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Rejig subgroups, add ideals (#79)

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Patrick Stevens
2019-11-19 19:56:01 +00:00
committed by GitHub
parent f0790e4f52
commit f2f4e867fc
8 changed files with 69 additions and 145 deletions
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9
Groups/Subgroups/Normal/Definition.agda
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@@ -20,10 +20,7 @@ open import Groups.QuotientGroup.Definition
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Groups.Subgroups.Normal.Definition where
module Groups.Subgroups.Normal.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} (G : Group S _+_) where
record NormalSubgroup {a} {b} {c} {d} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) (H : Group T _·B_) {f : B → A} (hom : GroupHom H G f) : Set (a ⊔ b ⊔ c ⊔ d) where
open Setoid S
field
subgroup : Subgroup G H hom
normal : {g : A} {h : B} → Sg B (λ fromH → (g ·A (f h)) ·A (Group.inverse G g) ∼ f fromH)
normalSubgroup : {c : _} {pred : A → Set c} (wd : {x y : A} → (Setoid._∼_ S x y) → (pred x → pred y)) (sub : subgroup G wd) → Set (a ⊔ c)
normalSubgroup {pred = pred} wd sub = {g k : A} → pred k → pred (g + (k + Group.inverse G g))

34
Groups/Subgroups/Normal/Lemmas.agda
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@@ -58,25 +58,29 @@ injectionFromKernelToGIsHom : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid
GroupHom.groupHom (injectionFromKernelToGIsHom {S = S} G hom) {kerOfElt x prX} {kerOfElt y prY} = Equivalence.reflexive (Setoid.eq S)
GroupHom.wellDefined (injectionFromKernelToGIsHom G hom) {kerOfElt x prX} {kerOfElt y prY} i = i
groupKernelGroupIsSubgroup : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) → Subgroup G (groupKernelGroup G hom) (injectionFromKernelToGIsHom G hom)
Subgroup.fInj (groupKernelGroupIsSubgroup {S = S} {T = T} G {f = f} hom) = record { wellDefined = λ {x} {y} → GroupHom.wellDefined (injectionFromKernelToGIsHom G hom) {x} {y} ; injective = λ {x} {y} → inj {x} {y} }
where
inj : {x : GroupKernelElement G hom} → {y : GroupKernelElement G hom} → Setoid._∼_ S (injectionFromKernelToG G hom x) (injectionFromKernelToG G hom y) → Setoid._∼_ (groupKernel G hom) x y
inj {kerOfElt x prX} {kerOfElt y prY} = id
groupKernelGroupPred : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) → A → Set d
groupKernelGroupPred {T = T} G {H = H} {f = f} hom a = Setoid._∼_ T (f a) (Group.0G H)
groupKernelGroupIsNormalSubgroup : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) → NormalSubgroup G (groupKernelGroup G hom) (injectionFromKernelToGIsHom G hom)
NormalSubgroup.subgroup (groupKernelGroupIsNormalSubgroup G hom) = groupKernelGroupIsSubgroup G hom
NormalSubgroup.normal (groupKernelGroupIsNormalSubgroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {f = f} hom) {g} {kerOfElt h prH} = kerOfElt ((g ·A h) ·A Group.inverse G g) ans , Equivalence.reflexive (Setoid.eq S)
groupKernelGroupPredWd : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) → {x y : A} → (Setoid._∼_ S x y) → (groupKernelGroupPred G hom x → groupKernelGroupPred G hom y)
groupKernelGroupPredWd {S = S} {T = T} G hom {x} {y} x=y fx=0 = Equivalence.transitive (Setoid.eq T) (GroupHom.wellDefined hom (Equivalence.symmetric (Setoid.eq S) x=y)) fx=0
groupKernelGroupIsSubgroup : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) → subgroup G (groupKernelGroupPredWd G hom)
_&_&_.one (groupKernelGroupIsSubgroup {S = S} {T = T} G {H = H} hom) g=0 h=0 = Equivalence.transitive (Setoid.eq T) (GroupHom.groupHom hom) (Equivalence.transitive (Setoid.eq T) (Group.+WellDefined H g=0 h=0) (Group.identLeft H))
_&_&_.two (groupKernelGroupIsSubgroup G hom) = imageOfIdentityIsIdentity hom
_&_&_.three (groupKernelGroupIsSubgroup {S = S} {T = T} G {H = H} hom) g=0 = Equivalence.transitive (Setoid.eq T) (homRespectsInverse hom) (Equivalence.transitive (Setoid.eq T) (inverseWellDefined H g=0) (invIdent H))
groupKernelGroupIsNormalSubgroup : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) → normalSubgroup G (groupKernelGroupPredWd G hom) (groupKernelGroupIsSubgroup G hom)
groupKernelGroupIsNormalSubgroup {T = T} G {H = H} hom k=0 = transitive groupHom (transitive (+WellDefined reflexive groupHom) (transitive (+WellDefined reflexive (transitive (+WellDefined k=0 reflexive) identLeft)) (transitive (symmetric groupHom) (transitive (wellDefined (Group.invRight G)) (imageOfIdentityIsIdentity hom)))))
where
open Setoid T
open Group H
open Equivalence eq
ans : f ((g ·A h) ·A Group.inverse G g) ∼ Group.0G H
ans = transitive (GroupHom.groupHom hom) (transitive (Group.+WellDefined H (GroupHom.groupHom hom) reflexive) (transitive (Group.+WellDefined H (Group.+WellDefined H reflexive prH) reflexive) (transitive (Group.+WellDefined H (Group.identRight H) reflexive) (transitive (symmetric (GroupHom.groupHom hom)) (transitive (GroupHom.wellDefined hom (Group.invRight G)) (imageOfIdentityIsIdentity hom))))))
open GroupHom hom
abelianGroupSubgroupIsNormal : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {underG : Group S _+A_} (G : AbelianGroup underG) {H : Group T _+B_} {f : B → A} {hom : GroupHom H underG f} (s : Subgroup underG H hom) → NormalSubgroup underG H hom
NormalSubgroup.subgroup (abelianGroupSubgroupIsNormal G H) = H
NormalSubgroup.normal (abelianGroupSubgroupIsNormal {S = S} {underG = G} record { commutative = commutative } H) {g} {h} = h , transitive (+WellDefined commutative reflexive) (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight))
abelianGroupSubgroupIsNormal : {a b c : _} {A : Set a} {B : Set b} {S : Setoid {a} {b} A} {_+_ : A → A → A} {G : Group S _+_} {pred : A → Set c} (predWd : {x y : A} → (Setoid._∼_ S x y) → (pred x → pred y)) → (s : subgroup G predWd) → AbelianGroup G → normalSubgroup G predWd s
abelianGroupSubgroupIsNormal {S = S} {_+_ = _+_} {G = G} predWd record { one = respectsPlus ; two = respectsId ; three = respectsInv } abelian {k} {l} prK = predWd (transitive (transitive (transitive (symmetric identLeft) (+WellDefined (symmetric invRight) reflexive)) (symmetric +Associative)) (+WellDefined reflexive commutative)) prK
where
open Setoid S
open Group G
open Equivalence (Setoid.eq S)
open AbelianGroup abelian
open Setoid S
open Equivalence eq