On the Characterization of Sporadic Simple Groups by Codegrees

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 $\mathrm{cod}(\chi ):=\frac{|G:\ker (\chi )|}{\chi (1)}$ 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$(H)$.