Multiplicities of conjugacy class sizes of finite groups

It has been proved recently by Moreto and Craven 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 Zaikin-Zapirain. 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 and 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 $m$.

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