Show the sub group $\langle r^k\rangle$ is normal in the dihedral group of order $2n$ when $k | n$ and $r$ is the rotation by $2\pi/n$

Question

let $ k | n ; n,k \in \Bbb Z $ let $r$ be the rotation in the plane by an angle $2 \pi \over n$

prove the subgroup $ \langle r^k\rangle $ of $D_n$ is normal.

Further is there a normal subgroup of a normal subgroup $ K \le H \le G $ such that K is not normal to G. I imagine it would be a subgroup of a dihedral since they are not abelian groups

Answer 1

Let $s$ be an axial symmetry in $D_n$. Then, $r$ and $s$ generate $D_n$, so it is sufficient to show that $sr^ks\in \langle r^k\rangle$ and $rr^kr^{-1}\in \langle r^k\rangle$.

By definition, $$sr^ks=r^{-k}=r^{n-k}=r^{(j-1)k}\in \langle r^k\rangle$$ where $jk=n$. Moreover, $rr^kr^{-1}=r^k\in \langle r^k\rangle $. Thus $\langle r^k\rangle$ is normal.