This comes from the proof of the third Sylow Theorem in Michael Artin's "Algebra".
Let S be the set of Sylow p-groups in a given group G of order $p^em$. Let H be any Sylow group. If we decompose S into orbits for the operation of conjugation by H, then to establish $s=1$ modulo p, we only must show no element of S is fixed, since if H is a p-group, the order of any H-orbit is a power of p.
I'm having trouble with this last claim, I can't see why it is true. I don't know how much superfluous information I gave because I really don't understand the inference. Can anyone help?