On the Characterization of Sporadic Simple Groups by Codegrees

TL;DR

This paper proves that each of the 26 sporadic simple groups can be uniquely identified by its set of codegrees, providing a new characterization of these groups.

Contribution

It establishes that the codegree set uniquely determines each sporadic simple group among all finite groups.

Findings

Each sporadic simple group is uniquely characterized by its codegree set.

The codegree set serves as a complete invariant for identifying sporadic simple groups.

The result enhances understanding of the structure of sporadic groups through character theory.

Abstract

Let $G$ be a finite group and $\mathrm{Irr}(G)$ the set of all irreducible complex characters of $G$. Define the codegree of $\chi \in \mathrm{Irr}(G)$ as $\mathrm{cod}(\chi):=\frac{|G:\mathrm{ker}(\chi) |}{\chi(1)}$ and denote by $\mathrm{cod}(G):=\{\mathrm{cod}(\chi)|\chi\in \mathrm{Irr}(G)\}$ the codegree set of $G$. Let $H$ be one of the $26$ sporadic simple groups. In this paper, we show that $H$ is determined up to isomorphism by cod$(H)$.