Let $D_{2n}$ be the dihedral group of order $2n$. Let $H$ be the set of rotations of the regular $n$-gon. Is $H\lhd D_{2n}$?

Question

Let $D_{2n}$ be the dihedral group of order $2n$, i.e., the group of symmetries of the regular $n$-gon.

Let $H$ be the set of rotations of the regular $n$-gon.

Is $H\lhd D_{2n}$?

Answer 1

Yes, it is. Moreover, the dihedral group happens to be a semidirect product of the subgroup of rotations of regular $n$-gon and a subgroup, generated by a mirror symmetry. You can find more about semidirect products here: https://en.wikipedia.org/wiki/Semidirect_product