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

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

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