[Math] If $S$ is the cyclic subgroup of order n in the dihedral group $D_n$, show that $D_n/S$ is isomorphic to $Z_2$

Question

If $S$ is the cyclic subgroup of order $n$ in the dihedral group $D_n$, show that $D_n/S$ is isomorphic to $Z_2$.

I know I'm supposed to find an epimorphism from $D_n$ to $Z_2$, such that $S$ is its Kernel, so that the quotient group $D_n/S$ will be isomorphic to $Z_2$. But I have no idea of how to find such an isomorphism. Besides, I don't even know what $n$ is, so I have no way of finding the elements of $D_n$ and defining an epimorphism on them.

Any help will be appreciated!!!!

Answer 1

$D_n$ has order $2n$. If you have a subgroup $S$ of order $n$, then $D_n/S$ will have order 2 and hence is isomorphic to $\mathbb{Z}_2$.