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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:
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