The $p$-nilpotency of finite groups with some weakly pronormal subgroups

Liu, Jianjun, Chang, Jian, and Chen, Guiyun. "The $p$-nilpotency of finite groups with some weakly pronormal subgroups." Czechoslovak Mathematical Journal 70.3 (2020): 805-816. <http://eudml.org/doc/297011>.

@article{Liu2020,
abstract = {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 $P\cap G^\{\prime \}$ with order $p$ lies in the center of $N_G(P)$ and when $p=2$, either every element of $P\cap G^\{\prime \}$ 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 $G$.},
author = {Liu, Jianjun, Chang, Jian, Chen, Guiyun},
journal = {Czechoslovak Mathematical Journal},
keywords = {weakly pronormal subgroup; normalizer; minimal subgroup; formation; $p$-nilpotency},
language = {eng},
number = {3},
pages = {805-816},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The $p$-nilpotency of finite groups with some weakly pronormal subgroups},
url = {http://eudml.org/doc/297011},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Liu, Jianjun
AU - Chang, Jian
AU - Chen, Guiyun
TI - The $p$-nilpotency of finite groups with some weakly pronormal subgroups
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 3
SP - 805
EP - 816
AB - 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 $P\cap G^{\prime }$ with order $p$ lies in the center of $N_G(P)$ and when $p=2$, either every element of $P\cap G^{\prime }$ 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 $G$.
LA - eng
KW - weakly pronormal subgroup; normalizer; minimal subgroup; formation; $p$-nilpotency
UR - http://eudml.org/doc/297011
ER -