The Fixed Point Theorem in Artin's book
Solution 1:
Assume there is no fixed point. Since |G|=|stab(s)||orbit(s)| and |G|=p^a then the |orbit(s)|=p^m where m is not 0. But |S| is the sum of the orbits. Each orbit is divisible by p, which makes |S| divisible by p. Contradiction.