A note on the codegree of finite groups (Q6624913)

Let \(G\) be a finite group and let \(\mathrm{Irr}(G)\) be the set of complex irreducible characters of \(G\). The codegree of \(\chi \in \mathrm{Irr}(G)\) is the number \(\mathrm{cod}(\chi)=|G: \ker \chi | \cdot \chi(1)^{-1}\). Let \(S_{c}(G)=\sum_{\chi \in \mathrm{Irr}(G)} \mathrm{cod}(\chi)\) and let \(\mathrm{fcod}(G)=S_{c}(G) \cdot |G|^{-1}\).\N\NThe main result in the paper under review is Theorem 1.1: Let \(G\) be a non-solvable group. Then \(S_{c}(G)\geq S_{c}(A_{5})=68\), with equality if and only if \(G \simeq A_{5}\).\N\NFurthermore, the authors show that there exist families of non-solvable groups \(\{G \}\) and \(\{ H \}\) such that \(\mathrm{fcod}(G) \rightarrow 0\) as \(|G| \rightarrow \infty\) and \(\mathrm{fcod}(H) \rightarrow \infty\) as \(|H| \rightarrow \infty\). Moreover, they discuss the special case when \(\mathrm{fcod}(G) =1\) and provide some observations and open questions.