**Q.** Prove that any subgroup of the center $Z(G)$ of $G$ is a normal subgroup of $G$.

When proving "If $G$ is a $p$-group of order $p^l$, then $G$ has a normal subgroup
of order $p^k$ for every $k ≤ l$.", the author seems to use the statement that any subgroup of the center $Z(G)$ of $G$ is a normal subgroup of $G$.

I have proved that $Z(G)$ is a normal subgroup of $G$, if we pick any subgroup of $Z(G)$, how to show that it is also a normal subgroup of $G$ ? I want to take *conjugates* but I am stuck here