Let G be a finite group. Regard G as a G-set where G acts on itself by conjugation. 

a. Show that G_G is the center Z(G) of G.  

b. Use Theorem 36.1 to show that the center of a finite nontrivial p-group is nontrivial.

Data from Theorem 36.1

Let G be a group of order pⁿ and let X be a finite G-set. Then |X| ≡ |X_G| (mod p ). 

Proof In the notation of Eq. (2), we know that |G_xi| divides |G| by Theorem 16.16. Consequently p divides |G_xi| for s + 1≤ i ≤ r. Equation (2) then shows that |X| ≡ |X_G|is divisible by p, so |X| ≡ |X_G| (mod p).