Order of Subgroup of Cyclic Group

Theorem
Let $C_n = \left \langle {g} \right \rangle$ be the cyclic group of order $n$ which is generated by $g$ whose identity is $e$.

Let $a \in C_n: a = g^i$.

Let $H = \left\langle{a}\right\rangle$.

Then:
 * $\left\vert{H}\right\vert = \dfrac n {\gcd \left\{{n, i}\right\} }$

where:
 * $\left\vert{H}\right\vert$ denotes the order of $H$
 * $\gcd \left\{{n, i}\right\}$ denotes the greatest common divisor of $n$ and $i$.

Proof
The fact that $H$ is cyclic follows from Subgroup of Cyclic Group is Cyclic.

We need to show that $H$ has $\dfrac n d$ elements.

Let $\left\vert{H}\right\vert = k$.

By Non-Trivial Group has Non-Trivial Cyclic Subgroup:
 * $k = \left\vert{a}\right\vert$

where $\left\vert{a}\right\vert$ denotes the order of $a$.

That is:
 * $a^k = e$

We have that $a = g^i$.

So:

We now need to calculate the smallest $k$ such that:
 * $n \mathop \backslash i k$

where $\backslash$ denotes divisibility.

That is, the smallest $t \in \N$ such that $n t = i k$.

Let $d = \gcd \left\{{n, i}\right\}$.

Thus:
 * $t = \dfrac {k \left({i / d}\right)} {n / d}$

From Integers Divided by GCD are Coprime, $\dfrac n d$ and $\dfrac i d$ are coprime.

Thus from Euclid's Lemma:
 * $\dfrac n d \mathop \backslash k$

As $a \mathrel \backslash b \implies a \le b$, the smallest value of $k$ such that $\dfrac k {\left({n / d}\right)} \in \Z$ is $\dfrac n d$.

Hence the result.