Order of element in the center of $G$ If $G$ is a group of order $p^n$, where $p$ is prime, then by the class equation, the center of $G$, $Z(G)$, is nontrivial. But must the center specifically contain an element of order $p$? 

If $P$ is a $p$-group and $g\in P$ is nontrivial, then $g$ has order $p^k$, in which case $g^{p^{\large k-1}}$ has order $p$. We conclude every $p$-group has an element of order $p$. This applies in particular with the center $Z(P)$ of any $p$-group $P$, which is a nontrivial subgroup.
