Order of Cyclic Group equals Order of Generator

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

Then:


 * $\left\lvert{a}\right\vert = \left\lvert{G}\right\vert$

where:
 * $\left\lvert{a}\right\vert$ denotes the order of $a$
 * $\left\lvert{G}\right\vert$ denotes the order of $G$.

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

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

Hence the result.