The one-prime hypothesis on the co-degrees of irreducible characters

Abstract

Let G be a finite group. For a character χ of G, the number $\text{cod}(\chi )=\frac{[G:\text{ker}\chi ]}{\chi (1)}$ is called the co-degree of χ. Let N be a non-trivial normal subgroup of G and set $\text{Irr}(G|N)=\text{Irr}(G)-\text{Irr}(G/N).$ In this paper, we show that if for every $\chi ,\psi \in \text{Irr}(G|N)$ with $\text{cod}(\chi )\ne \text{cod}(\psi ),$ the greatest common divisor of $\text{cod}(\chi )$ and $\text{cod}(\psi )$ is either 1 or a prime, then N is solvable.

KEYWORD:

• The co-degree of a character
• the one-prime hypothesis
• the regular character

2020 MATHEMATICS subject Classification:

• 2010
• 20C15
• 20D05

Acknowledgments

The author would like to express her gratitude to the referee for careful reading and several useful comments, which improved the manuscript. Also, the second and the third paragraphs in the proof of Theorem A, in the current form, is due to the referee.

Additional information

Funding

Thanks are due to Shahrekord University for financial support (Grant 99GRN31M1023).