Set of Rotations Cyclic?

Question

For the dihedral group $D_{n}$ of order $2n$, is the group $R$ formed by its $n$ rotations cyclic in general? Or is the factor group $D_{n}/R$ cyclic? I am trying to show the series $D_{n}>R>(1)$ has abelian factors.

Answer 1

Yes, the rotations are cyclic, generated by a rotation of angle $2\pi/n$.

Answer 2

Yes, the rotation group $C_n$ is a cyclic group, and it is a subgroup of index $2$ in $D_n$, hence a normal subgroup. We can form the quotient $D_n/C_n$. The quotient group has $[D_n:C_n]=2$ elements, and hence must be cyclic, too.