Does Every Factor of 2n Form a Subgroup in Dihedral Groups?

[Kalinka35]

Let n be a positive integer and let m be a factor of 2n. Show that D_n (the dihedral group) contains a subgroup of order m.

I'm not really sure where to start with this one. I know that D_n is generated by two types of rotations: flipping the n-gon over about an axis, and rotating it 2π/n around an axis through the center. But how do I show that you can have a subgroup of order m for every factor of 2n?

Thanks.

 

[matt grime]

What if m divides n? What if m does not divide n? What is a polygon with m, or m/2 sides compared to one with n sides?