I am trying to prove, or understand the proof of, the following part of the sylow theorems (according to my notes anyway):
Let G be a finite group and let p be a prime which divides the order of G. Then
Every p-subgroup of G is contained in a Sylow p-subgroup of G.
Proof: Let P be a Sylow p-subgroup. Let Q be any p-subgroup. Let be the set of all conjugates of P. Then G acts on by conjugation. By the orbit stabiliser theorem, the number of conjugates of P is r = | | = [G: ]. Note that p does not divide r.
but I don't understand why p does not divide r! could anyone help me with out with this? thanks very much