A note on the codegree of finite groups

Abstract

Let $\chi$ be an irreducible character of a group $G,$ and $S_c(G)={\sum }_{\chi \in \mathrm{Irr}(G)}\mathrm{cod}(\chi )$ be the sum of the codegrees of the irreducible characters of $G.$ Write $\mathrm{fcod}(G)=\frac{S_c(G)}{|G|}.$ We aim to explore the structure of finite groups in terms of $\mathrm{fcod}(G).$ On the other hand, we determine the lower bound of $S_c(G)$ for nonsolvable groups and prove that if $G$ is nonsolvable, then $S_c(G)\ge S_c(A_5)=68,$ with equality if and only if $G\cong A_5.$ Additionally, we show that there is a solvable group so that it has the codegree sum as $A_5.$