Czechoslovak Mathematical Journal, Vol. 72, No. 3, pp. 747-750, 2022
On the average number of Sylow subgroups in finite groups
Alireza Khalili Asboei, Seyed Sadegh Salehi Amiri
Received April 10, 2021. Published online September 6, 2021.
Abstract: We prove that if the average number of Sylow subgroups of a finite group is less than $\tfrac{41}5$ and not equal to $\tfrac{29}4$, then $G$ is solvable or $G/F(G)\cong A_5$. In particular, if the average number of Sylow subgroups of a finite group is $\tfrac{29}4$, then $G/N\cong A_5$, where $N$ is the largest normal solvable subgroup of $G$. This generalizes an earlier result by Moretó et al.
References: [1] A. K. Asboei, M. R. Darafsheh: On sums of Sylow numbers of finite groups. Bull. Iran. Math. Soc. 44 (2018), 1509-1518. DOI 10.1007/s41980-018-0104-z | MR 3878407 | Zbl 1452.20009
[2] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Wilson: Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups. Clarendon, Oxford (1985). MR 827219 | Zbl 0568.20001
[3] M. Hall, Jr.: The Theory of Groups. Macmillan, New York (1959). MR 0103215 | Zbl 0084.02202
[4] J. Lu, W. Meng, A. Moretó, K. Wu: Notes on the average number of Sylow subgroups of finite groups. To appear in Czech. Math. J. DOI 10.21136/CMJ.2021.0229-20
[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
[6] J. Zhang: Sylow numbers of finite groups. J. Algebra 176 (1995), 111-123. DOI 10.1006/jabr.1995.1235 | MR 1345296 | Zbl 0832.20042
Affiliations: Alireza Khalili Asboei, Department of Mathematics, Farhangian University, Tarbiat-e-Moallem St, Shahid Farahzadi Blv, Tehran, Iran, e-mail: a.khalili@cfu.ac.ir; Seyed Sadegh Salehi Amiri (corresponding author), Department of Mathematics, Babol Branch, Islamic Azad University, Babol, Iran, e-mail: salehisss@baboliau.ac.ir
