ON THE CHARACTERISATION 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 $\mathrm {cod}(\chi ):={|G:\mathrm {ker}(\chi ) |}/{\chi (1)}$ and let $\mathrm {cod}(G):=\{\mathrm {cod}(\chi ) \mid \chi \in \mathrm {Irr}(G)\}$ be the codegree set of G. Let $\mathrm {A}_n$ be an alternating group of degree $n \ge 5$ . We show that $\mathrm {A}_n$ is determined up to isomorphism by $\operatorname {cod}(\mathrm {A}_n)$ .