373 lines
13 KiB
Text
373 lines
13 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, Egbert Rijke
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Constructions with groups
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-/
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import hit.set_quotient .subgroup
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open eq algebra is_trunc set_quotient relation sigma sigma.ops prod trunc function equiv
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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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variables {A B : CommGroup}
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/- Quotient 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 (g h : G) : Prop := N (g * h⁻¹)
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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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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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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 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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show N (g * k⁻¹), by rewrite [-H2]; exact H1
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theorem is_equivalence_quotient_rel : is_equivalence (quotient_rel N) :=
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is_equivalence.mk quotient_rel_refl
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(λg h, quotient_rel_symm)
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(λg h k, quotient_rel_trans)
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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 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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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 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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... = (g * g') * (h * h')⁻¹ : by rewrite [mul_inv],
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show N ((g * g') * (h * h')⁻¹), from transport (λx, N x) H2 H1
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local attribute is_equivalence_quotient_rel [instance]
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variable (N)
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definition qg : Type := set_quotient (quotient_rel N)
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variable {N}
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local attribute qg [reducible]
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definition quotient_one [constructor] : qg N := class_of one
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definition quotient_inv [unfold 3] : qg N → qg N :=
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quotient_unary_map has_inv.inv (λg g' r, quotient_rel_resp_inv r)
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definition quotient_mul [unfold 3 4] : qg N → qg N → qg N :=
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quotient_binary_map has_mul.mul (λg g' r h h' r', quotient_rel_resp_mul r r')
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section
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local notation 1 := quotient_one
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local postfix ⁻¹ := quotient_inv
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local infix * := quotient_mul
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theorem quotient_mul_assoc (g₁ g₂ g₃ : qg N) : g₁ * g₂ * g₃ = g₁ * (g₂ * g₃) :=
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begin
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refine set_quotient.rec_prop _ g₁,
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refine set_quotient.rec_prop _ g₂,
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refine set_quotient.rec_prop _ g₃,
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clear g₁ g₂ g₃, intro g₁ g₂ g₃,
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exact ap class_of !mul.assoc
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end
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theorem quotient_one_mul (g : qg N) : 1 * g = g :=
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begin
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refine set_quotient.rec_prop _ g, clear g, intro g,
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exact ap class_of !one_mul
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end
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theorem quotient_mul_one (g : qg N) : g * 1 = g :=
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begin
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refine set_quotient.rec_prop _ g, clear g, intro g,
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exact ap class_of !mul_one
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end
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theorem quotient_mul_left_inv (g : qg N) : g⁻¹ * g = 1 :=
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begin
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refine set_quotient.rec_prop _ g, clear g, intro g,
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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 : CommGroup} {N : normal_subgroup_rel G} (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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refine set_quotient.rec_prop _ h, clear h, intro h,
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apply ap class_of, esimp, apply mul.comm
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end
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end
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variable (N)
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definition group_qg [constructor] : group (qg N) :=
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group.mk quotient_mul _ quotient_mul_assoc quotient_one quotient_one_mul quotient_mul_one
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quotient_inv quotient_mul_left_inv
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definition quotient_group [constructor] : Group :=
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Group.mk _ (group_qg N)
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definition comm_group_qg [constructor] {G : CommGroup} (N : normal_subgroup_rel G)
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: comm_group (qg N) :=
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⦃comm_group, group_qg N, mul_comm := quotient_mul_comm⦄
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definition quotient_comm_group [constructor] {G : CommGroup} (N : subgroup_rel G)
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: CommGroup :=
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CommGroup.mk _ (comm_group_qg (normal_subgroup_rel_comm N))
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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 comm_gq_map {G : CommGroup} (N : subgroup_rel G) : G →g quotient_comm_group N :=
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begin
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fapply homomorphism.mk,
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exact class_of,
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exact λ g h, idp
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end
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namespace quotient
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notation `⟦`:max a `⟧`:0 := qg_map a _
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end quotient
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open quotient
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variable {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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apply eq_of_rel,
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have e : (g * 1⁻¹ = g),
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from calc
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g * 1⁻¹ = g * 1 : one_inv
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... = g : mul_one,
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unfold quotient_rel, rewrite e, exact H
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end
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definition comm_gq_map_eq_one {K : subgroup_rel A} (g :A) (H : K g) : comm_gq_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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from calc
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g * 1⁻¹ = g * 1 : one_inv
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... = g : mul_one,
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unfold quotient_rel, xrewrite e, exact H
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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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begin
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have e : (g * 1⁻¹ = g),
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from calc
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g * 1⁻¹ = g * 1 : one_inv
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... = g : mul_one,
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rewrite (inverse e),
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apply rel_of_eq _ 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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(g : quotient_group N) : G' :=
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begin
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refine set_quotient.elim f _ g,
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intro g h K,
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apply eq_of_mul_inv_eq_one,
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have e : f (g * h⁻¹) = f g * (f h)⁻¹,
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from calc
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f (g * h⁻¹) = f g * (f h⁻¹) : to_respect_mul
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... = f g * (f h)⁻¹ : to_respect_inv,
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rewrite (inverse e),
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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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begin
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fapply homomorphism.mk,
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-- define function
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{ exact quotient_group_elim_fun f H },
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{ intro g h, induction g using set_quotient.rec_prop with g,
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induction h using set_quotient.rec_prop with h,
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krewrite (inverse (to_respect_mul (qg_map N) g h)),
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unfold qg_map, esimp, exact to_respect_mul f g h }
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end
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definition quotient_group_compute (f : G →g G') (H : Π⦃g⦄, N g → f g = 1) :
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quotient_group_elim f H ∘g qg_map N ~ f :=
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begin
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intro g, 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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: ( 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 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 ∘g qg_map N = f) :=
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begin
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fapply is_contr.mk,
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-- give center of contraction
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{ fapply sigma.mk, exact quotient_group_elim f H, apply homomorphism_eq, exact quotient_group_compute f H },
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-- give contraction
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{ intro pair, induction pair with g p, fapply sigma_eq,
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{esimp, apply homomorphism_eq, symmetry, exact gelim_unique f H g (homotopy_of_homomorphism_eq p)},
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{fapply is_prop.elimo} }
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end
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definition comm_group_quotient_homomorphism (A B : CommGroup)(K : subgroup_rel A)(L : subgroup_rel B) (f : A →g B)
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(p : Π(a:A), K(a) → L(f a)) : quotient_comm_group K →g quotient_comm_group L :=
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begin
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fapply quotient_group_elim,
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exact (comm_gq_map L) ∘g f,
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intro a,
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intro k,
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exact @comm_gq_map_eq_one B L (f a) (p a k),
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end
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definition comm_group_first_iso_thm (A B : CommGroup) (f : A →g B) : quotient_comm_group (kernel_subgroup f) ≃g comm_image f :=
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begin
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fapply isomorphism.mk,
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fapply quotient_group_elim,
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fapply image_lift,
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intro a,
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intro k,
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fapply image_incl_eq_one,
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exact k,
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exact sorry,
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-- show that the above map is injective and surjective.
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end
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definition comm_group_kernel_factor {A B C: CommGroup} (f : A →g B)(g : A →g C){i : C →g B}(H : f = i ∘g g )
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: Π a:A , kernel_subgroup(g)(a) → kernel_subgroup(f)(a) :=
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begin
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intro a,
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intro p,
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exact calc
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f a = i (g a) : homotopy_of_eq (ap group_fun H) a
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... = i 1 : ap i p
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... = 1 : respect_one i
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end
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definition comm_group_kernel_equivalent {A B : CommGroup} (C : CommGroup) (f : A →g B)(g : A →g C)(i : C →g B)(H : f = i ∘g g )(K : is_embedding i)
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: Π a:A , kernel_subgroup(g)(a) ↔ kernel_subgroup(f)(a) :=
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begin
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intro a,
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fapply iff.intro,
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exact comm_group_kernel_factor f g H a,
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intro p,
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apply @is_injective_of_is_embedding _ _ i _ (g a) 1,
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exact calc
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i (g a) = f a : (homotopy_of_eq (ap group_fun H) a)⁻¹
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... = 1 : p
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... = i 1 : (respect_one i)⁻¹
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end
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definition comm_group_kernel_image_lift (A B : CommGroup) (f : A →g B)
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: Π a : A, kernel_subgroup(image_lift(f))(a) ↔ kernel_subgroup(f)(a) :=
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begin
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fapply comm_group_kernel_equivalent (comm_image f) (f) (image_lift(f)) (image_incl(f)),
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exact image_factor f,
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exact is_embedding_of_is_injective (image_incl_injective(f)),
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end
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definition comm_group_kernel_quotient_to_image {A B : CommGroup} (f : A →g B)
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: quotient_comm_group (kernel_subgroup f) →g comm_image (f) :=
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begin
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fapply quotient_group_elim (image_lift f), intro a, intro p,
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apply iff.mpr (comm_group_kernel_image_lift _ _ f a) p
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end
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definition is_surjective_kernel_quotient_to_image {A B : CommGroup} (f : A →g B)
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: is_surjective (comm_group_kernel_quotient_to_image f) :=
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begin
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intro b, exact sorry
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-- have H : is_surjective (image_lift f)
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end
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print iff.mpr
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/- set generating normal subgroup -/
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section
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parameters {A₁ : CommGroup} (S : A₁ → Prop)
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variable {A₂ : CommGroup}
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inductive generating_relation' : A₁ → Type :=
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| rincl : Π{g}, S g → generating_relation' g
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| rmul : Π{g h}, generating_relation' g → generating_relation' h → generating_relation' (g * h)
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| rinv : Π{g}, generating_relation' g → generating_relation' g⁻¹
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| rone : generating_relation' 1
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open generating_relation'
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definition generating_relation (g : A₁) : Prop := ∥ generating_relation' g ∥
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local abbreviation R := generating_relation
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definition gr_one : R 1 := tr (rone S)
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definition gr_inv (g : A₁) : R g → R g⁻¹ :=
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trunc_functor -1 rinv
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definition gr_mul (g h : A₁) : R g → R h → R (g * h) :=
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trunc_functor2 rmul
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definition normal_generating_relation : subgroup_rel A₁ :=
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⦃ subgroup_rel,
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R := R,
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Rone := gr_one,
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Rinv := gr_inv,
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Rmul := gr_mul⦄
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parameter (A₁)
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definition quotient_comm_group_gen : CommGroup := quotient_comm_group normal_generating_relation
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definition gqg_map [constructor] : A₁ →g quotient_comm_group_gen :=
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qg_map _
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parameter {A₁}
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definition gqg_eq_of_rel {g h : A₁} (H : S (g * h⁻¹)) : gqg_map g = gqg_map h :=
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eq_of_rel (tr (rincl H))
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definition gqg_elim [constructor] (f : A₁ →g A₂) (H : Π⦃g⦄, S g → f g = 1)
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: quotient_comm_group_gen →g A₂ :=
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begin
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apply quotient_group_elim f,
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intro g r, induction r with r,
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induction r with g s g h r r' IH1 IH2 g r IH,
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{ exact H s },
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{ exact !respect_mul ⬝ ap011 mul IH1 IH2 ⬝ !one_mul },
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{ exact !respect_inv ⬝ ap inv IH ⬝ !one_inv },
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{ apply respect_one }
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end
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definition gqg_elim_compute (f : A₁ →g A₂) (H : Π⦃g⦄, S g → f g = 1)
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: gqg_elim f H ∘g gqg_map ~ f :=
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begin
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intro g, reflexivity
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end
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definition gqg_elim_unique (f : A₁ →g A₂) (H : Π⦃g⦄, S g → f g = 1)
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(k : quotient_comm_group_gen →g A₂) : ( k ∘g gqg_map ~ f ) → k ~ gqg_elim f H :=
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!gelim_unique
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end
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end group
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