Normal subgroups and quotient groups #2854
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| Normal : ∀ {c′ ℓ′} → Subgroup c′ ℓ′ → Set (c ⊔ ℓ ⊔ c′) | ||
| Normal subgroup = ∀ n g → ∃[ n′ ] ι n′ G.∙ g G.≈ g G.∙ ι n |
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On the Structures/Bundles convention, should we call this IsNormal?
| Normal subgroup = ∀ n g → ∃[ n′ ] ι n′ G.∙ g G.≈ g G.∙ ι n | ||
| where open Subgroup subgroup | ||
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| record NormalSubgroup c′ ℓ′ : Set (c ⊔ ℓ ⊔ suc (c′ ⊔ ℓ′)) where |
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And hence this becomes (simply?) Normal, as the Bundled version of IsNormal?
| record NormalSubgroup c′ ℓ′ : Set (c ⊔ ℓ ⊔ suc (c′ ⊔ ℓ′)) where | ||
| field | ||
| subgroup : Subgroup c′ ℓ′ | ||
| normal : Normal subgroup |
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| normal : Normal subgroup | |
| isNormal : IsNormal subgroup |
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| open Subgroup subgroup public | ||
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| abelian⇒subgroup-normal : ∀ {c′ ℓ′} → Commutative G._≈_ G._∙_ → (subgroup : Subgroup c′ ℓ′) → Normal subgroup |
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Again
| abelian⇒subgroup-normal : ∀ {c′ ℓ′} → Commutative G._≈_ G._∙_ → (subgroup : Subgroup c′ ℓ′) → Normal subgroup | |
| abelian⇒subgroup-normal : ∀ {c′ ℓ′} → Commutative G._≈_ G._∙_ → (subgroup : Subgroup c′ ℓ′) → IsNormal subgroup |
| 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
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| -- every element of the subgroup commutes in G | ||
| Normal : ∀ {c′ ℓ′} → Subgroup c′ ℓ′ → Set (c ⊔ ℓ ⊔ c′) | ||
| Normal subgroup = ∀ n g → ∃[ n′ ] ι n′ G.∙ g G.≈ g G.∙ ι n |
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How does flipping the old definition improve matters?
Curious...
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If you look at the diff of the commit that flipped this (e3baf59#diff-4b2ec40733e56b6ad2e835149fee88672cc51d5ec131be90521ea0098f4813cc) in the quotient groups module, previously every time I was using normal I had to sym it, this way I can use it directly
Builds off #2852, continuing towards #2729 in bitesize chunks.