Dihedral groups are solvable

Question

I'm trying to prove the dihedral groups are solvable for any Dn. I use the normal subgroup of all rotations, since the quotient of Dn/{rotations} is isomorphic to Z2 so it's abelian as well, so we get a solvable group.

Answer 1

I think your idea is correct: There is an exact sequence $$ 1 \to \mathbb{Z}/n\mathbb{Z} \to D_n \to \mathbb{Z}/2\mathbb{Z} \to 1, $$ where the map $\mathbb{Z}/n\mathbb{Z} \to D_n$ identifies $\mathbb{Z}/n\mathbb{Z}$ with the rotations and $D_n \to \mathbb{Z}/2\mathbb{Z}$ is the quotient map. Since $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}$ are solvable, so is $D_n$.