The effect of the sum of the inverse-power of element orders on finite groups

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics, Payamenoor University, Tehran, Iran.

2 Department of Mathematics, Firoozabad Branch, Islamic Azad University, Firoozabad, Iran.

Abstract

Let $G$ be a finite group, and let $m(G)=\sum_{g \in G}1/o(g)$, where $o(g)$ denotes the order of $g$. In this paper, we determine all groups such that $m(G)<4$. Additionally, for a finite group $G$ of odd order, we investigate the influence of $m(G)$ on supersolvability. Finally, we provide a criterion for $p$-solvability using the function $m(G)$, where $ p \in \{7,11\}$.

Keywords

Main Subjects

References

[1] C. S. Anabanti and A. K. Asboei, On the integral of the Sylow polynomial of a finite simple group, Sib. Math. J., 65 No. 6 (2024) 1402-1406.
[2] M. Archita, Exact bounds for the sum of the inverse-power of element orders in non-cyclic finite groups, arXiv preprint arXiv:2404.07500, (2024).
[3] M. Baniasad Azad, Simple Bψ-groups, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A Mat., 118 (2024) 133.
[4] M. Baniasad Azad, B. Khosravi and H. Rashidi, On the sum of the inverses of the element orders in finite groups, Commun. Algebra, 51 No. 2 (2023) 694-698.
[5] M. Baniasad Azad and B. Khosravi, A criterion for solvability of a finite group by the sum of element orders, J. Algebra, 516 (2018) 115-124.
[6] M. Baniasad Azad and B. Khosravi, On two conjectures about the sum of element orders, Can. Math. Bull., 65 No. 1 (2022) 30-38.
[7] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford University Press, Oxford, 1985.
[8] The GAP Group, GAP-Groups, Algorithms, and Programming, Version 4.12.2, 2022. https://www.gap-system.org
[9] M. Garonzi and M. Patassini, Inequalities detecting structural properties of a finite group, Commun. Algebra, 45 No. 2 (2017) 677-687.
[10] I. M. Isaacs, Finite Group Theory, American Mathematical Society, 2008.
[11] M. Herzog, P. Longobardi and M. Maj, Properties of Finite and Periodic Groups Determined by their Element Orders (A survey), In: Group Theory and Computation, eds. N. Sastry and M. Yadav, pp. 59-90, Springer, Berlin, 2018.
[12] M. Kuang, J. Lu, W. Meng and B. Zhang, Some criteria for solvability in finite groups, Ital. J. Pure Appl. Math., 52 (2024) 86-92.
[13] A. Lucchini, On the order of transitive permutation groups with cyclic point-stabilizer, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl., 9 No. 4 (1998) 241-243.
[14] I. C. Pleşca and M. Tărnăuceanu, Finite groups with integer harmonic mean of element orders, arXivpreprint arXiv:2310.00181, (2023).
[15] H. Xu, G. Chen and Y. Yan, A new characterization of simple K3-groups by their orders and large degrees of their irreducible characters, Comm. Algebra, 42 No. 12 (2014) 5374-5380.
Algebraic Structures and Their Applications

Articles in Press, Corrected Proof
Available Online from 14 June 2026

Files

Share

How to cite

Statistics