A characterization of $A_5$ by its average order

Tarnauceanu, Marius

doi:10.22108/ijgt.2023.138637.1866

تعداد نشریات	44
تعداد شماره‌ها	1,877
تعداد مقالات	15,278
تعداد مشاهده مقاله	43,730,722
تعداد دریافت فایل اصل مقاله	17,563,035

	A characterization of $A_5$ by its average order
International Journal of Group Theory
مقاله 2، دوره 14، شماره 3، آذر 2025، صفحه 117-123 اصل مقاله (427.14 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2023.138637.1866
نویسنده
Marius Tarnauceanu^*
Faculty of Mathematics, "Al.I. Cuza" University of Iasi
چکیده
Let $o(G)$ be the average order of a finite group $G$. M. Herzog, P. Longobardi and M. Maj [M. Herzog, P. Longobardi and M. Maj, Another criterion for solvability of finite groups, J. Algebra, 597 (2022) 1-23.] showed that if $G$ is non-solvable and $o(G)=o(A_5)$, then $G\cong A_5$. In this note, we prove that the equality $o(G)=o(A_5)$ does not hold for any finite solvable group $G$. Consequently, up to isomorphism, $A_5$ is determined by its average order.
کلیدواژه‌ها
average order؛ sum of element orders؛ solvable group

مراجع
[1] 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. [2] 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. [3] 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. [4] M. Baniasad Azad and B. Khosravi, On two conjectures about the sum of element orders, Canad. Math. Bull., 65 no. 1 (2022) 30–38. [5] M. Herzog, P. Longobardi and M. Maj, Two new criteria for solvability of finite groups, J. Algebra, 511 (2018) 215–226. [6] M. Herzog, P. Longobardi and M. Maj, An exact upper bound for sums of element orders in non-cyclic finite groups, J. Pure Appl. Algebra, 222 no. 7 (2018) 1628–1642. [7] M. Herzog, P. Longobardi and M. Maj, Another criterion for solvability of finite groups, J. Algebra, 597 (2022) 1-23. [8] M. Herzog, P. Longobardi and M. Maj, On groups with average element orders equal to the average order of the alternating group of degree 5, to appear in Glas. Mat. III. Ser, (2023). [9] I. M. Isaacs, Finite group theory, Graduate Studies in Mathematics, 92, American Mathematical Society, Providence, RI, 2008 [10] A. Jaikin-Zapirain, On the number of conjugacy classes of finite nilpotent groups, Adv. Math., no. 3 227 (2011) 1129–1143. [11] E. I. Khukhro, A. Moretó and M. Zarrin, The average element order and the number of conjugacy classes of finite groups, J. Algebra, 569 (2021) 1–11. [12] T. J. Laffey, The number of solution of xp = 1 in a finite group, Math. Proc. Cambridge Philos. Soc., 80 no. 2 (1976) 229–231. [13] M. S. Lazorec and M. Tărnăuceanu, On the average order of a finite group, J. Pure Appl. Algebra, 227 no. 4 (2023) 9 pp. [14] G. A. Miller, On the minimum number of operators whose orders exceed two in any finite group, Bull. Amer. Math. Soc., 13 no. 5 (1907) 235–239. [15] M. Tărnăuceanu, Detecting structural properties of finite groups by the sum of element orders, Israel J. Math., 238 no. 2 (2020) 629–637. [16] M. Tărnăuceanu, A criterion for nilpotency of a finite group by the sum of element orders, Comm. Algebra, 49 no. 4 (2021) 1571–1577. [17] M. Tărnăuceanu, Another criterion for supersolvability of finite groups, J. Algebra, 604 (2022) 682–693. [18] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.11.0, 2020, https://www.gap-system.org.
آمار تعداد مشاهده مقاله: 507 تعداد دریافت فایل اصل مقاله: 719

سامانه مدیریت نشریات علمی. طراحی و پیاده سازی از سیناوب

پیوندهای مفید

اخبار و اعلانات

آمار

A characterization of $A_5$ by its average order