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

Year 2024, Volume: 36 Issue: 36, 206 - 214, 12.07.2024

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

References

• L. An and Q. Zhang, Finite metahamiltonian p-groups, J. Algebra, 442 (2015), 23-35.
• V. A. Belonogov, Finite groups with three classes of maximal subgroups, Math. USSR-Sb., 59(1) (1988), 223-236.
• M. Brescia, M. Ferrara and M. Trombetti, The structure of metahamiltonian groups, Jpn. J. Math., 18(1) (2023), 1-65.
• L. Cui, W. Meng, J. Lu and W. Zheng, A new criterion for solvability of a finite group by the sum of orders of non-normal subgroups, Colloq. Math., 174(2) (2023), 169-176.
• T. De Medts and M. Tarnauceanu, Finite groups determined by an inequality of the orders of their subgroups, Bull. Belg. Math. Soc. Simon Stevin, 15(4) (2008), 699-704.
• R. Dedekind, Ueber Gruppen, deren sammtliche Theiler Normaltheiler sind, Math. Ann., 48 (1897), 548-561.
• X. Fang and L. An, A classification of finite metahamiltonian p-groups, Commun. Math. Stat., 9(2) (2021), 239-260.
• M. Garonzi and M. Patassini, Inequalities detecting structural properties of a finite group, Comm. Algebra, 45(2) (2017), 677-687.
• I. N. Herstein, A remark on finite groups, Proc. Amer. Math. Soc, 9(2) (1958), 255-257.
• M. Herzog, P. Longobardi and M. Maj, On a criterion for solvability of a finite group, Comm. Algebra, 49(5) (2021), 2234-2240.
• W. Meng and J. Lu, On the sum of non-cyclic subgroups order in a finite group, Comm. Algebra, 52(3) (2024), 1084-1096.
• M. Suzuki, Group Theory II, Fundamental Principles of Mathematical Sciences, 248, Springer-Verlag, New York, 1986.
• M. Tarnauceanu, Finite groups determined by an inequality of the orders of their subgroups II, Comm. Algebra, 45(11) (2017), 4865-4868.
• M. Tarnauceanu, On the solvability of a finite group by the sum of subgroup orders, Bull. Korean Math. Soc., 57 (2020), 1475-1479.
• M. Tarnauceanu, On the supersolvability of a finite group by the sum of subgroup orders, J. Algebra Appl., 21 (2022), 2250232 (7 pp).