Third Sylow theorem

As proven in the previous section, the number of Sylow p-subgroups of G is given by |G : N_G(P)|, where P is any Sylow p-subgroup of G. This number, denoted by n_p is a factor of |G|. In addition, we have the following information about n_p.

Theorem 1
For each prime p, n_p º 1 (mod p).

Proof
Recall that for any p-subgroup H of G, we have |G : H| º |N_G(H) : H| (mod p). Let P be any Sylow p-subgroup, and we have |G : P| º |N_G(P) : P| (mod p). Note that P < N_G(P) < G. Also, since P is a Sylow p-subgroup, |G : P| is relatively prime to p Þ |G : N_G(P)| = |G : P| / |N_G(P) : P| º 1 (mod p).