Order of Cyclic Group equals Order of Generator

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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.

$\blacksquare$


Sources