Let $n$ be an even number. Prove that $D_n/Z(D_n)$ is isomorphic to $D_{n/2}$.
To avoid confusion say: $$D_n=\{a^ib^j:i\in\{0,1\}, j\in\{0,\ldots,n-1\}, ba=ab^{-1}\}.$$ Through some calculation we obtain that: $$Z(D_n)=\{e,b^{n/2}\}$$
I feel like this uses first isomorphism theorem, but I can't figure out the correct homomorphism $\phi$ where $\phi(D_n)=D_{n/2}$ and $\text{Ker}\ \phi=Z(D_n)$.
Having $\text{Ker}\ \phi=Z(D_n)$ just means that $e$ and $b^{n/2}$ are the only elements mapped to $e$ in $D_{n/2}$.