Finite groups with some bounded codegrees

Abstract

For a character $\chi$ of a finite group $G$, the number cod$(\chi ):=|G:\ker (\chi )|/\chi (1)$ is called the codegree of $\chi$.In this paper, we give a solvability criterion for a finite group $G$ depending on the minimum of the ratio $\chi (1{)}^2/\mathrm{cod}(\chi )$, when $\chi$ varies among the irreducible characters of $G$.