List of Elements in Finite Cyclic Group

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

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

Let $\left \langle {a} \right \rangle$ be the cyclic subgroup generated by $a$.

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

That is, let $\left[{0 \,.\,.\, k-1}\right]$ be the integer interval from $0$ to $k-1$.

Then the mapping:
 * $\left[{0 \,.\,.\, k-1}\right] \to \left \langle {a} \right \rangle$:
 * $n \mapsto g^n$

is a bijection.

Proof
By Element to 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.