On the sum of orders of non-cyclic and non-normal subgroups in a finite group

Abstract

Let $G$ be a finite group and $\mathcal{C}(G)$ denote the set of all non-normal non-cyclic subgroups of $G$. In this paper, the function $\delta_c(G) =\frac{1}{|G|}\sum\limits_{H\in\mathcal{C}(G)}|H|$ is introduced. In fact, we prove that, if $\delta_c(G)\leq \frac{10}{3}$, then either $G\cong A_5$, or $G$ is solvable. We also find some examples of finite groups $G$ with $\delta_c(G)\leq \frac{10}{3}$.