List of Elements in Finite Cyclic Group

Theorem
Let $G$ be a group whose identity is $e$.

Let $a \in G$ have finite order such that $\left|{a}\right| = k$.

Then:
 * $\left\{{a^0, a^1, a^2, \ldots, a^{k - 1}}\right\}$ is a complete repetition-free list of the elements of $\left \langle {a} \right \rangle$

where $\left \langle {a} \right \rangle$ is the cyclic group generated by $a$.

Proof
By Element to the Power of Remainder, every power of $a$ is equal to one appearing in the list $a^0, a^1, a^2, \ldots, a^{k - 1}$.

This list has to be repetition free, otherwise it would contain $a^m = a^n$ with $0 \le m < n < k$ which contradicts the hypothesis.