A new lower bound for the number of conjugacy classes

In 2000, Héthelyi and Külshammer [Bull. London Math. Soc. 32 (2000), pp. 668–672] proposed that if G G is a finite group, p p is a prime dividing the group order, and k ( G ) k(G) is the number of conjugacy classes of G G , then k ( G ) ≥ 2 p − 1 k(G)\geq 2\sqrt {p-1} , and they proved this conjecture for solvable G G and showed that it is sharp for those primes p p for which p − 1 \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 p , and we prove it for solvable groups, and when p p is large, also for arbitrary groups.