The center of a group

Subgroup. $e\in Z(G)$ trivially. If $g_1,g_2\in Z(G)$ and $h\in G$: $(g_1g_2)h=g_1hg_2=hg_1g_2$ (using commutation of $g_2$, then of $g_1$). So $g_1g_2\in Z(G)$. If $g\in Z(G)$, then $gh=hg$ for all $h$, so $g^{-1}h=hg^{-1}$ for all $h$, hence $g^{-1}\in Z(G)$.

Abelian. By definition, elements of $Z(G)$ commute with each other.

Normal. For $g\in Z(G)$ and $x\in G$: $xgx^{-1}=gxx^{-1}=g\in Z(G)$.

Conclusion. $Z(G)$ is an abelian normal subgroup of $G$.