 Solution

The center of a group consists of its elements commuting with every element. First check it is a subgroup. The identity element is central. If $a,b\in Z(G)$ and $x\in G$, then $b^{-1}$ commutes with $x$ andThus $a{b}^{-1}\in Z(G)$, proving closure under the subgroup criterion. Moreover, if $z\in Z(G)$, then $xz{x}^{-1}=z$ for every $x\in G$. Hence conjugation fixes the center elementwise, soTherefore the center of a group is a normal subgroup.