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

Authors : Haowen Chen, Boru Zhang, Wei Meng

Pages : 206-214

Doi:10.24330/ieja.1476690

View : 79 | Download : 61

Publication Date : 2024-07-12

Article Type : Research Paper

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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