Solution
= Solution
The centre is
$$
Z(G)=\{z\in G:zg=gz\text{ for every }g\in G\}.
$$
Every <subgroup> $H\le Z(G)$ is normal, since $ghg^{-1}=h$ for all $g\in G$ and $h\in H$.
Write $D_{2n}=\langle r,s:r^n=s^2=1, srs=r^{-1}\rangle$. A central rotation must satisfy $r^k=r^{-k}$, and no reflection is central for $n\ge3$. Hence
$$
Z(D_{2n})=\begin{cases}\{1\},&n\text{ odd},\\\{1,r^{n/2}\},&n\text{ even}.\end{cases}
$$