Czechoslovak Mathematical Journal, Vol. 74, No. 4, pp. 1083-1095, 2024
Some results on Sylow numbers of finite groups
Yang Liu, Jinjie Zhang
Received October 22, 2023. Published online November 12, 2024.
Abstract: 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 ${\rm asn}(G)\neq29/4$, 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.
Affiliations: Yang Liu (corresponding author), School of Mathematical Sciences, Tianjin Normal University, 393 Binshui W Ave, Xiqing District, Tianjin, P. R. China; Institute of Mathematics and Interdisciplinary Sciences, Tianjin Normal University, Tianjin, P. R. China, e-mail: yliu@tjnu.edu.cn; Jinjie Zhang, School of Mathematical Sciences, Tianjin Normal University, 393 Binshui W Ave, Xiqing District, Tianjin, P. R. China, e-mail: zhangjinjie0101@163.com
