Group Actions, Fixed Point Principle
Fixed Point Principle
Let G have order pk, and act on a finite set S.
Recall that orbit size times stabilizer size = |G|.
If the orbit is nontrivial its size is divisible by p,
hence p divides the number of elements not fixed by G.
Write |S| equals |S0| mod p,
where S0 is the subset of elements fixed by G.
This is the fixed point principle.
For example, Macay's theorem (previous section) has Zp acting on a set S,
where |S| is 0 mod p, hence |S0| is divisible by p,
and S0 cannot be a singel element.