Order of Cyclic Group equals Order of Generator

Theorem
Let $G$ be a finite cyclic group which is generated by $a \in G$.

Then:


 * $\order a = \order G$

where:
 * $\order a$ denotes the order of $a$ in $G$
 * $\order G$ denotes the order of $G$.

Proof
Let $\left\lvert{a}\right\vert = n$.

From List of Elements in Finite Cyclic Group:
 * $G = \set {a_0, a_1, \ldots, a_{n - 1} }$

Hence the result.