
cyclic
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}

Let $\displaystyle \left< x\text{Z}(G) \right> = G/\text{Z}(G)$. If $\displaystyle \alpha \in G$ then form $\displaystyle \alpha \text{Z}(G) \in G/\text{Z}(G) = x^a \text{Z}(G)$. Thus, $\displaystyle \alpha = x^a z_1$ for $\displaystyle z_1 \in \text{Z}(G)$. Likewise $\displaystyle \beta = x^b z_2$.
But then $\displaystyle \alpha \beta = x^a z_1 x^b z_2 = x^{a+b} z_1 z_2 = x^{b} z_2 x^a z_1 = \beta \alpha$.