= Solution
The <center of a group> contains $e$. If $z,w$ lie in the center, then for every $g\in G$,
$$
g(zw)g^{-1}=(gzg^{-1})(gwg^{-1})=zw,
\qquad
gz^{-1}g^{-1}=(gzg^{-1})^{-1}=z^{-1}.
$$
It is therefore closed under the <group operation> and <inverse elements>, and inherits <associativity>, proving it is a <subgroup>. Each of its elements commutes with every element of $G$, in particular with every other central element. Hence
$$
\boxed{Z(G)\leq G\text{ and }zw=wz\text{ for all }z,w\in Z(G).}
$$
Thus the center is an <abelian subgroup>.
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