On Solvability and Average Codegree of a Finite Group

Abstract

Let $$\mathrm{Irr}_2(G)$$ be the set of all complex irreducible characters $$\chi$$ such that their  codegrees $$\chi^c(1)=\frac{|G:\ker\chi|}{\chi(1)}$$ are 1 or even, and let

$$\mathrm{acod}_2(G)=\frac{1}{|\mathrm{Irr}_2(G)|}\sum\limits_{\chi\in\mathrm{Irr}_2(G)}\chi^c(1).$$

We show that, a group $$G$$ with $$\mathrm{acod}_2(G)<\frac{53}{4}$$ is solvable.