Center of Group is Abelian Subgroup

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

Proof
By Center of Group is Subgroup, $Z \left({G}\right)$ is a subgroup of $G$.

The definition of the center $Z \left({G}\right)$ grants that all elements of $Z \left({G}\right)$ commute with all elements of $G$.

In particular, all elements of $Z \left({G}\right)$ commute with all elements of $Z \left({G}\right)$ as $Z \left({G}\right) \subseteq G$.

Therefore $Z \left({G}\right)$ is abelian.

Also see

 * Group equals Center iff Abelian