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.