Groups in which the co-degrees of the irreducible characters are distinct

Abstract

Let $G$ be a finite group and let $\mathrm{Irr}(\mathrm{G})$ be the set of all irreducible complex characters of $G$. For a character $\chi \in \mathrm{Irr}(\mathrm{G})$, the number $\mathrm{cod}(\chi ):=|G:\ker \chi |/\chi (1)$ is called the co-degree of $\chi$. The set of co-degrees of all irreducible characters of $G$ is denoted by $\mathrm{cod}(\mathrm{G})$. In this paper, we show that for a non-trivial finite group $G$, $|\mathrm{Irr}(\mathrm{G})|=|\mathrm{cod}(\mathrm{G})|$ if and only if $G$ is isomorphic to the cyclic group ${\mathbb{Z}}_2$ or the symmetric group $S_3$.