Cyclic Group is Abelian/Proof 1

Theorem
A cyclic group is always abelian.

Proof
Let $G$ be a cyclic group.

All elements of $G$ are of the form $a^n$, where $n \in \Z$.

Let $x, y \in G: x = a^p, y = a^q$.

From Powers of Group Elements: Sum of Indices:
 * $x y = a^p a^q = a^{p + q} = a^{q + p} = a^q a^p = y x$

Thus:
 * $\forall x, y \in G: x y = y x$

and $G$ is by definition abelian.