Cyclic Group is Abelian

Theorem
A cyclic group is always abelian.

Proof
This follows from Epimorphism from Integers to a Cyclic Group and Epimorphism Preserves Commutativity, as Integer Addition forms Abelian Group.


 * Alternatively, let $$G$$ be a cyclic group.

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

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

From Powers of Group Elements, $$x y = a^p a^q = a^{p + q} = a^{q + p} = a^q a^p = y x$$.

Thus $$\forall x, y \in G: x y = y x$$ and $$G$$ is abelian.