Prove that every subgroup of the dihedral groups that has od

Prove that every subgroup of the dihedral groups that has odd order is cyclic.

Solution

Dn all the subgroups containing reflections have even order so they can\'t be made into a subgroup of D of odd order. This leaves us with the rotations.
Let H=<R^(360/n)>

If n is odd then the order of H is odd so every subgroup of H would be cyclic. In particular every subgroup of odd order would be cyclic.
If n is even it would still be the same because all the subgroups of odd order would be cyclic.