Question:
Determine the number of subgroups of index $2$ in $D_{2n}$.
Consider $D_{2n}=\langle r,s|r^n=1, s^2=1, sr=r^{-1}s\rangle$.
Take $N=\langle r\rangle$ and consider canonical homomorphism $G$ to $G/N$.
Then $\langle s\rangle$ is isomorphic to $G/N$. Hence $D_n$ is semidirect product of $C_2$ and $C_n$.
So my claim is there is unique subgroups $N$ of index 2.
Thanks
my argument correct? If so how do I prove the claim.