Conjugacy class numbers and nilpotent subgroups of finite groups

Let 𝐺 be a finite group, k ⁢ ( G ) k(G) the number of conjugacy classes of 𝐺, and 𝐵 a nilpotent subgroup of 𝐺. In this paper, we prove that | B ⁢ O π ⁢ ( G ) / O π ⁢ ( G ) | ≤ | G | / k ⁢ ( G ) \lvert BO_{\pi}(G)/O_{\pi}(G)\rvert\leq\lvert G\rvert/k(G) if 𝐺 is solvable and that 15 7 ⁢ | B ⁢ O π ⁢ ( G ) / O π ⁢ ( G ) | ≤ | G | / k ⁢ ( G ) \frac{15}{7}\lvert BO_{\pi}(G)/O_{\pi}(G)\rvert\leq\lvert G\rvert/k(G) if 𝐺 is nonsolvable, where π = π ⁢ ( B ) \pi=\pi(B) is the set of prime divisors of | B | \lvert B\rvert . Both bounds are best possible.