63 lines
2.1 KiB
Text
63 lines
2.1 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 algebra.group_theory hit.set_quotient types.list types.sum .subgroup .quotient_group
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open eq algebra is_trunc set_quotient relation sigma sigma.ops prod prod.ops sum list trunc function
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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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{A B : CommGroup}
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/- Binary products (direct product) of Groups -/
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definition product_one [constructor] : G × G' := (one, one)
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definition product_inv [unfold 3] : G × G' → G × G' :=
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λv, (v.1⁻¹, v.2⁻¹)
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definition product_mul [unfold 3 4] : G × G' → G × G' → G × G' :=
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λv w, (v.1 * w.1, v.2 * w.2)
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section
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local notation 1 := product_one
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local postfix ⁻¹ := product_inv
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local infix * := product_mul
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theorem product_mul_assoc (g₁ g₂ g₃ : G × G') : g₁ * g₂ * g₃ = g₁ * (g₂ * g₃) :=
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prod_eq !mul.assoc !mul.assoc
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theorem product_one_mul (g : G × G') : 1 * g = g :=
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prod_eq !one_mul !one_mul
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theorem product_mul_one (g : G × G') : g * 1 = g :=
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prod_eq !mul_one !mul_one
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theorem product_mul_left_inv (g : G × G') : g⁻¹ * g = 1 :=
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prod_eq !mul.left_inv !mul.left_inv
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theorem product_mul_comm {G G' : CommGroup} (g h : G × G') : g * h = h * g :=
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prod_eq !mul.comm !mul.comm
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end
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variables (G G')
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definition group_prod [constructor] : group (G × G') :=
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group.mk product_mul _ product_mul_assoc product_one product_one_mul product_mul_one
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product_inv product_mul_left_inv
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definition product [constructor] : Group :=
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Group.mk _ (group_prod G G')
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definition comm_group_prod [constructor] (G G' : CommGroup) : comm_group (G × G') :=
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⦃comm_group, group_prod G G', mul_comm := product_mul_comm⦄
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definition comm_product [constructor] (G G' : CommGroup) : CommGroup :=
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CommGroup.mk _ (comm_group_prod G G')
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infix ` ×g `:30 := group.product
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end group
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