| Title:
|
On the average number of Sylow subgroups in finite groups (English) |
| Author:
|
Khalili Asboei, Alireza |
| Author:
|
Salehi Amiri, Seyed Sadegh |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
72 |
| Issue:
|
3 |
| Year:
|
2022 |
| Pages:
|
747-750 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Sylow number |
| Keyword:
|
non-solvable group |
| MSC:
|
20D15 |
| MSC:
|
20D20 |
| idZBL:
|
Zbl 07584099 |
| idMR:
|
MR4467939 |
| DOI:
|
10.21136/CMJ.2021.0131-21 |
| . |
| Date available:
|
2022-08-22T08:21:20Z |
| Last updated:
|
2024-10-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/150614 |
| . |
| Reference:
|
[1] Asboei, A. K., Darafsheh, M. R.: On sums of Sylow numbers of finite groups.Bull. Iran. Math. Soc. 44 (2018), 1509-1518. Zbl 1452.20009, MR 3878407, 10.1007/s41980-018-0104-z |
| Reference:
|
[2] Conway, J. H., Curtis, R. T., Norton, S. P., Wilson, R. A.: Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups.Clarendon, Oxford (1985). Zbl 0568.20001, MR 827219 |
| Reference:
|
[3] M. Hall, Jr.: The Theory of Groups.Macmillan, New York (1959). Zbl 0084.02202, MR 0103215 |
| Reference:
|
[4] Lu, J., Meng, W., Moretó, A., Wu, K.: Notes on the average number of Sylow subgroups of finite groups.(to appear) in Czech. Math. J. MR 4339115, 10.21136/CMJ.2021.0229-20 |
| Reference:
|
[5] Moretó, A.: The average number of Sylow subgroups of a finite group.Math. Nachr. 287 (2014), 1183-1185. Zbl 1310.20026, MR 3231532, 10.1002/mana.201300064 |
| Reference:
|
[6] Zhang, J.: Sylow numbers of finite groups.J. Algebra 176 (1995), 111-123. Zbl 0832.20042, MR 1345296, 10.1006/jabr.1995.1235 |
| . |
