A nilpotency criterion for finite groups by the sum of element orders

Document Type : Research Paper

Authors

1 Department of Mathematics, Shabestar Branch, Islamic Azad University, Shabestar, Iran

2 Department of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran

Abstract

Let $G$ be a finite group and $\psi(G)=\sum_{g\in G}o(g)$, where $o(g)$ denotes the order of $g\in G$. We give a criterion for nilpotency of finite groups $G$ based on the sum of element orders of $G$. We prove that if $\psi(G)>\frac{13}{21}\psi(C_n)$ then $G$ is a nilpotent group.

Keywords

[1] H. Amiri and S.M. Jafarian Amiri, Sum of element orders on finite groups of the same order, J. Algebra Appl. 10 (2011), no. 2, 187–190.
[2] H. Amiri and S.M. Jafarian Amiri, Sum of element orders of maximal subgroups of the symmetric group, Commun. Algebra Appl. 40 (2012), no. 2, 770–778.
[3] H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs, Sums of element orders in finite groups, Comm. Algebra 37 (2009), 2978–2980.
[4] M. Baniasad and B. Khosravi, A creterian for solvability of a finite group by the sum of element orders, J. Algebra 516 (2018), 115–124.
[5] M. Herzog, P. Longobardi and M. Maj, An exact upper bound for sums of element orders in non-cyclic finite groups, J. Algebra 222 (2018), no. 7, 1628–1642.
[6] M. Herzog, P. Longobardi and M. Maj, Two new criteria for solvability of finite groups, J. pure Appl. Algebra 511 (2018), 215–226.
[7] M. Herzog, P. Longobardi and M. Maj, Sums of element orders groups of order 2m with m odd, Commun. Algebra 47 (2019), no. 5, 2035–2048.
[8] S.M. Jafarian Amiri, Characterization of a5 and psl(2, 7) by sum of element orders, Int. J. Group Theory 2 (2013), no. 2, 35–39.
[9] S.M. Jafarian Amiri, Maximum sum of element orders of all proper subroups of psl(2, q), Bull. Iran. Math. Soc. 39 (2013), no. 3, 501–505.
[10] S.M. Jafarian Amiri, Second maximum sum of element orders on finite nilpotent groups, Commun. Algebra 41 (2013), no. 6, 2055–2059.
[11] S.M. Jafarian Amiri and M. Amiri, Second maximum sum of element orders on finite groups, J. Pure Appl. Algebra 218 (2014), no. 3, 531–539.
[12] S.M. Jafarian Amiri and M. Amiri, Sum of the products of the orders of two distinct elements in finite groups, Commun. Algebra 42 (2014), no. 12, 531–539.
[13] S.M. Jafarian Amiri and M. Amiri, Characterization of p-groups by sum of element orders, Publ. Math. Debrecen 86 (2015), no. 1–2, 31–37.
[14] S.M. Jafarian Amiri and M. Amiri, Sum of element orders in groups with the square-free orders, Bull. Malays. Math. Sci. Soc. 40 (2017), no. 3, 1025–1034.
[15] Y. Marefat, A. Iranmanesh and A. Tehranian, On the sum of element orders of finite simple groups, J. Algebra Appl. 12 (2013), no. 7.
[16] R. Shen, G. Chen and C. Wu, On groups with the second largest value of the sum of element orders, Commun. Algebra 43 (2015), no. 6, 2618–2631.
Volume 14, Issue 1
January 2023
Pages 2931-2937
  • Receive Date: 14 May 2022
  • Revise Date: 02 July 2022
  • Accept Date: 24 July 2022