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 violates the hypothesis.