Let be an odd prime, the dihedral group with elements and its identity element.
A -Sylow in has an order of , so it is cyclic: indeed, the order of an element of such a -Sylow subgroups is either or (because it divides ) so if the element is different than , then it's a generator, and since is different from , there are generators.
Furthermore, has elements. The number of -Sylow and are congruent modulo , and divides , so divides . So . But all -Sylow subgroups are conjugate, so when there is only one -Sylow , it is normal.
Indeed, has the same order than . Thus it's a -Sylow, and .