> [!NOTE] Theorem (Cyclic groups are abelian) > [[Cyclic Group|Cyclic groups]] are [[Groups|abelian]]. *Proof*. Let $G$ be a cyclic group generated by $g.$ Let $a,b\in G.$ Then $a=g^{m}$ and $b=g^{n}$ for some $m,n\in \mathbb{Z}.$ Now by [[Product of Powers of a Group Element]], $ab=g^{m}g^{n}=g^{m+n}$ and $ba=g^{n}g^{m}=g^{n+m}.$ But $m+n=n+m$ by commutativity of integer addition. So $ab=ba.$