On the Characterization of Alternating Groups by Codegrees

Let $G$ be a finite group and $\mathrm{I}\mathrm{r}\mathrm{r}(G)$ the set of all irreducible complex characters of $G$. Define the codegree of $\chi \in \mathrm{I}\mathrm{r}\mathrm{r}(G)$ as $\mathrm{c}\mathrm{o}\mathrm{d}(\chi ):=\frac{\mathrm{\mid }G:\mathrm{k}\mathrm{e}\mathrm{r}(\chi )\mathrm{\mid }}{\chi (1)}$ and denote by $\mathrm{c}\mathrm{o}\mathrm{d}(G):=\{\mathrm{c}\mathrm{o}\mathrm{d}(\chi )\mid \chi \in \mathrm{I}\mathrm{r}\mathrm{r}(G)\}$ 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 $\mathrm{c}\mathrm{o}\mathrm{d}({\mathrm{A}}_n)$.