Number of Powers of Cyclic Group Element

Theorem
Let $G$ be a cyclic group of order $n$, generated by $g$.

Let $d \backslash n$.

Then the element $g^{n/d}$ has $d$ distinct powers.

Proof
Follows directly from Order of Subgroup of Cyclic Group:


 * $\displaystyle \left \langle {g^{n/d}}\right \rangle = \frac n {\gcd \left\{{n, n/d}\right\}} = d$

Thus from List of Elements in Finite Cyclic Group:
 * $\left \langle {g^{n/d}}\right \rangle = \left\{{e, g^{n/d}, \left({g^{n/d}}\right)^2, \ldots, \left({g^{n/d}}\right)^{d-1}}\right\}$

and the result follows.