# 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 .