Normal subgroups and quotient groups #2854
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| quotientGroup : Group c (c ⊔ ℓ ⊔ c′) | ||
| quotientGroup = record { isGroup = quotientIsGroup } | ||
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| project : Group.Carrier G → Group.Carrier quotientGroup |
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My personal preference is for π (for projection), echoing ι (for inclusion), but I won't fight for it...
... maybe I should?
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I've got no strong feeling here
Taneb
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I noted that every time I used normal it was under sym This felt like a good reason to reverse it
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| data _≋_ (x y : Carrier) : Set (c ⊔ ℓ ⊔ c′) where | ||
| _by_ : ∀ g → ι g ∙ x ≈ y → x ≋ y |
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Are you sticking with this version, or the symmetric one which multiplies on each side of the equation?
| data _≋_ (x y : Carrier) : Set (c ⊔ ℓ ⊔ c′) where | |
| _by_ : ∀ g → ι g ∙ x ≈ y → x ≋ y | |
| data _≋_ (x y : Carrier) : Set (c ⊔ ℓ ⊔ c′) where | |
| by: ∀ g h → ι g ∙ x ≈ ι h ∙ y → x ≋ y |
plus some smart constructors/pattern synonyms to achieve the various properties?
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I want to experiment with the symmetric version more, but:
- it means we need to carry around more data, operationally (minor issue)
- I think it works out pretty close to being equivalent to this construction on the Grothendieck group for a cancellative monoid, and if that's the case I'd rather have that in two steps explicitly
| private | ||
| module N = NormalSubgroup N | ||
| open NormalSubgroup N using (ι; module ι; conjugate; normal) |
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Can simplify this to:
| private | |
| module N = NormalSubgroup N | |
| open NormalSubgroup N using (ι; module ι; conjugate; normal) | |
| private | |
| open module N = NormalSubgroup N using (ι; module ι; conjugate; normal) |
| abelian⇒subgroup-normal : ∀ {c′ ℓ′} → Commutative G._≈_ G._∙_ → (subgroup : Subgroup c′ ℓ′) → IsNormal subgroup | ||
| abelian⇒subgroup-normal ∙-comm subgroup = record { normal = λ n g → ∙-comm (ι n) g } | ||
| where open Subgroup subgroup |
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I'd maybe be tempted to lift this out to Algebra.Construct.Quotient.AbelianGroup to bundle up the quotient-out-of-any-subgroup construction, and simplify the downstream imports in #2855 ?
Builds off #2852, continuing towards #2729 in bitesize chunks.