On the Characterization of Alternating 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 $\mathrm{cod}(G):=\{\mathrm{cod}(\chi )\mid \chi \in \mathrm{Irr}(G)\}$ the codegree set of $G$. Let ${\mathrm{A}}_n$ be an alternating group of degree $n\ge 5$. In this paper, we show that ${\mathrm{A}}_n$ is determined up to isomorphism by $\mathrm{cod}({\mathrm{A}}_n)$.

arXiv commentarXiv admin note: substantial text overlap with arXiv:2301.02365