Cyclic group

$Exponent$

a^{0} := e

,

a^{1} := a

,

a^{n + 1} := a^{n} a

a^{- n} := (a^{- 1})^{n}

Group generated by

a

⟨ a ⟩ := {a^{k} ∣ k \in Z}

An order of

a

or d a := ∣ ⟨ a ⟩ ∣

or d a

is finite

\Rightarrow or d a

becomes the smallest natural number

n

that

a^{n} = e

x^{t} = e \Leftrightarrow n ∣ t

G

is a cyclic group :

\exists a \in G (⟨ a ⟩ = G)

a

is a generator of

G

⟨ a ⟩ = G

Subgroup of cyclic group id cyclic group

The subgroup of

G

with order

m

uniquely exists