A Note on the Codegree of Finite Groups

Abstract

Let \(\chi \) be an irreducible character of a group G,  and \(S_c(G)=\sum _{\chi \in \textrm{Irr}(G)}\textrm{cod}(\chi )\) be the sum of the codegrees of the irreducible characters of G. Write \(\textrm{fcod} (G)=\frac{S_c(G)}{|G|}.\) We aim to explore the structure of finite groups in terms of \(\textrm{fcod} (G).\) On the other hand, we determine the lower bound of \(S_c(G)\) for nonsolvable groups and prove that if G is nonsolvable, then \(S_c(G)\geqslant S_c(A_5)=68,\) 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 \(A_5.\)