Center of Group is Normal Subgroup/Proof 1

Theorem

Let $G$ be a group

The center $\map Z G$ of $G$ is a normal subgroup of $G$.

Proof

Recall that Center of Group is Abelian Subgroup.

Since $g x = x g$ for each $g \in G$ and $x \in \map Z G$:

$g \map Z G = \map Z G g$

Thus:

$\map Z G \lhd G$

$\blacksquare$