If $G$ is a group, then the center of $G$, assume its $Z(G)$, is the set of all elements of $G$ which commute with everything. So $Z(G)$ $=$ {$x$ $\in$ $G$: $xy$ $=$ $yx$ for all $y$ $\in$ $G$}. Prove that $Z(G)$ is a normal subgroup of $G$ for any group $G$.
First I need to show $G$ is a subgroup; and then show it's normal.
To show its a subgroup I know I need to prove that it has an identity, all elements contain inverses and it is closed under the operation