On a question of Jaikin-Zapirain about the average order elements of finite groups

Document Type : Research Paper

Authors

Department of Mathematical Sciences, Isfahan University of Technology, P.O.Box 8415683111, Isfahan, Iran.

Abstract

For a finite group $G$, the average order $o(G)$ is defined to be the average of all order elements in $G$, that is $o( G)=\frac{1}{|G|}\sum_{x\in G}o(x)$, where $o(x)$ is the order of element $x$ in $G$. Jaikin-Zapirain in [On the number of conjugacy classes of finite nilpotent groups, Advances in Mathematics, \textbf{227} (2011) 1129-1143] asked the following question: if $G$ is a finite ($p$-) group and $N$ is a normal (abelian) subgroup of $G$, is it true that $o(N)^{\frac{1}{2}}\leq o(G) $? We say that $G$ satisfies the average condition if $o(H)\leq o(G)$, for all subgroups $H$ of $G$. In this paer we show that every finite abelian group satisfies the average condition. This result confirms and improves the question of Jaikin-Zapirain for finite abelian groups.

Keywords

Main Subjects

References

[1] H. Amiri, S. M. Jafarian Amiri and I. M. Isaacs, Sums of element orders in finite groups, Comm. Algebra, 37 no. 9 (2009) 2978–2980.
[2] H. Amiri and S. M. Jafarian Amiri, Sum of element orders on finite groups of the same order, J. Algebra Appl., 10 no. 2 (2011) 187–190.
[3] C. Y. Chew, A. Y. M. Chin and C. S. Lim, A recursive formula for the sum of element orders of finite abelian groups, Results Math., 72 (2017) 1897–1905.
[4] D. Gorenstein, Finite groups, Chelsea publishing company, New York , 1980.
[5] M. Tărnăuceanu and G. Fodor, On the sum of element orders of finite abelian groups, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat., 60 (2013) 1–7.
[6] A. Jaikin-Zapirain, On the number of conjugacy classes of finite nilpotent groups, Adv. Math., 227 no. 3 (2011) 1129–1143.
[7] E. I. Khokhro, A. Moreto and M. Zarrin, The average element order and the number of conjugacy classes of finite groups, J. Algebra, 569 (2021) 1–11.
[8] Wolfram Research, Inc., Mathematica, Version 9.0, Wolfram Research, Inc. Place of publication: Cham-paign, Illinois, 2012.
International Journal of Group Theory
Volume 14, Issue 3 - Serial Number 3
September 2025
Pages 139-147

Files

History

  • Receive Date: 21 October 2023
  • Revise Date: 28 June 2024
  • Accept Date: 04 July 2024
  • Published Online: 04 August 2024

Share

How to cite

Statistics