Tidy up groups more (#68)

Groups/Cyclic/Definition.agda

@@ -0,0 +1,39 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Setoids.Setoids
+open import Sets.EquivalenceRelations
+open import Functions
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+open import Numbers.Naturals.Naturals
+open import Numbers.Integers.Integers
+open import Numbers.Integers.Addition
+open import Sets.FinSet
+open import Groups.Homomorphisms.Definition
+open import Groups.Groups
+open import Groups.Subgroups.Definition
+open import Groups.Abelian.Definition
+open import Groups.Definition
+
+module Groups.Cyclic.Definition {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) where
+
+open Setoid S
+open Group G
+open Equivalence eq
+
+positiveEltPower : (x : A) (i : ℕ) → A
+positiveEltPower x 0 = Group.0G G
+positiveEltPower x (succ i) = x · (positiveEltPower x i)
+
+positiveEltPowerCollapse : (x : A) (i j : ℕ) → Setoid._∼_ S ((positiveEltPower x i) · (positiveEltPower x j)) (positiveEltPower x (i +N j))
+positiveEltPowerCollapse x zero j = Group.identLeft G
+positiveEltPowerCollapse x (succ i) j = transitive (symmetric +Associative) (+WellDefined reflexive (positiveEltPowerCollapse x i j))
+
+elementPower : (x : A) (i : ℤ) → A
+elementPower x (nonneg i) = positiveEltPower x i
+elementPower x (negSucc i) = Group.inverse G (positiveEltPower x (succ i))
+
+record CyclicGroup : Set (m ⊔ n) where
+  field
+    generator : A
+    cyclic : {a : A} → (Sg ℤ (λ i → Setoid._∼_ S (elementPower generator i) a))