Fix up the structure of Actions and SymmetricGroups (#67)

Groups/Actions/Definition.agda

@@ -0,0 +1,26 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Setoids.Setoids
+open import Functions
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+open import Numbers.Naturals.Naturals
+open import Sets.FinSet
+open import Groups.Definition
+open import Groups.Lemmas
+open import Groups.Groups
+open import Groups.Groups2
+open import Sets.EquivalenceRelations
+
+module Groups.Actions.Definition where
+
+record GroupAction {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·_ : A → A → A} {B : Set n} (G : Group S _·_) (X : Setoid {n} {p} B) : Set (m ⊔ n ⊔ o ⊔ p) where
+  open Group G
+  open Setoid S renaming (_∼_ to _∼G_)
+  open Setoid X renaming (_∼_ to _∼X_)
+  field
+    action : A → B → B
+    actionWellDefined1 : {g h : A} → {x : B} → (g ∼G h) → action g x ∼X action h x
+    actionWellDefined2 : {g : A} → {x y : B} → (x ∼X y) → action g x ∼X action g y
+    identityAction : {x : B} → action 0G x ∼X x
+    associativeAction : {x : B} → {g h : A} → action (g · h) x ∼X action g (action h x)