Theorem 7.3.2 Let G be a p-group, and let S be a finite set on which G operates. If the order of S is not divisible by p, there is a fixed point for the operation of G on S - an element s whose stabilizer is the whole group.
Do not how to prove it..
2026-03-29 05:49:41.1774763381
The Fixed Point Theorem in Artin's book
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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.