Multiplicities of conjugacy class sizes of finite groups

It has been proved recently by Moretó (2007) [8], and Craven (2008) [3], that the order of a finite group is bounded in terms of the largest multiplicity of its irreducible character degrees. A conjugacy class version of this result was proved for solvable groups by Jaikin-Zapirain (2005) [7]. In this note, we prove that if G is a finite simple group then the order of G, denoted by |G|, is bounded in terms of the largest multiplicity of its conjugacy class sizes. As a consequence, we prove that if the largest multiplicity of conjugacy class sizes of any quotient of a finite group G is m, then |G| is bounded in terms of m.