Theorem 6.1

Statement:

Every cyclic group is abelian.

Proof:

Let $G$ be a cyclic group and let $a$ be a generator of $G$ so that

(1)
\begin{align} G = \langle a \rangle = \{a^{n}\ |\ n \in \mathbb{Z}\}. \end{align}

If $g_{1}$ and $g_{2}$ are any two elements of $G$, there exists integers $r$ and $s$ such that $g_{1} = a^{r}$ and $g_{2} = a^{s}$. Then

(2)
$$g_{1}g_{2} = a^{r}a^{s} = a^{r+s} = a^{s+r} = a^{s}a^{r} = g_{2}g_{1},$$

so $G$ is abelian.