Dihedral Group of even order

Question

Prove for odd n, $D_{2n}$, dihedral group of 2n, isomorphic to $D_{n} \times Z_{2}$. Is it true for even n?

Answer 1

Hint

$$D_{2n}=\langle r,s\;|\; r^{2n}=s^2=1, \; rs=sr^{-1}\rangle$$ Consider $H=\langle r^2,s\rangle$ and $K=\langle r^n\rangle$.
Prove that $D_{2n}=H\times K$, $H\cong D_n$ and $K\cong \Bbb{Z}_2$.

Counterexample for $n$ even

$D_{16}\not\cong D_8\times \Bbb{Z_2}$

This can be done by considering the number of elements of order 2 in each group.