On the sum of non-cyclic subgroups order in a finite group

Abstract

Let G be a finite group and $\delta (G)=\frac{1}{\left|G\right|}\sum _{H\le G}\left\{\right|H\left|\right|H$ is non-cyclic} . In this paper, we show that some arithmetical conditions of $\delta (G)$ influence the structure of G. Firstly, we prove that if $\delta (G)<\frac{13}{3}$, then G is solvable. Secondly, we determine the structure of finite groups with $\delta (G)\le 2$. Moreover, we prove that if $\delta (G)<1+\frac{4}{\left|G\right|}$, then G is supersolvable, and we also determine the structure of finite groups G with $\delta (G)=1+\frac{4}{\left|G\right|}$. Finally, we show that $\delta (G)<c$ does not imply the supersolvability of G for any constant $c\in (1,\infty )$.

KEYWORDS:

• Non-cyclic subgroups
• subgroup orders
• supersolvable groups

2020 Mathematics Subject Classification:

• 20D10
• 20D20

Acknowledgments

The authors are grateful to the referee, who provided her/his valuable suggestions and detailed reports.

Disclosure statement

The authors declare that they have non conflict of interest.

Additional information

Funding

W. Meng is supported by National Natural Science Foundation of China10.13039/501100001809 (12161021), Guangxi Natural Science Foundation Program (2021GXNSFAA220117), Center for Applied Mathematics of Guangxi (GUET) and Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation. J.K. Lu is supported by National Natural Science Foundation of China10.13039/501100001809 (11861015) and Guangxi Natural Science Foundation Program (2020GXNSFAA238045).