Multiplicities of conjugacy class sizes of finite groups

TL;DR

This paper establishes bounds on the order of finite groups based on the maximum multiplicity of their conjugacy class sizes, extending previous results related to character degrees.

Contribution

It proves that for finite simple groups, the group order is bounded by the largest multiplicity of conjugacy class sizes, and similarly for quotients of finite groups.

Findings

Finite simple groups have bounded order in terms of conjugacy class size multiplicity.

The order of any finite group is bounded by the maximum conjugacy class size multiplicity of its quotients.

Extends bounds from character degrees to conjugacy class sizes.

Abstract

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 $m$.