Conjugacy class numbers and nilpotent subgroups of finite groups

Let G be a finite group, k(G) the number of conjugacy classes of G, and B a nilpotent subgroup of G. In this paper, we prove that vertical bar BO pi (G)/O-pi(G)vertical bar <= vertical bar G vertical bar/k(G) if G is solvable and that 15/7 vertical bar BO pi (G)/O-pi(G)vertical bar <= vertical bar G vertical bar/k(G) if G is nonsolvable, where pi = pi(B) is the set of prime divisors of vertical bar B vertical bar. Both bounds are best possible.