Finite groups whose codegrees are almost prime

Abstract

For a character χ of a finite group G, the number $\text{cod}(\chi )=|G:\text{ker}(\chi )|/\chi (1)$ is called the codegree of χ. In this paper we show that if G is nonsolvable, then the codegree graph $\Gamma (G)$ has a triangle. Also by using this result we show that if G has one composite codegree, then G is solvable and $|\text{cod}(G)|\le 5.$

Keywords:

• Finite group
• codegree graph
• irreducible character

2010 MATHEMATICAS SUBJECT CLASSIFICATION:

• Primary 20C15
• Secondary 20D05

Acknowledgments

The authors gratefully appreciate anonymous referee for comments and recommendations which definitely helped to improve the readability and quality of the paper.