Geodesic
On the average number of Sylow subgroups in finite groups
Czechoslovak Mathematical Journal, Tome 72 (2022) no. 3, pp. 747-750
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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.
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.
DOI : 10.21136/CMJ.2021.0131-21
Classification : 20D15, 20D20
Keywords: Sylow number; non-solvable group 
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Khalili Asboei, Alireza; Salehi Amiri, Seyed Sadegh. On the average number of Sylow subgroups in finite groups. Czechoslovak Mathematical Journal, Tome 72 (2022) no. 3, pp. 747-750. doi: 10.21136/CMJ.2021.0131-21

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