Finite groups with some bounded codegrees

TL;DR

This paper establishes a solvability criterion for finite groups based on the minimum ratio of the square of irreducible character degrees to their codegrees, linking character theory to group structure.

Contribution

It introduces a new solvability criterion for finite groups using bounds on character codegrees and degrees, advancing understanding of group structure via character ratios.

Findings

Provides a criterion for solvability based on character ratios.

Connects character codegrees with group solvability.

Offers new insights into the structure of finite groups.

Abstract

For a character $\chi$ of a finite group $G$, the number cod$(\chi):=|G:\mathrm{ker}(\chi)|/\chi(1)$ 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 $G$.