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

Abstract

Let G be a finite group and let $\text{Irr}(\mathrm{G})$ be the set of all irreducible complex characters of G. For a character $\chi \in \text{Irr}(\mathrm{G}),$ the number $\text{cod}(\chi ):=|\mathrm{G}:\text{ker}(\chi )|/\chi (1)$ is called the co-degree of χ. The set of co-degrees of all irreducible characters of G is denoted by $\text{cod}(\mathrm{G}).$ In this paper, we show that for a nontrivial finite group G, $|\text{Irr}(\mathrm{G})|=|\text{cod}(\mathrm{G})|$ if and only if G is isomorphic to ${\mathbb{Z}}_2$ or S₃.

Keywords:

• Camina pair
• co-degree
• irreducible character

2020 AMS Subject Classification:

• 20C15
• 05C25

Additional information

Funding

This research was supported in part by a grant from School of Mathematics, Institute for Research in Fundamental Sciences (IPM).