Show that a Sylow p-subgroup of the dihedral group is cyclic and normal for every odd prime.
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 .
clic-clac, the dihedral group is not necessarily in the form it could be with i'll prove that in fact every sugroup of odd order in is normal and cyclic. to see this, let
and suppose is a subgroup with odd order. then because have order 2 and hence they cannot be in
H. so H is cyclic. besides a cyclic group has only one subgroup of any order, i.e. H is normal. Q.E.D.