Center of Group is Abelian Subgroup

Theorem
Let $G$ be a group.

Let $\map Z G$ be the center of $G$.

Then $\map Z G$ is an abelian subgroup of $G$.

Proof
By Center of Group is Subgroup, $\map Z G$ is a subgroup of $G$.

The definition of the center $\map Z G$ grants that all elements of $\map Z G)$ commute with all elements of $G$.

In particular, all elements of $\map Z G$ commute with all elements of $\map Z G$ as $\map Z G \subseteq G$.

Therefore $\map Z G$ is abelian.

Also see

 * Group equals Center iff Abelian