revise quotient_group
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@ -1,7 +1,7 @@
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/-
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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, Egbert Rijke
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Authors: Floris van Doorn, Egbert Rijke, Jeremy Avigad
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Constructions with groups
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-/
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@ -9,11 +9,14 @@ Constructions with groups
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import hit.set_quotient .subgroup ..move_to_lib types.equiv
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open eq algebra is_trunc set_quotient relation sigma sigma.ops prod trunc function equiv is_equiv
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open property
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namespace group
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variables {G G' : Group} (H : subgroup_rel G) (N : normal_subgroup_rel G) {g g' h h' k : G}
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{N' : normal_subgroup_rel G'}
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variables {G G' : Group}
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(H : property G) [is_subgroup G H]
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(N : property G) [is_normal_subgroup G N]
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{g g' h h' k : G}
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(N' : property G') [is_normal_subgroup G' N']
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variables {A B : AbGroup}
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/- Quotient Group -/
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@ -21,20 +24,21 @@ namespace group
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definition homotopy_of_homomorphism_eq {f g : G →g G'}(p : f = g) : f ~ g :=
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λx : G , ap010 group_fun p x
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definition quotient_rel [constructor] (g h : G) : Prop := N (g * h⁻¹)
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definition quotient_rel [constructor] (g h : G) : Prop := g * h⁻¹ ∈ N
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variable {N}
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-- We prove that quotient_rel is an equivalence relation
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theorem quotient_rel_refl (g : G) : quotient_rel N g g :=
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transport (λx, N x) !mul.right_inv⁻¹ (subgroup_has_one N)
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transport (λx, N x) !mul.right_inv⁻¹ (subgroup_one_mem N)
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theorem quotient_rel_symm (r : quotient_rel N g h) : quotient_rel N h g :=
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transport (λx, N x) (!mul_inv ⬝ ap (λx, x * _) !inv_inv) (subgroup_respect_inv N r)
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transport (λx, N x) (!mul_inv ⬝ ap (λx, x * _) !inv_inv)
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begin apply subgroup_inv_mem r end
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theorem quotient_rel_trans (r : quotient_rel N g h) (s : quotient_rel N h k)
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: quotient_rel N g k :=
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have H1 : N ((g * h⁻¹) * (h * k⁻¹)), from subgroup_respect_mul N r s,
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have H1 : N ((g * h⁻¹) * (h * k⁻¹)), from subgroup_mul_mem r s,
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have H2 : (g * h⁻¹) * (h * k⁻¹) = g * k⁻¹, from calc
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(g * h⁻¹) * (h * k⁻¹) = ((g * h⁻¹) * h) * k⁻¹ : by rewrite [mul.assoc (g * h⁻¹)]
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... = g * k⁻¹ : by rewrite inv_mul_cancel_right,
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@ -48,18 +52,18 @@ namespace group
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-- We prove that quotient_rel respects inverses and multiplication, so
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-- it is a congruence relation
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theorem quotient_rel_resp_inv (r : quotient_rel N g h) : quotient_rel N g⁻¹ h⁻¹ :=
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have H1 : N (g⁻¹ * (h * g⁻¹) * g), from
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is_normal_subgroup' N g (quotient_rel_symm r),
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have H1 : g⁻¹ * (h * g⁻¹) * g ∈ N, from
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is_normal_subgroup' g (quotient_rel_symm r),
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have H2 : g⁻¹ * (h * g⁻¹) * g = g⁻¹ * h⁻¹⁻¹, from calc
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g⁻¹ * (h * g⁻¹) * g = g⁻¹ * h * g⁻¹ * g : by rewrite -mul.assoc
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... = g⁻¹ * h : inv_mul_cancel_right
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... = g⁻¹ * h⁻¹⁻¹ : by rewrite algebra.inv_inv,
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show N (g⁻¹ * h⁻¹⁻¹), by rewrite [-H2]; exact H1
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show g⁻¹ * h⁻¹⁻¹ ∈ N, by rewrite [-H2]; exact H1
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theorem quotient_rel_resp_mul (r : quotient_rel N g h) (r' : quotient_rel N g' h')
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: quotient_rel N (g * g') (h * h') :=
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have H1 : N (g * ((g' * h'⁻¹) * h⁻¹)), from
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normal_subgroup_insert N r' r,
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have H1 : g * ((g' * h'⁻¹) * h⁻¹) ∈ N, from
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normal_subgroup_insert r' r,
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have H2 : g * ((g' * h'⁻¹) * h⁻¹) = (g * g') * (h * h')⁻¹, from calc
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g * ((g' * h'⁻¹) * h⁻¹) = g * (g' * (h'⁻¹ * h⁻¹)) : by rewrite [mul.assoc]
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... = (g * g') * (h'⁻¹ * h⁻¹) : mul.assoc
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@ -114,7 +118,7 @@ namespace group
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exact ap class_of !mul.left_inv
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end
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theorem quotient_mul_comm {G : AbGroup} {N : normal_subgroup_rel G} (g h : qg N)
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theorem quotient_mul_comm {G : AbGroup} {N : property G} [is_normal_subgroup G N] (g h : qg N)
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: g * h = h * g :=
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begin
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refine set_quotient.rec_prop _ g, clear g, intro g,
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@ -132,21 +136,21 @@ namespace group
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definition quotient_group [constructor] : Group :=
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Group.mk _ (group_qg N)
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definition ab_group_qg [constructor] {G : AbGroup} (N : normal_subgroup_rel G)
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definition ab_group_qg [constructor] {G : AbGroup} (N : property G) [is_normal_subgroup G N]
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: ab_group (qg N) :=
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⦃ab_group, group_qg N, mul_comm := quotient_mul_comm⦄
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definition quotient_ab_group [constructor] {G : AbGroup} (N : subgroup_rel G)
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definition quotient_ab_group [constructor] {G : AbGroup} (N : property G) [is_subgroup G N]
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: AbGroup :=
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AbGroup.mk _ (ab_group_qg (normal_subgroup_rel_ab N))
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AbGroup.mk _ (@ab_group_qg G N (is_normal_subgroup_ab _))
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definition qg_map [constructor] : G →g quotient_group N :=
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homomorphism.mk class_of (λ g h, idp)
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definition ab_qg_map {G : AbGroup} (N : subgroup_rel G) : G →g quotient_ab_group N :=
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qg_map _
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definition ab_qg_map {G : AbGroup} (N : property G) [is_subgroup G N] : G →g quotient_ab_group N :=
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@qg_map _ N (is_normal_subgroup_ab _)
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definition is_surjective_ab_qg_map {A : AbGroup} (N : subgroup_rel A) : is_surjective (ab_qg_map N) :=
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definition is_surjective_ab_qg_map {A : AbGroup} (N : property A) [is_subgroup A N] : is_surjective (ab_qg_map N) :=
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begin
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intro x, induction x,
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fapply image.mk,
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@ -159,7 +163,7 @@ namespace group
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end quotient
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open quotient
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variable {N}
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variables {N N'}
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definition qg_map_eq_one (g : G) (H : N g) : qg_map N g = 1 :=
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begin
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@ -171,7 +175,7 @@ namespace group
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unfold quotient_rel, rewrite e, exact H
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end
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definition ab_qg_map_eq_one {K : subgroup_rel A} (g :A) (H : K g) : ab_qg_map K g = 1 :=
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definition ab_qg_map_eq_one {K : property A} [is_subgroup A K] (g :A) (H : K g) : ab_qg_map K g = 1 :=
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begin
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apply eq_of_rel,
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have e : (g * 1⁻¹ = g),
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@ -182,7 +186,7 @@ namespace group
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end
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--- there should be a smarter way to do this!! Please have a look, Floris.
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definition rel_of_qg_map_eq_one (g : G) (H : qg_map N g = 1) : N g :=
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definition rel_of_qg_map_eq_one (g : G) (H : qg_map N g = 1) : g ∈ N :=
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begin
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have e : (g * 1⁻¹ = g),
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from calc
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@ -192,14 +196,15 @@ namespace group
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apply rel_of_eq _ H
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end
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definition rel_of_ab_qg_map_eq_one {K : subgroup_rel A} (a :A) (H : ab_qg_map K a = 1) : K a :=
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definition rel_of_ab_qg_map_eq_one {K : property A} [is_subgroup A K] (a :A) (H : ab_qg_map K a = 1) : a ∈ K :=
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begin
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have e : (a * 1⁻¹ = a),
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from calc
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a * 1⁻¹ = a * 1 : one_inv
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... = a : mul_one,
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rewrite (inverse e),
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apply rel_of_eq _ H
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have is_normal_subgroup A K, from is_normal_subgroup_ab _,
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apply rel_of_eq (quotient_rel K) H
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end
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definition quotient_group_elim_fun [unfold 6] (f : G →g G') (H : Π⦃g⦄, N g → f g = 1)
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@ -216,7 +221,7 @@ namespace group
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apply H, exact K
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end
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definition quotient_group_elim [constructor] (f : G →g G') (H : Π⦃g⦄, N g → f g = 1) : quotient_group N →g G' :=
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definition quotient_group_elim [constructor] (f : G →g G') (H : Π⦃g⦄, g ∈ N → f g = 1) : quotient_group N →g G' :=
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begin
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fapply homomorphism.mk,
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-- define function
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@ -227,24 +232,30 @@ namespace group
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unfold qg_map, esimp, exact to_respect_mul f g h }
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end
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example {K : property A} [is_subgroup A K] :
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quotient_ab_group K = @quotient_group A K (is_normal_subgroup_ab _) := rfl
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definition quotient_ab_group_elim [constructor] {K : property A} [is_subgroup A K] (f : A →g B)
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(H : Π⦃g⦄, g ∈ K → f g = 1) : quotient_ab_group K →g B :=
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@quotient_group_elim A B K (is_normal_subgroup_ab _) f H
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definition quotient_group_compute (f : G →g G') (H : Π⦃g⦄, N g → f g = 1) (g : G) :
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quotient_group_elim f H (qg_map N g) = f g :=
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begin
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reflexivity
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end
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definition gelim_unique (f : G →g G') (H : Π⦃g⦄, N g → f g = 1) (k : quotient_group N →g G')
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definition gelim_unique (f : G →g G') (H : Π⦃g⦄, g ∈ N → f g = 1) (k : quotient_group N →g G')
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: ( k ∘g qg_map N ~ f ) → k ~ quotient_group_elim f H :=
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begin
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intro K cg, induction cg using set_quotient.rec_prop with g,
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exact K g
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end
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definition ab_gelim_unique {K : subgroup_rel A} (f : A →g B) (H : Π (a :A), K a → f a = 1) (k : quotient_ab_group K →g B)
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: ( k ∘g ab_qg_map K ~ f) → k ~ quotient_group_elim f H :=
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begin
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fapply gelim_unique,
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end
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definition ab_gelim_unique {K : property A} [is_subgroup A K] (f : A →g B) (H : Π (a :A), a ∈ K → f a = 1) (k : quotient_ab_group K →g B)
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: ( k ∘g ab_qg_map K ~ f) → k ~ quotient_ab_group_elim f H :=
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--@quotient_group_elim A B K (is_normal_subgroup_ab _) f H :=
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@gelim_unique _ _ K (is_normal_subgroup_ab _) f H _
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definition qg_universal_property (f : G →g G') (H : Π⦃g⦄, N g → f g = 1) :
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is_contr (Σ(g : quotient_group N →g G'), g ∘ qg_map N ~ f) :=
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{fapply is_prop.elimo} }
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end
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definition ab_qg_universal_property {K : subgroup_rel A} (f : A →g B) (H : Π (a :A), K a → f a = 1) :
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definition ab_qg_universal_property {K : property A} [is_subgroup A K] (f : A →g B) (H : Π (a :A), K a → f a = 1) :
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is_contr ((Σ(g : quotient_ab_group K →g B), g ∘g ab_qg_map K ~ f) ) :=
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begin
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fapply qg_universal_property,
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fapply @qg_universal_property _ _ K (is_normal_subgroup_ab _),
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exact H
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end
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definition quotient_group_functor_contr {K L : subgroup_rel A} (H : Π (a : A), K a → L a) :
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definition quotient_group_functor_contr {K L : property A} [is_subgroup A K] [is_subgroup A L]
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(H : Π (a : A), K a → L a) :
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is_contr ((Σ(g : quotient_ab_group K →g quotient_ab_group L), g ∘g ab_qg_map K ~ ab_qg_map L) ) :=
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begin
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fapply ab_qg_universal_property,
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intro a p,
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fapply qg_map_eq_one,
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exact H a p,
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fapply ab_qg_map_eq_one,
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exact H a p
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end
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definition quotient_group_functor_id {K : subgroup_rel A} (H : Π (a : A), K a → K a) :
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center' (@quotient_group_functor_contr _ K K H) = ⟨gid (quotient_ab_group K), λ x, rfl⟩ :=
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definition quotient_group_functor_id {K : property A} [is_subgroup A K] (H : Π (a : A), K a → K a) :
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center' (@quotient_group_functor_contr _ K K _ _ H) = ⟨gid (quotient_ab_group K), λ x, rfl⟩ :=
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begin
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note p := @quotient_group_functor_contr _ K K H,
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note p := @quotient_group_functor_contr _ K K _ _ H,
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fapply eq_of_is_contr,
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end
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set_option pp.universes true
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definition subgroup_rel_eq' {K L : subgroup_rel A} (htpy : Π (a : A), K a ≃ L a) : K = L :=
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definition subgroup_rel_eq' {K L : property A} [HK : is_subgroup A K] [HL : is_subgroup A L] (htpy : Π (a : A), K a ≃ L a) : K = L :=
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begin
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induction K with K', induction L with L', esimp at *,
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assert q : K' = L',
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induction HK with Rone Rmul Rinv, induction HL with Rone' Rmul' Rinv', esimp at *,
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assert q : K = L,
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begin
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fapply eq_of_homotopy,
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intro a,
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exact htpy a,
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end,
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induction q,
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assert q : Rone = Rone_1,
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assert q : Rone = Rone',
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begin
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fapply is_prop.elim,
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end,
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induction q,
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assert q2 : @Rmul = @Rmul_1,
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assert q2 : @Rmul = @Rmul',
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begin
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fapply is_prop.elim,
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end,
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induction q2,
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assert q : @Rinv = @Rinv_1,
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assert q : @Rinv = @Rinv',
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begin
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fapply is_prop.elim,
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end,
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reflexivity
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end
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definition subgroup_rel_eq {K L : subgroup_rel A} (K_in_L : Π (a : A), K a → L a) (L_in_K : Π (a : A), L a → K a) : K = L :=
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definition subgroup_rel_eq {K L : property A} [is_subgroup A K] [is_subgroup A L] (K_in_L : Π (a : A), a ∈ K → a ∈ L) (L_in_K : Π (a : A), a ∈ L → a ∈ K) : K = L :=
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begin
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have htpy : Π (a : A), K a ≃ L a,
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begin
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intro a,
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fapply equiv_of_is_prop,
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fapply K_in_L a,
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fapply L_in_K a,
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apply @equiv_of_is_prop (a ∈ K) (a ∈ L) _ _ (K_in_L a) (L_in_K a),
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end,
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exact subgroup_rel_eq' htpy,
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end
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definition eq_of_ab_qg_group' {K L : subgroup_rel A} (p : K = L) : quotient_ab_group K = quotient_ab_group L :=
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definition eq_of_ab_qg_group' {K L : property A} [HK : is_subgroup A K] [HL : is_subgroup A L] (p : K = L) : quotient_ab_group K = quotient_ab_group L :=
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begin
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induction p, reflexivity
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revert HK, revert HL, induction p, intros,
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have HK = HL, begin apply @is_prop.elim _ _ HK HL end,
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rewrite this
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end
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definition iso_of_eq {B : AbGroup} (p : A = B) : A ≃g B :=
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@ -336,29 +348,35 @@ namespace group
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induction p, fapply isomorphism.mk, exact gid A, fapply adjointify, exact id, intro a, reflexivity, intro a, reflexivity
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end
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definition iso_of_ab_qg_group' {K L : subgroup_rel A} (p : K = L) : quotient_ab_group K ≃g quotient_ab_group L :=
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definition iso_of_ab_qg_group' {K L : property A} [is_subgroup A K] [is_subgroup A L] (p : K = L) : quotient_ab_group K ≃g quotient_ab_group L :=
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iso_of_eq (eq_of_ab_qg_group' p)
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definition htpy_of_ab_qg_group' {K L : subgroup_rel A} (p : K = L) : (iso_of_ab_qg_group' p) ∘g ab_qg_map K ~ ab_qg_map L :=
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/-
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definition htpy_of_ab_qg_group' {K L : property A} [HK : is_subgroup A K] [HL : is_subgroup A L] (p : K = L) : (iso_of_ab_qg_group' p) ∘g ab_qg_map K ~ ab_qg_map L :=
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begin
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induction p, reflexivity
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revert HK, revert HL, induction p, intros HK HL, unfold iso_of_ab_qg_group', unfold ab_qg_map
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-- have HK = HL, begin apply @is_prop.elim _ _ HK HL end,
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-- rewrite this
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-- induction p, reflexivity
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end
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-/
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definition eq_of_ab_qg_group {K L : subgroup_rel A} (K_in_L : Π (a : A), K a → L a) (L_in_K : Π (a : A), L a → K a) : quotient_ab_group K = quotient_ab_group L :=
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definition eq_of_ab_qg_group {K L : property A} [is_subgroup A K] [is_subgroup A L] (K_in_L : Π (a : A), K a → L a) (L_in_K : Π (a : A), L a → K a) : quotient_ab_group K = quotient_ab_group L :=
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eq_of_ab_qg_group' (subgroup_rel_eq K_in_L L_in_K)
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definition iso_of_ab_qg_group {K L : subgroup_rel A} (K_in_L : Π (a : A), K a → L a) (L_in_K : Π (a : A), L a → K a) : quotient_ab_group K ≃g quotient_ab_group L :=
|
||||
definition iso_of_ab_qg_group {K L : property A} [is_subgroup A K] [is_subgroup A L] (K_in_L : Π (a : A), K a → L a) (L_in_K : Π (a : A), L a → K a) : quotient_ab_group K ≃g quotient_ab_group L :=
|
||||
iso_of_eq (eq_of_ab_qg_group K_in_L L_in_K)
|
||||
|
||||
definition htpy_of_ab_qg_group {K L : subgroup_rel A} (K_in_L : Π (a : A), K a → L a) (L_in_K : Π (a : A), L a → K a) : iso_of_ab_qg_group K_in_L L_in_K ∘g ab_qg_map K ~ ab_qg_map L :=
|
||||
/-
|
||||
definition htpy_of_ab_qg_group {K L : property A} [is_subgroup A K] [is_subgroup A L] (K_in_L : Π (a : A), K a → L a) (L_in_K : Π (a : A), L a → K a) : iso_of_ab_qg_group K_in_L L_in_K ∘g ab_qg_map K ~ ab_qg_map L :=
|
||||
begin
|
||||
fapply htpy_of_ab_qg_group'
|
||||
end
|
||||
|
||||
-/
|
||||
end quotient_group_iso_ua
|
||||
|
||||
section quotient_group_iso
|
||||
variables {K L : subgroup_rel A} (H1 : Π (a : A), K a → L a) (H2 : Π (a : A), L a → K a)
|
||||
variables {K L : property A} [is_subgroup A K] [is_subgroup A L] (H1 : Π (a : A), K a → L a) (H2 : Π (a : A), L a → K a)
|
||||
include H1
|
||||
include H2
|
||||
|
||||
|
@ -372,15 +390,15 @@ namespace group
|
|||
|
||||
definition quotient_group_iso_contr_KK :
|
||||
is_contr (Σ (g : quotient_ab_group K →g quotient_ab_group K), g ∘g ab_qg_map K ~ ab_qg_map K) :=
|
||||
@quotient_group_functor_contr A K K (λ a, H2 a ∘ H1 a)
|
||||
@quotient_group_functor_contr A K K _ _ (λ a, H2 a ∘ H1 a)
|
||||
|
||||
definition quotient_group_iso_contr_LK :
|
||||
quotient_ab_group L →g quotient_ab_group K :=
|
||||
pr1 (center' (@quotient_group_functor_contr A L K H2))
|
||||
pr1 (center' (@quotient_group_functor_contr A L K _ _ H2))
|
||||
|
||||
definition quotient_group_iso_contr_LL :
|
||||
quotient_ab_group L →g quotient_ab_group L :=
|
||||
pr1 (center' (@quotient_group_functor_contr A L L (λ a, H1 a ∘ H2 a)))
|
||||
pr1 (center' (@quotient_group_functor_contr A L L _ _ (λ a, H1 a ∘ H2 a)))
|
||||
|
||||
/-
|
||||
definition quotient_group_iso : quotient_ab_group K ≃g quotient_ab_group L :=
|
||||
|
@ -402,15 +420,15 @@ namespace group
|
|||
intro a, induction a with g h,
|
||||
fapply is_contr_of_inhabited_prop,
|
||||
fapply adjointify,
|
||||
rexact group_fun (pr1 (center' (@quotient_group_functor_contr A L K H2))),
|
||||
note htpy := homotopy_of_eq (ap group_fun (ap sigma.pr1 (@quotient_group_functor_id _ L (λ a, (H1 a) ∘ (H2 a))))),
|
||||
rexact group_fun (pr1 (center' (@quotient_group_functor_contr A L K _ _ H2))),
|
||||
note htpy := homotopy_of_eq (ap group_fun (ap sigma.pr1 (@quotient_group_functor_id _ L _ (λ a, (H1 a) ∘ (H2 a))))),
|
||||
have KK : is_contr ((Σ(g' : quotient_ab_group K →g quotient_ab_group K), g' ∘g ab_qg_map K ~ ab_qg_map K) ), from
|
||||
quotient_group_functor_contr (λ a, (H2 a) ∘ (H1 a)),
|
||||
-- have KK_path : ⟨g, h⟩ = ⟨id, λ a, refl (ab_qg_map K a)⟩, from eq_of_is_contr ⟨g, h⟩ ⟨id, λ a, refl (ab_qg_map K a)⟩,
|
||||
repeat exact sorry
|
||||
end
|
||||
/-
|
||||
definition quotient_group_iso_contr {K L : subgroup_rel A} (H1 : Π (a : A), K a → L a) (H2 : Π (a : A), L a → K a) :
|
||||
definition quotient_group_iso_contr {K L : property A} [is_subgroup A K] [is_subgroup A L] (H1 : Π (a : A), K a → L a) (H2 : Π (a : A), L a → K a) :
|
||||
is_contr (Σ (g : quotient_ab_group K ≃g quotient_ab_group L), g ∘g ab_qg_map K ~ ab_qg_map L) :=
|
||||
begin
|
||||
refine @is_trunc_equiv_closed (Σ(gh : Σ (g : quotient_ab_group K →g quotient_ab_group L), g ∘g ab_qg_map K ~ ab_qg_map L), is_equiv (group_fun (pr1 gh))) (Σ (g : quotient_ab_group K ≃g quotient_ab_group L), g ∘g ab_qg_map K ~ ab_qg_map L) -2 _ (quotient_group_iso_contr_aux H1 H2),
|
||||
|
@ -422,8 +440,7 @@ namespace group
|
|||
|
||||
end quotient_group_iso
|
||||
|
||||
|
||||
definition quotient_group_functor [constructor] (φ : G →g G') (h : Πg, N g → N' (φ g)) :
|
||||
definition quotient_group_functor [constructor] (φ : G →g G') (h : Πg, g ∈ N → φ g ∈ N') :
|
||||
quotient_group N →g quotient_group N' :=
|
||||
begin
|
||||
apply quotient_group_elim (qg_map N' ∘g φ),
|
||||
|
@ -435,17 +452,18 @@ namespace group
|
|||
-- FIRST ISOMORPHISM THEOREM
|
||||
------------------------------------------------
|
||||
|
||||
|
||||
definition kernel_quotient_extension {A B : AbGroup} (f : A →g B) : quotient_ab_group (kernel_subgroup f) →g B :=
|
||||
definition kernel_quotient_extension {A B : AbGroup} (f : A →g B) : quotient_ab_group (kernel f) →g B :=
|
||||
begin
|
||||
fapply quotient_group_elim f, intro a, intro p, exact p
|
||||
unfold quotient_ab_group,
|
||||
fapply @quotient_group_elim A B _ (@is_normal_subgroup_ab _ (kernel f) _) f,
|
||||
intro a, intro p, exact p
|
||||
end
|
||||
|
||||
definition kernel_quotient_extension_triangle {A B : AbGroup} (f : A →g B) :
|
||||
kernel_quotient_extension f ∘ ab_qg_map (kernel_subgroup f) ~ f :=
|
||||
kernel_quotient_extension f ∘ ab_qg_map (kernel f) ~ f :=
|
||||
begin
|
||||
intro a,
|
||||
apply quotient_group_compute
|
||||
apply @quotient_group_compute _ _ _ (@is_normal_subgroup_ab _ (kernel f) _)
|
||||
end
|
||||
|
||||
definition is_embedding_kernel_quotient_extension {A B : AbGroup} (f : A →g B) :
|
||||
|
@ -453,27 +471,27 @@ definition is_embedding_kernel_quotient_extension {A B : AbGroup} (f : A →g B)
|
|||
begin
|
||||
fapply is_embedding_of_is_mul_hom,
|
||||
intro x,
|
||||
note H := is_surjective_ab_qg_map (kernel_subgroup f) x,
|
||||
note H := is_surjective_ab_qg_map (kernel f) x,
|
||||
induction H, induction p,
|
||||
intro q,
|
||||
apply qg_map_eq_one,
|
||||
apply @qg_map_eq_one _ _ (@is_normal_subgroup_ab _ (kernel f) _),
|
||||
refine _ ⬝ q,
|
||||
symmetry,
|
||||
rexact kernel_quotient_extension_triangle f a
|
||||
end
|
||||
|
||||
definition ab_group_quotient_homomorphism (A B : AbGroup)(K : subgroup_rel A)(L : subgroup_rel B) (f : A →g B)
|
||||
(p : Π(a:A), K(a) → L(f a)) : quotient_ab_group K →g quotient_ab_group L :=
|
||||
definition ab_group_quotient_homomorphism (A B : AbGroup)(K : property A)(L : property B) [is_subgroup A K] [is_subgroup B L] (f : A →g B)
|
||||
(p : Π(a:A), a ∈ K → f a ∈ L) : quotient_ab_group K →g quotient_ab_group L :=
|
||||
begin
|
||||
fapply quotient_group_elim,
|
||||
fapply @quotient_group_elim,
|
||||
exact (ab_qg_map L) ∘g f,
|
||||
intro a,
|
||||
intro k,
|
||||
exact @ab_qg_map_eq_one B L (f a) (p a k),
|
||||
exact @ab_qg_map_eq_one B L _ (f a) (p a k),
|
||||
end
|
||||
|
||||
definition ab_group_kernel_factor {A B C: AbGroup} (f : A →g B)(g : A →g C){i : C →g B}(H : f = i ∘g g )
|
||||
: Π a:A , kernel_subgroup(g)(a) → kernel_subgroup(f)(a) :=
|
||||
: kernel g ⊆ kernel f :=
|
||||
begin
|
||||
intro a,
|
||||
intro p,
|
||||
|
@ -484,41 +502,36 @@ definition ab_group_kernel_factor {A B C: AbGroup} (f : A →g B)(g : A →g C){
|
|||
end
|
||||
|
||||
definition ab_group_triv_kernel_factor {A B C: AbGroup} (f : A →g B)(g : A →g C){i : C →g B}(H : f = i ∘g g ) :
|
||||
is_trivial_subgroup _ (kernel_subgroup(f)) → is_trivial_subgroup _ (kernel_subgroup(g)) :=
|
||||
begin
|
||||
intro p,
|
||||
intro a,
|
||||
intro q,
|
||||
fapply p,
|
||||
exact ab_group_kernel_factor f g H a q
|
||||
end
|
||||
kernel f ⊆ '{1} → kernel g ⊆ '{1} :=
|
||||
λ p, subproperty.trans (ab_group_kernel_factor f g H) p
|
||||
|
||||
definition triv_kern_is_embedding {A B : AbGroup} (f : A →g B):
|
||||
is_trivial_subgroup _ (kernel_subgroup(f)) → is_embedding(f) :=
|
||||
begin
|
||||
intro p,
|
||||
fapply is_embedding_of_is_mul_hom,
|
||||
intro a q,
|
||||
apply p,
|
||||
exact q
|
||||
end
|
||||
definition is_embedding_of_kernel_subproperty_one {A B : AbGroup} (f : A →g B) :
|
||||
kernel f ⊆ '{1} → is_embedding f :=
|
||||
λ p, is_embedding_of_is_mul_hom _
|
||||
(take x, assume h : f x = 1,
|
||||
show x = 1, from eq_of_mem_singleton (p _ h))
|
||||
|
||||
definition kernel_subproperty_one {A B : AbGroup} (f : A →g B) :
|
||||
is_embedding f → kernel f ⊆ '{1} :=
|
||||
λ h x hx,
|
||||
have x = 1, from eq_one_of_is_mul_hom hx,
|
||||
show x ∈ '{1}, from mem_singleton_of_eq this
|
||||
|
||||
definition ab_group_kernel_equivalent {A B : AbGroup} (C : AbGroup) (f : A →g B)(g : A →g C)(i : C →g B)(H : f = i ∘g g )(K : is_embedding i)
|
||||
: Π a:A , kernel_subgroup(g)(a) ↔ kernel_subgroup(f)(a) :=
|
||||
begin
|
||||
intro a,
|
||||
fapply iff.intro,
|
||||
exact ab_group_kernel_factor f g H a,
|
||||
intro p,
|
||||
apply @is_injective_of_is_embedding _ _ i _ (g a) 1,
|
||||
exact calc
|
||||
i (g a) = f a : (homotopy_of_eq (ap group_fun H) a)⁻¹
|
||||
... = 1 : p
|
||||
... = i 1 : (respect_one i)⁻¹
|
||||
end
|
||||
: Π a:A, a ∈ kernel g ↔ a ∈ kernel f :=
|
||||
exteq_of_subproperty_of_subproperty
|
||||
(show kernel g ⊆ kernel f, from ab_group_kernel_factor f g H)
|
||||
(show kernel f ⊆ kernel g, from
|
||||
take a,
|
||||
suppose f a = 1,
|
||||
have i (g a) = i 1, from calc
|
||||
i (g a) = f a : (homotopy_of_eq (ap group_fun H) a)⁻¹
|
||||
... = 1 : this
|
||||
... = i 1 : (respect_one i)⁻¹,
|
||||
is_injective_of_is_embedding this)
|
||||
|
||||
definition ab_group_kernel_image_lift (A B : AbGroup) (f : A →g B)
|
||||
: Π a : A, kernel_subgroup(image_lift(f))(a) ↔ kernel_subgroup(f)(a) :=
|
||||
: Π a : A, a ∈ kernel (image_lift f) ↔ a ∈ kernel f :=
|
||||
begin
|
||||
fapply ab_group_kernel_equivalent (ab_image f) (f) (image_lift(f)) (image_incl(f)),
|
||||
exact image_factor f,
|
||||
|
@ -526,14 +539,14 @@ definition ab_group_kernel_image_lift (A B : AbGroup) (f : A →g B)
|
|||
end
|
||||
|
||||
definition ab_group_kernel_quotient_to_image {A B : AbGroup} (f : A →g B)
|
||||
: quotient_ab_group (kernel_subgroup f) →g ab_image (f) :=
|
||||
: quotient_ab_group (kernel f) →g ab_image (f) :=
|
||||
begin
|
||||
fapply quotient_group_elim (image_lift f), intro a, intro p,
|
||||
fapply quotient_ab_group_elim (image_lift f), intro a, intro p,
|
||||
apply iff.mpr (ab_group_kernel_image_lift _ _ f a) p
|
||||
end
|
||||
|
||||
definition ab_group_kernel_quotient_to_image_domain_triangle {A B : AbGroup} (f : A →g B)
|
||||
: ab_group_kernel_quotient_to_image (f) ∘g ab_qg_map (kernel_subgroup (f)) ~ image_lift(f) :=
|
||||
: ab_group_kernel_quotient_to_image (f) ∘g ab_qg_map (kernel f) ~ image_lift(f) :=
|
||||
begin
|
||||
intros a,
|
||||
esimp,
|
||||
|
@ -551,10 +564,10 @@ definition ab_group_kernel_quotient_to_image_codomain_triangle {A B : AbGroup} (
|
|||
definition is_surjective_kernel_quotient_to_image {A B : AbGroup} (f : A →g B)
|
||||
: is_surjective (ab_group_kernel_quotient_to_image f) :=
|
||||
begin
|
||||
fapply is_surjective_factor (group_fun (ab_qg_map (kernel_subgroup f))),
|
||||
exact image_lift f,
|
||||
apply quotient_group_compute,
|
||||
exact is_surjective_image_lift f
|
||||
fapply is_surjective_factor (group_fun (ab_qg_map (kernel f))),
|
||||
exact image_lift f,
|
||||
apply @quotient_group_compute _ _ _ (@is_normal_subgroup_ab _ (kernel f) _),
|
||||
exact is_surjective_image_lift f
|
||||
end
|
||||
|
||||
definition is_embedding_kernel_quotient_to_image {A B : AbGroup} (f : A →g B)
|
||||
|
@ -566,7 +579,7 @@ definition is_embedding_kernel_quotient_to_image {A B : AbGroup} (f : A →g B)
|
|||
end
|
||||
|
||||
definition ab_group_first_iso_thm {A B : AbGroup} (f : A →g B)
|
||||
: quotient_ab_group (kernel_subgroup f) ≃g ab_image f :=
|
||||
: quotient_ab_group (kernel f) ≃g ab_image f :=
|
||||
begin
|
||||
fapply isomorphism.mk,
|
||||
exact ab_group_kernel_quotient_to_image f,
|
||||
|
@ -576,13 +589,13 @@ definition ab_group_first_iso_thm {A B : AbGroup} (f : A →g B)
|
|||
end
|
||||
|
||||
definition codomain_surjection_is_quotient {A B : AbGroup} (f : A →g B)( H : is_surjective f)
|
||||
: quotient_ab_group (kernel_subgroup f) ≃g B :=
|
||||
: quotient_ab_group (kernel f) ≃g B :=
|
||||
begin
|
||||
exact (ab_group_first_iso_thm f) ⬝g (iso_surjection_ab_image_incl f H)
|
||||
end
|
||||
|
||||
definition codomain_surjection_is_quotient_triangle {A B : AbGroup} (f : A →g B)( H : is_surjective f)
|
||||
: codomain_surjection_is_quotient (f)(H) ∘g ab_qg_map (kernel_subgroup f) ~ f :=
|
||||
: codomain_surjection_is_quotient (f)(H) ∘g ab_qg_map (kernel f) ~ f :=
|
||||
begin
|
||||
intro a,
|
||||
esimp
|
||||
|
@ -610,18 +623,17 @@ definition codomain_surjection_is_quotient_triangle {A B : AbGroup} (f : A →g
|
|||
definition gr_mul (g h : A₁) : R g → R h → R (g * h) :=
|
||||
trunc_functor2 rmul
|
||||
|
||||
definition normal_generating_relation : subgroup_rel A₁ :=
|
||||
⦃ subgroup_rel,
|
||||
R := R,
|
||||
Rone := gr_one,
|
||||
Rinv := gr_inv,
|
||||
Rmul := gr_mul⦄
|
||||
definition normal_generating_relation [instance] : is_subgroup A₁ generating_relation :=
|
||||
⦃ is_subgroup,
|
||||
one_mem := gr_one,
|
||||
inv_mem := gr_inv,
|
||||
mul_mem := gr_mul⦄
|
||||
|
||||
parameter (A₁)
|
||||
definition quotient_ab_group_gen : AbGroup := quotient_ab_group normal_generating_relation
|
||||
definition quotient_ab_group_gen : AbGroup := quotient_ab_group generating_relation
|
||||
|
||||
definition gqg_map [constructor] : A₁ →g quotient_ab_group_gen :=
|
||||
qg_map _
|
||||
ab_qg_map _
|
||||
|
||||
parameter {A₁}
|
||||
definition gqg_eq_of_rel {g h : A₁} (H : S (g * h⁻¹)) : gqg_map g = gqg_map h :=
|
||||
|
@ -634,7 +646,7 @@ definition codomain_surjection_is_quotient_triangle {A B : AbGroup} (f : A →g
|
|||
definition gqg_elim [constructor] (f : A₁ →g A₂) (H : Π⦃g⦄, S g → f g = 1)
|
||||
: quotient_ab_group_gen →g A₂ :=
|
||||
begin
|
||||
apply quotient_group_elim f,
|
||||
apply quotient_ab_group_elim f,
|
||||
intro g r, induction r with r,
|
||||
induction r with g s g h r r' IH1 IH2 g r IH,
|
||||
{ exact H s },
|
||||
|
@ -651,7 +663,7 @@ definition codomain_surjection_is_quotient_triangle {A B : AbGroup} (f : A →g
|
|||
|
||||
definition gqg_elim_unique (f : A₁ →g A₂) (H : Π⦃g⦄, S g → f g = 1)
|
||||
(k : quotient_ab_group_gen →g A₂) : ( k ∘g gqg_map ~ f ) → k ~ gqg_elim f H :=
|
||||
!gelim_unique
|
||||
!ab_gelim_unique
|
||||
|
||||
end
|
||||
|
||||
|
@ -659,16 +671,18 @@ end group
|
|||
|
||||
namespace group
|
||||
|
||||
variables {G H K : Group} {R : normal_subgroup_rel G} {S : normal_subgroup_rel H}
|
||||
{T : normal_subgroup_rel K}
|
||||
variables {G H K : Group} {R : property G} [is_normal_subgroup G R]
|
||||
{S : property H} [is_normal_subgroup H S]
|
||||
{T : property K} [is_normal_subgroup K T]
|
||||
|
||||
definition quotient_ab_group_functor [constructor] {G H : AbGroup} {R : subgroup_rel G}
|
||||
{S : subgroup_rel H} (φ : G →g H)
|
||||
(h : Πg, R g → S (φ g)) : quotient_ab_group R →g quotient_ab_group S :=
|
||||
quotient_group_functor φ h
|
||||
definition quotient_ab_group_functor [constructor] {G H : AbGroup}
|
||||
{R : property G} [is_subgroup G R]
|
||||
{S : property H} [is_subgroup H S] (φ : G →g H)
|
||||
(h : Πg, g ∈ R → φ g ∈ S) : quotient_ab_group R →g quotient_ab_group S :=
|
||||
@quotient_group_functor G H R (is_normal_subgroup_ab _) S (is_normal_subgroup_ab _) φ h
|
||||
|
||||
theorem quotient_group_functor_compose (ψ : H →g K) (φ : G →g H)
|
||||
(hψ : Πg, S g → T (ψ g)) (hφ : Πg, R g → S (φ g)) :
|
||||
(hψ : Πg, g ∈ S → ψ g ∈ T) (hφ : Πg, g ∈ R → φ g ∈ S) :
|
||||
quotient_group_functor ψ hψ ∘g quotient_group_functor φ hφ ~
|
||||
quotient_group_functor (ψ ∘g φ) (λg, proof hψ (φ g) qed ∘ hφ g) :=
|
||||
begin
|
||||
|
@ -681,18 +695,18 @@ namespace group
|
|||
intro g, induction g using set_quotient.rec_prop with g hg, reflexivity
|
||||
end
|
||||
|
||||
definition quotient_group_functor_mul.{u₁ v₁ u₂ v₂}
|
||||
{G H : AbGroup} {R : subgroup_rel.{u₁ v₁} G} {S : subgroup_rel.{u₂ v₂} H}
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(ψ φ : G →g H) (hψ : Πg, R g → S (ψ g)) (hφ : Πg, R g → S (φ g)) :
|
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definition quotient_group_functor_mul
|
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{G H : AbGroup} {R : property G} [is_subgroup G R] {S : property H} [is_subgroup H S]
|
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(ψ φ : G →g H) (hψ : Πg, g ∈ R → ψ g ∈ S) (hφ : Πg, g ∈ R → φ g ∈ S) :
|
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homomorphism_mul (quotient_ab_group_functor ψ hψ) (quotient_ab_group_functor φ hφ) ~
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quotient_ab_group_functor (homomorphism_mul ψ φ)
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(λg hg, subgroup_respect_mul S (hψ g hg) (hφ g hg)) :=
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(λg hg, is_subgroup.mul_mem (hψ g hg) (hφ g hg)) :=
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begin
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intro g, induction g using set_quotient.rec_prop with g hg, reflexivity
|
||||
end
|
||||
|
||||
definition quotient_group_functor_homotopy {ψ φ : G →g H} (hψ : Πg, R g → S (ψ g))
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||||
(hφ : Πg, R g → S (φ g)) (p : φ ~ ψ) :
|
||||
(hφ : Πg, g ∈ R → φ g ∈ S) (p : φ ~ ψ) :
|
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quotient_group_functor φ hφ ~ quotient_group_functor ψ hψ :=
|
||||
begin
|
||||
intro g, induction g using set_quotient.rec_prop with g hg,
|
||||
|
|
|
@ -22,6 +22,19 @@ namespace group
|
|||
(mul_mem : Π{g h}, g ∈ H → h ∈ H → g * h ∈ H)
|
||||
(inv_mem : Π{g}, g ∈ H → g⁻¹ ∈ H)
|
||||
|
||||
definition is_prop_is_subgroup [instance] (G : Group) (H : property G) : is_prop (is_subgroup G H) :=
|
||||
proof -- this results in a simpler choice of universe metavariables
|
||||
have 1 ∈ H × (Π{g h}, g ∈ H → h ∈ H → g * h ∈ H) × (Π{g}, g ∈ H → g⁻¹ ∈ H) ≃ is_subgroup G H,
|
||||
begin
|
||||
fapply equiv.MK,
|
||||
{ intro p, cases p with p1 p2, cases p2 with p2 p3, exact is_subgroup.mk p1 @p2 @p3 },
|
||||
{ intro p, split, exact is_subgroup.one_mem H, split, apply @is_subgroup.mul_mem G H p, apply @is_subgroup.inv_mem G H p},
|
||||
{ intro b, cases b, reflexivity },
|
||||
{ intro a, cases a with a1 a2, cases a2, reflexivity }
|
||||
end,
|
||||
is_trunc_equiv_closed _ this
|
||||
qed
|
||||
|
||||
/-- Every group G has at least two subgroups, the trivial subgroup containing only one, and the full subgroup. --/
|
||||
definition trivial_subgroup [instance] (G : Group) : is_subgroup G '{1} :=
|
||||
begin
|
||||
|
@ -103,7 +116,7 @@ namespace group
|
|||
/-- Next, we formalize some aspects of normal subgroups. Recall that a normal subgroup H of a
|
||||
group G is a subgroup which is invariant under all inner automorophisms on G. --/
|
||||
|
||||
definition is_normal [constructor] (G : Group) (N : property G) : Prop :=
|
||||
definition is_normal.{u v} [constructor] (G : Group) (N : property.{u v} G) : Prop :=
|
||||
trunctype.mk (Π{g} h, g ∈ N → h * g * h⁻¹ ∈ N) _
|
||||
|
||||
structure is_normal_subgroup [class] (G : Group) (N : property G) extends is_subgroup G N :=
|
||||
|
@ -122,7 +135,7 @@ section
|
|||
theorem is_normal_subgroup' (h : G) (r : g ∈ N) : h⁻¹ * g * h ∈ N :=
|
||||
inv_inv h ▸ subgroup_is_normal N h⁻¹ r
|
||||
|
||||
definition is_normal_subgroup_ab.{u} [constructor] {C : property A} (subgrpA : is_subgroup A C)
|
||||
definition is_normal_subgroup_ab [constructor] {C : property A} (subgrpA : is_subgroup A C)
|
||||
: is_normal_subgroup A C :=
|
||||
⦃ is_normal_subgroup, subgrpA,
|
||||
is_normal := abstract begin
|
||||
|
@ -219,6 +232,8 @@ section
|
|||
|
||||
definition Kernel {G H : Group} (f : G →g H) : Group := subgroup (kernel f)
|
||||
|
||||
set_option trace.class_instances true
|
||||
|
||||
definition ab_kernel {G H : AbGroup} (f : G →g H) : AbGroup := ab_subgroup (kernel f)
|
||||
|
||||
definition incl_of_subgroup [constructor] {G : Group} (H : property G) [is_subgroup G H] :
|
||||
|
@ -249,7 +264,7 @@ section
|
|||
fapply is_embedding_incl_of_subgroup,
|
||||
end
|
||||
|
||||
definition is_subgroup_of_subgroup {G : Group} {H1 H2 : property G} [is_subgroup G H1]
|
||||
definition is_subgroup_of_is_subgroup {G : Group} {H1 H2 : property G} [is_subgroup G H1]
|
||||
[is_subgroup G H2] (hyp : Π (g : G), g ∈ H1 → g ∈ H2) :
|
||||
is_subgroup (subgroup H2) {h | incl_of_subgroup H2 h ∈ H1} :=
|
||||
is_subgroup.mk
|
||||
|
|
|
@ -44,6 +44,10 @@ theorem eq_of_subproperty_of_subproperty {a b : property X} (h₁ : a ⊆ b) (h
|
|||
subproperty.antisymm h₁ h₂
|
||||
-/
|
||||
|
||||
theorem exteq_of_subproperty_of_subproperty {a b : property X} (h₁ : a ⊆ b) (h₂ : b ⊆ a) :
|
||||
∀ ⦃x⦄, x ∈ a ↔ x ∈ b :=
|
||||
λ x, iff.intro (λ h, h₁ h) (λ h, h₂ h)
|
||||
|
||||
theorem mem_of_subproperty_of_mem {s₁ s₂ : property X} {a : X} : s₁ ⊆ s₂ → a ∈ s₁ → a ∈ s₂ :=
|
||||
assume h₁ h₂, h₁ _ h₂
|
||||
|
||||
|
|
Loading…
Reference in a new issue