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The $p$-nilpotency of finite groups with some weakly pronormal subgroups
Czechoslovak Mathematical Journal, Tome 70 (2020) no. 3, pp. 805-816
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For a finite group $G$ and a fixed Sylow $p$-subgroup $P$ of $G$, Ballester-Bolinches and Guo proved in 2000 that $G$ is $p$-nilpotent if every element of $P\cap G'$ with order $p$ lies in the center of $N_G(P)$ and when $p=2$, either every element of $P\cap G'$ with order $4$ lies in the center of $N_G(P)$ or $P$ is quaternion-free and $N_G(P)$ is $2$-nilpotent. Asaad introduced weakly pronormal subgroup of $G$ in 2014 and proved that $G$ is $p$-nilpotent if every element of $P$ with order $p$ is weakly pronormal in $G$ and when $p=2$, every element of $P$ with order $4$ is also weakly pronormal in $G$. These results generalized famous Itô's Lemma. We are motivated to generalize Ballester-Bolinches and Guo's Theorem and Asaad's Theorem. It is proved that if $p$ is the smallest prime dividing the order of a group $G$ and $P$, a Sylow $p$-subgroup of $G$, then $G$ is $p$-nilpotent if $G$ is $S_4$-free and every subgroup of order $p$ in $P\cap P^x\cap G^{\mathfrak {N_p}}$ is weakly pronormal in $N_G(P)$ for all $x\in G\setminus N_G(P)$, and when $p=2$, $P$ is quaternion-free, where $G^{\mathfrak {N_p}}$ is the $p$-nilpotent residual of $G$.
For a finite group $G$ and a fixed Sylow $p$-subgroup $P$ of $G$, Ballester-Bolinches and Guo proved in 2000 that $G$ is $p$-nilpotent if every element of $P\cap G'$ with order $p$ lies in the center of $N_G(P)$ and when $p=2$, either every element of $P\cap G'$ with order $4$ lies in the center of $N_G(P)$ or $P$ is quaternion-free and $N_G(P)$ is $2$-nilpotent. Asaad introduced weakly pronormal subgroup of $G$ in 2014 and proved that $G$ is $p$-nilpotent if every element of $P$ with order $p$ is weakly pronormal in $G$ and when $p=2$, every element of $P$ with order $4$ is also weakly pronormal in $G$. These results generalized famous Itô's Lemma. We are motivated to generalize Ballester-Bolinches and Guo's Theorem and Asaad's Theorem. It is proved that if $p$ is the smallest prime dividing the order of a group $G$ and $P$, a Sylow $p$-subgroup of $G$, then $G$ is $p$-nilpotent if $G$ is $S_4$-free and every subgroup of order $p$ in $P\cap P^x\cap G^{\mathfrak {N_p}}$ is weakly pronormal in $N_G(P)$ for all $x\in G\setminus N_G(P)$, and when $p=2$, $P$ is quaternion-free, where $G^{\mathfrak {N_p}}$ is the $p$-nilpotent residual of $G$.
DOI : 10.21136/CMJ.2020.0546-18
Classification : 20D10, 20D20
Keywords: weakly pronormal subgroup; normalizer; minimal subgroup; formation; $p$-nilpotency 
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Liu, Jianjun; Chang, Jian; Chen, Guiyun. The $p$-nilpotency of finite groups with some weakly pronormal subgroups. Czechoslovak Mathematical Journal, Tome 70 (2020) no. 3, pp. 805-816. doi: 10.21136/CMJ.2020.0546-18

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