If $H \leq G$ and $H \subset Z(G)$, the center of $G$, is $H \trianglelefteq G$? This is probably a very dumb question. Is it true that, in general, if $H$ is a subgroup of a group $G$, and $H \subset Z(G)$, the center of $G$, does it follow that $H$ is normal in $G$?
What I know so far that could potentially be useful:


*

*$Z(G)$ is normal in $G$.

*$H$ is normal in $G \Leftrightarrow gHg^{-1} \subseteq H$ for all $g \in G$.


This is part of an intermediate step for a homework problem. 

Yes. As a hint for how to prove it, let $h \in H$ and use the fact that $h \in Z(G)$ to simplify $ghg^{-1}$.
