Prove that every subgroup of D_n of odd order is cyclic
Remember that
Notice that
This means that each have order .
Thus, this subgroup cannot contain these elements for then Lagrange's theorem would imply two divides its order.
This means if is a subgroup of of odd order it contains the elements . Thus, it is a subgroup of . But every subgroup of a cyclic group is cyclic. Thus, is cyclic itself.