Central Subgroup is Normal/Proof 1

Theorem
Let $G$ be a group.

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

Then $H$ is a normal subgroup of $G$.

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

By definition of central subgroup:
 * $H \subseteq Z \left({G}\right)$

where $Z \left({G}\right)$ is the center of $G$.

Thus we have that $H$ is a group which is a subset of $Z \left({G}\right)$.

Therefore by definition $H$ is a subgroup of $Z \left({G}\right)$.

We also have from Center is an Abelian Subgroup that $Z \left({G}\right)$ is an abelian group.

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