Is $P/Z(P)$ cyclic when $Z(P)$ is not trivial?

Question

I'm working my way through a proof given in Dummit and Foote.

Theorem: If $|P| = p^2$ for some prime $p$, then $P$ is abelian.

Proof: Since $Z(P)\not = 1$, it follows that $P/Z(P)$ is cyclic $\dots$

Im stuck at the very first line of the proof. I understand why $Z(P)$ isn't trivial (it follows from the class equation that $p\vert|Z(P)|$ so $|Z(P)|\not = 1$). However, I can't understand why that implies $P/Z(P)$ is cyclic.

Im not asking for a whole proof. I just want a few guidelines on how to show that $P/Z(P)$ is cyclic and I'll prove the rest myself.

Answer 1

Now that you know $Z(P)$ is not trivial, you will see that $P/Z(P)$ is of order $p$ or $1$. Then use Lagrange theorem to see that $P/Z(P)$ is cyclic.