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

Year 2024, Early Access, 1 - 9

Haowen Chen Boru Zhang Wei Meng

https://doi.org/10.24330/ieja.1476690

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 $\delta_c(G)\leq \frac{10}{3}$.

Keywords

Solvable group, non-normal subgroup, non-cyclic subgroup, subgroup order

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