Center of Group is Normal Subgroup/Proof 1

Theorem
The center $Z \left({G}\right)$ of any group $G$ is a normal subgroup of $G$ which is abelian.

Proof
We have the result Center is Abelian Subgroup.

Since $g x = x g$ for each $g \in G$ and $x \in Z \left({G}\right)$, we have $g Z \left({G}\right) = Z \left({G}\right) g$.

Thus, $Z \left({G}\right) \triangleleft G$.