| Title:
|
The $p$-nilpotency of finite groups with some weakly pronormal subgroups (English) |
| Author:
|
Liu, Jianjun |
| Author:
|
Chang, Jian |
| Author:
|
Chen, Guiyun |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
70 |
| Issue:
|
3 |
| Year:
|
2020 |
| Pages:
|
805-816 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $G$. (English) |
| Keyword:
|
weakly pronormal subgroup |
| Keyword:
|
normalizer |
| Keyword:
|
minimal subgroup |
| Keyword:
|
formation |
| Keyword:
|
$p$-nilpotency |
| MSC:
|
20D10 |
| MSC:
|
20D20 |
| idZBL:
|
07250691 |
| idMR:
|
MR4151707 |
| DOI:
|
10.21136/CMJ.2020.0546-18 |
| . |
| Date available:
|
2020-09-07T09:39:59Z |
| Last updated:
|
2022-10-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148330 |
| . |
| Reference:
|
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