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Keywords:
Sylow number; nonsolvable group
Sylow number; nonsolvable group
Summary:
We study the group structure in terms of the number of Sylow $p$-subgroups, which is denoted by $n_p(G)$. The first part is on the group structure of finite group $G$ such that $n_p(G)=n_p(G/N)$, where $N$ is a normal subgroup of $G$. The second part is on the average Sylow number ${\rm asn}(G)$ and we prove that if $G$ is a finite nonsolvable group with ${\rm asn}(G)<39/4$ and ${\rm asn}(G)\neq 29/4$, then $G/F(G)\cong A_5$, where $F(G)$ is the Fitting subgroup of $G$. In the third part, we study the nonsolvable group with small sum of Sylow numbers.
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