14.44 (1999)

Let $k(X)$ denote the number of conjugacy classes of a finite group $X$. Suppose that a finite group $G = AB$ is a product of two subgroups $A, B$ of coprime orders. Is it true that $k(AB) \leqslant k(A)k(B)$?

Note that one cannot drop the coprimeness condition and the answer is positive if one of the subgroups is normal, see 11.43.