On the average degree of linear and even degree characters of finite groups (Q6635312)

Let \(G\) be a finite group, \(\mathrm{Irr}(G)\) the set of all complex irreducible characters of \(G\) and let \(\mathrm{Irr}_{2}(G)=\{ \chi \in \mathrm{Irr}(G) \mid \chi(1)=1 \text{ or } \chi(1) \text{ is even} \}\). Let \(\operatorname{acd}(G)=|\mathrm{Irr}(G)|^{-1}\cdot \sum_{\chi \in \mathrm{Irr}(G)} \chi(1)\), which is the average of the degrees of \(G\). In [\textit{I. M. Isaacs} et. al., Isr. J. Math. 197, 55--67 (2013; Zbl 1290.20006)] it is proved that if \(\operatorname{acd}(G) \leq 3\), then \(G\) is solvable.\N\NThe author defines \(\operatorname{acd}_{2}(G)=|\mathrm{Irr}_{2}(G)|^{-1}\cdot \sum_{\chi \in \mathrm{Irr}_{2}(G)} \chi(1)\). The main result is that, if \(N \trianglelefteq G\), \(\mathrm{Irr}_{2}(G | N) \not =\emptyset\) and \(\operatorname{acd}_{2}(G | N) \leq 5/2\), then \(N\) is solvable. Moreover, \(G\) is solvable if \(\operatorname{acd}_{2}(G|N) < 5/2\). The bound is sharp since, if \(G_{n}=A_{5} \times C_{n}\), then \(\operatorname{acd}_{2}(G_{n} | G_{n})=(5n-1)/(2n-1)\) and the ratio would converge to \(5/2\) as \(n\) tends to infinity. Also \(\operatorname{acd}_{2}(G_{2} | C_{2})=5/2\).