Exercise 8, § 5

Let A and B be subgroups of a finite group G, N_A and N_B the normalizers of A and B, $\nu_A$ and $\nu_B$ the indices of N_A and N_B in G, $r_A$ the number of conjugates of A which contain B and $r_B$ the number of conjugates of B which contain A; show that $\nu_A r_B = \nu_B r_A$.