On the Characterization of Alternating Groups by Codegrees

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 $\mathrm{cod}(G):=\{\mathrm{cod}(\chi) \mid \chi\in \mathrm{Irr}(G)\}$ 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 $\mathrm{cod}(\mathrm{A}_n)$.