Group Theory/The action by conjugation and p-groups

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 . p-group Let be a prime number. Then a -group is a group of order for some . 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 . 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 . p-groups have nontrivial center Let be a -group. Then , where denotes the identity. 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 .