Properties of the center of a group

Let $G$ be a finite group and $Z (G)$ denote its center.

Prove that if $G / Z (G)$ is cyclic, then $G$ is abelian.
Prove that if $G$ is nonabelian, then $| Z (G) | \leq \frac{1}{4} | G |$ .

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Let $G$ be a finite group and $\operatorname{Z}(G)$ denote its center.
\begin{enumerate}[label=\alph*)]
	\item Prove that if $G/\operatorname{Z}(G)$ is cyclic, then $G$ is abelian.
	\item Prove that if $G$ is nonabelian, then $|\operatorname{Z}(G)|\leq \frac{1}{4}|G|$.
\end{enumerate}