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:

$H \subseteq \map Z G$

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.

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

$\blacksquare$