ind (λ b, ker_coset_hom a b) bK) aK
|
|
|
|
lemma ker_coset_inj (a b : A) : (ker_natural_map ⟦a⟧ = ker_natural_map ⟦b⟧) → ⟦a⟧ = ⟦b⟧ :=
|
|
assume Pfeq : f a = f b,
|
|
assert Painb : a ∈ b ∘> ker f, from calc
|
|
f (b⁻¹*a) = (f b⁻¹) * (f a) : by rewrite [is_hom f]
|
|
... = (f b)⁻¹ * (f a) : by rewrite (hom_map_inv f)
|
|
... = (f a)⁻¹ * (f a) : by rewrite Pfeq
|
|
... = 1 : by rewrite (mul.left_inv (f a)),
|
|
quot.sound (@subg_in_lcoset_same_lcoset _ _ (ker f) _ a b Painb)
|
|
|
|
lemma ker_map_is_inj : injective (ker_natural_map : coset_of (ker f) → B) :=
|
|
take aK bK,
|
|
quot.ind (λ a, quot.ind (λ b, ker_coset_inj a b) bK) aK
|
|
|
|
-- a special case of the fundamental homomorphism theorem or the first isomorphism theorem
|
|
theorem first_isomorphism_theorem : isomorphic (ker_natural_map : coset_of (ker f) → B) :=
|
|
and.intro ker_map_is_inj ker_map_is_hom
|
|
|
|
end hom_theorem
|
|
end group_theory
|