p-Group Camina pairs

Let $(G,Z(G))$ be a Camina pair. We prove that $G$ must be a $p$-group for some prime $p$. We also prove that $\mathrm{\mid }Z(G)\mathrm{\mid }<\mathrm{\mid }G:Z(G){\mathrm{\mid }}^{3\mathrm{/}4}$. Also, we discuss how one might build examples with $\mathrm{\mid }Z(G)\mathrm{\mid }>\mathrm{\mid }G:Z(G){\mathrm{\mid }}^{1\mathrm{/}2}$, although we are not able to prove the existence of such examples.