A New Lower Bound for the Number of Conjugacy Classes

Abstract

In 2000, Hethelyi and K & uuml;lshammer [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) >= 2 root p - 1, and they proved this conjecture for solvable G and showed that it is sharp for those primes p for which root 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.