A new lower bound for the number of conjugacy classes

Abstract:

In 2000, Héthelyi and Külshammer [Bull. London Math. Soc. 32 (2000), pp. 668–672] proposed that if $G$ is a finite group, $p$ is a prime dividing the group order, and $k(G)$ is the number of conjugacy classes of $G$, then $k(G)\geq 2\sqrt {p-1}$, and they proved this conjecture for solvable $G$ and showed that it is sharp for those primes $p$ for which $\sqrt {p-1}$ is an integer. This initiated a flurry of activity, leading to many generalizations and variations of the result; in particular, today the conjecture is known to be true for all finite groups. In this note, we put forward a natural new and stronger conjecture, which is sharp for all primes $p$, and we prove it for solvable groups, and when $p$ is large, also for arbitrary groups.