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

Taeri, Bijan; Tooshmalani, Ziba

doi:10.22108/ijgt.2024.139508.1879

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	On a question of Jaikin-Zapirain about the average order elements of finite groups
International Journal of Group Theory
مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 14 مرداد 1403 اصل مقاله (429.58 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2024.139508.1879
نویسندگان
Bijan Taeri^* ؛ Ziba Tooshmalani
Department of Mathematical Sciences, Isfahan University of Technology, P.O.Box 8415683111, Isfahan, Iran.
چکیده
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.
کلیدواژه‌ها
‎Abelian groups؛ Group element orders؛ Sum of element orders؛ Average order

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