Article
Full entry |
PDF
(0.1 MB)
Feedback
PDF
(0.1 MB)
Feedback
Keywords:
Sylow number; non-solvable group
Sylow number; non-solvable group
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.
References:
[1] Asboei, A. K., Darafsheh, M. R.: 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] 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). MR 827219 | Zbl 0568.20001
[3] M. Hall, Jr.: The Theory of Groups. Macmillan, New York (1959). MR 0103215 | Zbl 0084.02202
[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. DOI 10.21136/CMJ.2021.0229-20 | MR 4339115
[5] Moretó, A.: 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] Zhang, J.: Sylow numbers of finite groups. J. Algebra 176 (1995), 111-123. DOI 10.1006/jabr.1995.1235 | MR 1345296 | Zbl 0832.20042
Search
Partner of
