14.65 (1999)

(Well-known problem). For a finite group $G$ let $\rho(G)$ denote the set of prime numbers dividing the order of some conjugacy class, and $\sigma(G)$ the maximum number of primes dividing the order of some conjugacy class. Is it true that $\lvert\rho(G)\rvert \leqslant 3\sigma(G)$?

A possible linear bound for $\lvert\rho(G)\rvert$ in terms of $\sigma(G)$ cannot be better than $3\sigma(G)$, since there is a family of groups $\{G_n\}$ such that $\lim_{n \to \infty} \lvert\rho(G_n)\rvert/\sigma(G_n) = 3$ (C. Casolo, S. Dolfi, Rend. Sem. Mat. Univ. Padova, 96 (1996), 121–130).