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

TL;DR

This paper characterizes finite groups where the number of irreducible characters equals the number of distinct co-degrees, showing this occurs only for groups isomorphic to Z_2 or S_3.

Contribution

It provides a complete classification of finite groups with equal number of irreducible characters and co-degrees, identifying only two specific group structures.

Findings

The equality |Irr(G)|=|cod(G)| holds only for G ≅ Z_2 or S_3.

The set of co-degrees uniquely determines these groups among all finite groups.

The result links group structure to properties of irreducible characters and their co-degrees.

Abstract

Let $G$ be a finite group and let $\rm{Irr}(G)$ be the set of all irreducible complex characters of $G$. For a character $\chi \in \rm{Irr}(G)$, the number $\rm{cod}(\chi):=|G:\rm{ker}\chi|/\chi(1)$ is called the co-degree of $\chi$. The set of co-degrees of all irreducible characters of $G$ is denoted by $\rm{cod}(G)$. 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 $S_3$.