Central Subgroup is Normal/Proof 1

Proof
Let $H$ be a central subgroup of $G$.

By definition of central subgroup:
 * $H \subseteq \map Z G$

where $\map Z G$ is the center of $G$.

Thus we have that $H$ is a group which is a subset of $\map Z G$.

Therefore by definition $H$ is a subgroup of $\map Z G$.

We also have from Center of Group is Abelian Subgroup that $\map Z G$ is an abelian group.

It follows from Subgroup of Abelian Group is Normal that $Z$ is a normal subgroup of $G$.