Group Theory/The action by conjugation and p-groups

Definition (global stabilizer):

Let be a group that acts on , where belongs to some algebraic variety . Let be a subset. Then the global stabilizer of is the set


where the notation stands for the set .

Definition (p-group):

Let be a prime number. Then a -group is a group of order for some .

Proposition (cardinality of fixed point set of a p-group equals cardinality of set mod p):

Let be a -group that acts on a set . Then


Proof: By the class equation,


where for each orbit of the action of on we pick one representative of that orbit. Since is a -group, whenever is not , it is divisible by by Lagrange's theorem. Hence, by taking the above equation , we get


where is the number of those for which . But means precisely that the orbit of is trivial, that is, that is fixed by all of .

Proposition (p-groups have nontrivial center):

Let be a -group. Then , where denotes the identity.

Proof: acts on itself via conjugation. Furthermore,


so that is precisely the fixed point set of under that action. But since the cardinality of the fixed point set of a p-group equals the cardinality of the whole set mod p, we get that


which would be impossible if .