On the largest character codegree of a finite group (Q6617369)

Let \(G\) be a finite group and let \(\mathrm{Irr}(G)\) be the set of all irreducible complex characters of \(G\). Also, let \(\operatorname{cod}(G)=\{ \operatorname{cod}(\chi) \mid \chi \in \mathrm{Irr}(G) \}\), where the codegree \(\operatorname{cod}(\chi)\) of \(\chi\) is defined by \(\operatorname{cod}(\chi) =\frac{|G : \ker(\chi)|}{\chi(1)}\). Let \(b^{c}(G)\) be the largest codegree of \(G\).\N\NThe authors study how the structure of a group \(G\) is bounded by \(b^{c}(G)\). Firstly, they give a criterion for solvability: if \(b^{c}(G) < 20\), then \(G\) is solvable (such a result is optimal in that \(b^{c}(A_{5})=20\)). Secondly, they consider the non-solvable groups \(G\) with small \(b^{c}(G)\) and prove that, if \(b^{c}(G) < 56\), then \(G\) is isomorphic to \(A_{5}\), \(S_{5}\) or \(A_{5} \times A\), where \(A\) is an elementary abelian \(2\)-group (such a result is optimal in that \(b^{c}(\mathrm{PSL}(2,7))=56\)).