Prove that any subgroup of of odd order is cyclic.

So far I have:

Let

Let denote a simple rotation such that and let denote a vertical flip

if , we're done since , which is cyclic

otherwise let for some non-negative

Consider .

If for some , then . Since the order of any element must divide the order of the group, we have that cannot be in .

So the only elements of are of the form .

But I can't seem to show that must be generated by an element of the form . Any help would be greatly appreciated.