2016-10-13 19:04:57 +00:00
|
|
|
|
/-
|
|
|
|
|
|
Copyright (c) 2015 Floris van Doorn. All rights reserved.
|
|
|
|
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
|
|
|
|
|
Authors: Floris van Doorn, Egbert Rijke
|
|
|
|
|
|
|
|
|
|
|
|
Constructions with groups
|
|
|
|
|
|
-/
|
|
|
|
|
|
|
|
|
|
|
|
import algebra.group_theory hit.set_quotient types.list types.sum .subgroup .quotient_group
|
|
|
|
|
|
|
|
|
|
|
|
open eq algebra is_trunc set_quotient relation sigma sigma.ops prod prod.ops sum list trunc function
|
|
|
|
|
|
equiv
|
|
|
|
|
|
namespace group
|
|
|
|
|
|
|
|
|
|
|
|
variables {G G' : Group} (H : subgroup_rel G) (N : normal_subgroup_rel G) {g g' h h' k : G}
|
2016-11-24 04:54:57 +00:00
|
|
|
|
{A B : AbGroup}
|
2016-10-13 19:04:57 +00:00
|
|
|
|
|
|
|
|
|
|
/- Binary products (direct product) of Groups -/
|
|
|
|
|
|
definition product_one [constructor] : G × G' := (one, one)
|
|
|
|
|
|
definition product_inv [unfold 3] : G × G' → G × G' :=
|
|
|
|
|
|
λv, (v.1⁻¹, v.2⁻¹)
|
|
|
|
|
|
definition product_mul [unfold 3 4] : G × G' → G × G' → G × G' :=
|
|
|
|
|
|
λv w, (v.1 * w.1, v.2 * w.2)
|
|
|
|
|
|
|
|
|
|
|
|
section
|
|
|
|
|
|
local notation 1 := product_one
|
|
|
|
|
|
local postfix ⁻¹ := product_inv
|
|
|
|
|
|
local infix * := product_mul
|
|
|
|
|
|
|
|
|
|
|
|
theorem product_mul_assoc (g₁ g₂ g₃ : G × G') : g₁ * g₂ * g₃ = g₁ * (g₂ * g₃) :=
|
|
|
|
|
|
prod_eq !mul.assoc !mul.assoc
|
|
|
|
|
|
|
|
|
|
|
|
theorem product_one_mul (g : G × G') : 1 * g = g :=
|
|
|
|
|
|
prod_eq !one_mul !one_mul
|
|
|
|
|
|
|
|
|
|
|
|
theorem product_mul_one (g : G × G') : g * 1 = g :=
|
|
|
|
|
|
prod_eq !mul_one !mul_one
|
|
|
|
|
|
|
|
|
|
|
|
theorem product_mul_left_inv (g : G × G') : g⁻¹ * g = 1 :=
|
|
|
|
|
|
prod_eq !mul.left_inv !mul.left_inv
|
|
|
|
|
|
|
2016-11-24 04:54:57 +00:00
|
|
|
|
theorem product_mul_comm {G G' : AbGroup} (g h : G × G') : g * h = h * g :=
|
2016-10-13 19:04:57 +00:00
|
|
|
|
prod_eq !mul.comm !mul.comm
|
|
|
|
|
|
|
|
|
|
|
|
end
|
|
|
|
|
|
|
|
|
|
|
|
variables (G G')
|
|
|
|
|
|
definition group_prod [constructor] : group (G × G') :=
|
|
|
|
|
|
group.mk product_mul _ product_mul_assoc product_one product_one_mul product_mul_one
|
|
|
|
|
|
product_inv product_mul_left_inv
|
|
|
|
|
|
|
|
|
|
|
|
definition product [constructor] : Group :=
|
|
|
|
|
|
Group.mk _ (group_prod G G')
|
|
|
|
|
|
|
2016-11-24 04:54:57 +00:00
|
|
|
|
definition ab_group_prod [constructor] (G G' : AbGroup) : ab_group (G × G') :=
|
|
|
|
|
|
⦃ab_group, group_prod G G', mul_comm := product_mul_comm⦄
|
2016-10-13 19:04:57 +00:00
|
|
|
|
|
2016-11-24 04:54:57 +00:00
|
|
|
|
definition ab_product [constructor] (G G' : AbGroup) : AbGroup :=
|
|
|
|
|
|
AbGroup.mk _ (ab_group_prod G G')
|
2016-10-13 19:04:57 +00:00
|
|
|
|
|
|
|
|
|
|
infix ` ×g `:30 := group.product
|
|
|
|
|
|
|
|
|
|
|
|
end group
|
