Article
| 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 |
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| Category: | math |
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| 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 |
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| Date available: | 2022-08-22T08:21:20Z |
| Last updated: | 2022-12-27 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/150614 |
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| 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 |
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