34 lines
1.1 KiB
Text
34 lines
1.1 KiB
Text
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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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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 .product_group .free_group .free_commutative_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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variables (X : Set) {l l' : list (X ⊎ X)}
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section
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parameters {I : Set} (Y : I → CommGroup)
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definition dirsum_carrier : CommGroup := free_comm_group (trunctype.mk (Σi, Y i) _)
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local abbreviation ι := @free_comm_group_inclusion
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inductive dirsum_rel : dirsum_carrier → Type :=
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| rmk : Πi y₁ y₂, dirsum_rel (ι ⟨i, y₁⟩ * ι ⟨i, y₂⟩ * (ι ⟨i, y₁ * y₂⟩)⁻¹)
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definition direct_sum : CommGroup := quotient_comm_group_gen dirsum_carrier (λg, ∥dirsum_rel g∥)
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end
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end group
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