Recall definitions for group center and subgroup.

Let $G$ be a group.
The center $Z(G)=\{z\in G:\mathrm{\forall }g\in G,zg=gz\}$

$Z(G)\le G$

Proof

1. Non-empty
   $eg=g=ge$ for all $g\in G$
   so $e\in Z(G)$
   so $Z(G)\ne \varnothing$
2. Inverses
   Let $z\in Z(G)$. Then $zg=gz$ for all $g\in G$.
   $z^{-1}zg=z^{-1}gz$
   $g=z^{-1}gz$
   $g{z}^{-1}=z^{-1}gz{z}^{-1}$
   $g{z}^{-1}=z^{-1}g$
   so $z^{-1}\in Z(G)$
3. Closure
   Let $x$ and $y$ in $Z(G)$.
   $(xy)g=x(yg)=x(gy)=(xg)y=(gx)y=g(xy)$
   so $(xy)\in Z(G)$

QED