On the characterization of some non-abelian simple groups using codegree set

TL;DR

This paper demonstrates that the set of codegrees uniquely identifies certain non-abelian simple groups, providing a new characterization tool based on codegree sets.

Contribution

It establishes that the codegree set uniquely determines various non-abelian simple groups, advancing the understanding of group characterization via codegrees.

Findings

The codegree set of each listed group uniquely determines the group up to isomorphism.

The paper extends the characterization of simple groups through codegree sets.

It provides explicit codegree set descriptions for several complex simple groups.

Abstract

Let $G$ be a finite group and $\chi\in \irr(G)$. The codegree of $\chi$ is defined as $\cod(\chi)=\frac{|G:\ker(\chi)|}{\chi(1)}$ and $\cod(G)=\{\cod(\chi) \ |\ \chi\in \irr(G)\}$ is called the set of codegrees of $G$. In this paper, we show that the set of codegrees of $\Sy_4(4), \U_4(2)$, $\Sy_4(q)\ (q \geq 4)$, $\U_4(3)$, ${}^2\F_4(2)'$, $\J_3$, $\G_2(3)$, $\A_9$, $\J_2$, $\PSL(4,3)$, $\McL$, $\Sy_4(5)$, $\G_2(4)$, $\HS$, $\ON$ and $\M_{24}$ determines the group up to isomorphism.