Czechoslovak Mathematical Journal, Vol. 71, No. 4, pp. 1129-1132, 2021


Notes on the average number of Sylow subgroups of finite groups

Jiakuan Lu, Wei Meng, Alexander Moretó, Kaisun Wu

Received June 4, 2020.   Published online March 19, 2021.

Abstract:  We show that if the average number of (nonnormal) Sylow subgroups of a finite group is less than $\frac{29}4$ then $G$ is solvable or $G/F(G)\cong A_5$. This generalizes an earlier result by the third author.
Keywords:  Fitting subgroup; Sylow subgroup; composition factor
Classification MSC:  20D20


References:
[1] GAP Group: GAP - Groups, Algorithms, Programming - a System for Computational Discrete Algebra. Version 4.4. Available at http://www.gap-system.org (2005). SW 00320
[2] R. M. Guralnick, G. R. Robinson: On the commuting probability in finite groups. J. Algebra 300 (2006), 509-528; addendum ibid 319 (2008), 1822. DOI 10.1016/j.jalgebra.2005.09.044 | MR 2228209 | Zbl 1100.20045
[3] M. Hall, Jr.: On the number of Sylow subgroups in a finite group. J. Algebra 7 (1967), 363-371. DOI 10.1016/0021-8693(67)90076-2 | MR 0222159 | Zbl 0178.02102
[4] I. M. Isaacs, M. Loukaki, A. Moretó: The average degree of an irreducible character of a finite group. Isr. J. Math. 197 (2013), 55-67. DOI 10.1007/s11856-013-0013-z | MR 3096606 | Zbl 1290.20006
[5] A. Moretó: The average number of Sylow subgroups of a finite group. Math. Nachr. 287 (2014), 1183-1185. DOI 10.1002/mana.201300064 | MR 3231532 | Zbl 1310.20026

Affiliations:   Jiakuan Lu (corresponding author), School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, Guangxi, P. R. China, e-mail: jklu@gxnu.edu.cn; Wei Meng, School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541006, Guangxi, P. R. China, e-mail: mlwhappyhappy@163.com; Alexander Moretó, Departament de Matemàtiques,Universitat de València, 46100 Burjassot, València, Spain; e-mail: Alexander.Moreto@uv.es; Kaisun Wu, School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, Guangxi, P. R. China, e-mail: 1547049416@qq.com


 
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