I might be totally totally wrong here.

First, elements might have different orders. So you cannot say that each element has the same order. But I think I know what you want to say.... By Lagrange's theorem the order of the element divides the order of the group, so you are probably claiming that the divisors of are the different orders. But this the the converse of Lagrange's theorem. So it can fail. If the dihedral group was abelian then it would be safe to say that but it is not. So maybe there is so strange property about dihedral groups that gaurentte of an existence of every order dividing .