Prove that if $\displaystyle G/Z(G)$ is cyclic, then $\displaystyle G$ is abelian. (If $\displaystyle G/Z(G)$ is cyclic with generator $\displaystyle xZ(G)$, show that every element of $\displaystyle G$ can be written in the form $\displaystyle x^az$ for some integer $\displaystyle a$ and some $\displaystyle z$ in $\displaystyle Z(G)$)

Z(G) = { g in G | gx = xg for every x in G}