Multiplicities of fields of values of conjugacy classes in finite groups

TL;DR

This paper investigates the relationship between the size of a finite group and the maximum multiplicity of fields of values of its conjugacy classes, establishing bounds and classifying groups with low multiplicity.

Contribution

It proves that the order of a finite group is bounded by its maximum field multiplicity and classifies groups with multiplicity at most three.

Findings

The order of a finite group is bounded in terms of the maximum field multiplicity.

Groups with field multiplicity less than or equal to three are classified.

The concept of field multiplicity of conjugacy classes is introduced and analyzed.

Abstract

Given a group $G$ we write $h(G)$ to denote the maximum number of times that a field extension of $\mathbb{Q}$ appears as the field of values of a conjugacy class of a group. In this work, we prove that $|G|$ is bounded in terms of $h(G)$. Moreover, we classify the groups with $h(G)\leq 3$.