A Note on the Codegree of Finite Groups

Let χ χ be an irreducible character of a group G , and Sc(G)=∑ χ ∈ Irr(G)cod(χ ) Sc(G)=∑χ∈Irr(G)cod(χ) be the sum of the codegrees of the irreducible characters of G . Write fcod (G)=Sc(G)/|G|. fcod(G)=Sc(G)|G|. We aim to explore the structure of finite groups in terms of fcod (G). fcod(G). On the other hand, we determine the lower bound of Sc(G) Sc(G) for nonsolvable groups and prove that if G is nonsolvable, then Sc(G) Sc(A₅)=68, Sc(G)⩾Sc(A5)=68, with equality if and only if G A₅. G≅A5. Additionally, we show that there is a solvable group so that it has the codegree sum as A₅. A5.