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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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46
Groups/Actions/Stabiliser.agda
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@@ -16,47 +16,25 @@ open import Groups.Groups2
open import Sets.EquivalenceRelations
open import Groups.Actions.Definition
module Groups.Actions.Stabiliser where
module Groups.Actions.Stabiliser {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+_ : A → A → A} {G : Group S _+_} (act : GroupAction G T) where
data Stabiliser {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+_ : A → A → A} {G : Group S _+_} (action : GroupAction G T) (x : B) : Set (a ⊔ b ⊔ c ⊔ d) where
stab : (g : A) → Setoid._∼_ T (GroupAction.action action g x) x → Stabiliser action x
open GroupAction act
open Setoid T
stabiliserSetoid : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+_ : A → A → A} {G : Group S _+_} (action : GroupAction G T) (x : B) → Setoid (Stabiliser action x)
Setoid._∼_ (stabiliserSetoid {S = S} action x) (stab g gx=x) (stab h hx=x) = Setoid._∼_ S g h
Equivalence.reflexive (Setoid.eq (stabiliserSetoid {S = S} action x)) {stab g _} = Equivalence.reflexive (Setoid.eq S)
Equivalence.symmetric (Setoid.eq (stabiliserSetoid {S = S} action x)) {stab g _} {stab h _} = Equivalence.symmetric (Setoid.eq S)
Equivalence.transitive (Setoid.eq (stabiliserSetoid {S = S} action x)) {stab g _} {stab h _} {stab i _} = Equivalence.transitive (Setoid.eq S)
stabiliserPred : (x : B) → (g : A) → Set d
stabiliserPred x g = (action g x) ∼ x
stabiliserGroupOp : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+_ : A → A → A} {G : Group S _+_} (action : GroupAction G T) {x : B} → Stabiliser action x → Stabiliser action x → Stabiliser action x
stabiliserGroupOp {T = T} {_+_ = _+_} action (stab p px=x) (stab q qx=x) = stab (p + q) (transitive (GroupAction.associativeAction action) (transitive (GroupAction.actionWellDefined2 action qx=x) px=x))
stabiliserWellDefined : (x : B) → {g h : A} → Setoid._∼_ S g h → (stabiliserPred x g) → stabiliserPred x h
stabiliserWellDefined x {g} {h} g=h gx=x = transitive (actionWellDefined1 (Equivalence.symmetric (Setoid.eq S) g=h)) gx=x
where
open Setoid T
open Equivalence eq
stabiliserGroup : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+_ : A → A → A} {G : Group S _+_} (action : GroupAction G T) {x : B} → Group (stabiliserSetoid action x) (stabiliserGroupOp action)
Group.+WellDefined (stabiliserGroup {T = T} {G = G} action {x}) {stab m mx=x} {stab n nx=x} {stab r rx=x} {stab s sx=x} m=r n=s = Group.+WellDefined G m=r n=s
stabiliserSubgroup : (x : B) → subgroup G (stabiliserWellDefined x)
_&_&_.one (stabiliserSubgroup x) gx=x hx=x = transitive associativeAction (transitive (actionWellDefined2 hx=x) gx=x)
where
open Setoid T
open Equivalence eq
Group.0G (stabiliserGroup {G = G} action) = stab (Group.0G G) (GroupAction.identityAction action)
Group.inverse (stabiliserGroup {T = T} {_+_ = _+_} {G = G} action {x}) (stab g gx=x) = stab (Group.inverse G g) (transitive {_} {GroupAction.action action ((inverse g) + g) x} (symmetric (transitive (GroupAction.associativeAction action) (GroupAction.actionWellDefined2 action gx=x))) (transitive (GroupAction.actionWellDefined1 action invLeft) (GroupAction.identityAction action)))
_&_&_.two (stabiliserSubgroup x) = identityAction
_&_&_.three (stabiliserSubgroup x) {g = g} gx=x = transitive (transitive (transitive (actionWellDefined2 (symmetric gx=x)) (symmetric associativeAction)) (actionWellDefined1 (invLeft {g}))) identityAction
where
open Equivalence eq
open Group G
open Setoid T
open Equivalence eq
Group.+Associative (stabiliserGroup {G = G} action) {stab m mx=x} {stab n nx=x} {stab o ox=x} = Group.+Associative G
Group.identRight (stabiliserGroup {G = G} action) {stab m mx=x} = Group.identRight G
Group.identLeft (stabiliserGroup {G = G} action) {stab m mx=x }= Group.identLeft G
Group.invLeft (stabiliserGroup {G = G} action) {stab m mx=x} = Group.invLeft G
Group.invRight (stabiliserGroup {G = G} action) {stab m mx=x} = Group.invRight G
stabiliserInjection : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+_ : A → A → A} {G : Group S _+_} (action : GroupAction G T) {x : B} → Stabiliser action x → A
stabiliserInjection action (stab g gx=x) = g
stabiliserInjectionIsHom : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+_ : A → A → A} {G : Group S _+_} (action : GroupAction G T) {x : B} → GroupHom (stabiliserGroup action {x}) G (stabiliserInjection action {x})
GroupHom.groupHom (stabiliserInjectionIsHom {S = S} action) {stab g gx=x} {stab h hx=x} = Equivalence.reflexive (Setoid.eq S)
GroupHom.wellDefined (stabiliserInjectionIsHom action) {stab g gx=x} {stab h hx=x} g=h = g=h
stabiliserIsSubgroup : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+_ : A → A → A} {G : Group S _+_} (action : GroupAction G T) {x : B} → Subgroup G (stabiliserGroup action) (stabiliserInjectionIsHom action {x})
SetoidInjection.wellDefined (Subgroup.fInj (stabiliserIsSubgroup action)) {stab g gx=x} {stab h hx=x} g=h = g=h
SetoidInjection.injective (Subgroup.fInj (stabiliserIsSubgroup action)) {stab g gx=x} {stab h hx=x} g=h = g=h