Nonsolvable groups with three nonlinear irreducible character codegrees

TL;DR

This paper classifies finite nonsolvable groups with exactly three nonlinear irreducible character codegrees, identifying specific groups such as (2^f), (q), and .

Contribution

It provides a complete classification of nonsolvable groups with three nonlinear irreducible character codegrees, expanding understanding of their character theory.

Findings

Identifies (2^f) for f 2 as such groups

Includes (q) for odd q 5

Adds  as a specific example

Abstract

For an irreducible character $\chi$ of a finite group $G$, the codegree of $\chi$ is defined as $|G:\ker(\chi)|/\chi(1)$. In this paper, we determine finite nonsolvable groups with exactly three nonlinear irreducible character codegrees, and they are $\mathrm{L}_2(2^f)$ for $f\ge 2$, $\mathrm{PGL}_2(q)$ for odd $q\ge 5$ or $\mathrm{M}_{10}$.