# List of Elements in Finite Cyclic Group

## Theorem

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

Let $a \in G$ have finite order $\order a = k$.

Let $\gen a$ be the cyclic subgroup generated by $a$.

Then:

$\set {a^0, a^1, a^2, \ldots, a^{k - 1} }$ is a complete repetition-free list of the elements of $\gen a$

That is, let $\closedint 0 {k - 1}$ be the integer interval from $0$ to $k - 1$.

Then the mapping:

$\closedint 0 {k - 1} \to \gen a$:
$n \mapsto g^n$

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.

$\blacksquare$