Prove that if $G/Z(G)$ is cyclic, then $G$ is abelian. (If $G/Z(G)$ is cyclic with generator $xZ(G)$, show that every element of $G$ can be written in the form $x^az$ for some integer $a$ and some $z$ in $Z(G)$)
2. Let $\left< x\text{Z}(G) \right> = G/\text{Z}(G)$. If $\alpha \in G$ then form $\alpha \text{Z}(G) \in G/\text{Z}(G) = x^a \text{Z}(G)$. Thus, $\alpha = x^a z_1$ for $z_1 \in \text{Z}(G)$. Likewise $\beta = x^b z_2$.
But then $\alpha \beta = x^a z_1 x^b z_2 = x^{a+b} z_1 z_2 = x^{b} z_2 x^a z_1 = \beta \alpha$.