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:

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

Thus from Results concerning Order of Element, $$\left \langle {g^{n/d}}\right \rangle = \left\{{e, g^{n/d}, \left({g^{n/d}}\right)^2, \ldots, \left({g^{n/d}}\right)^{d-1}}\right\}$$ and the result follows.