$m$-embedded Subgroups and $p$-nilpotency of Finite Groups

Canadian mathematical bulletin

Xinjian Zhang¹  Yong Xu² 

[1] School of Mathematics, Huaiyin Normal University (Huaian), Jiangsu, 223300, China;School of Mathematics and Statistics, Henan University of Science and Technology (Luoyang), Henan, 471023, China

关键词: finite group;    $p$-nilpotent group;    $m$-embedded subgroup;   

DOI  :  10.4153/CMB-2014-033-2

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Let $A$ be a subgroup of a finite group $G$ and $Sigma : G_0leqG_1leqcdots leq G_n$ some subgroup series of $G$. Suppose thatfor each pair $(K,H)$ such that $K$ is a maximal subgroup of $H$ and$G_{i-1}leq K lt Hleq G_i$, for some $i$, either $Acap H = Acap K$or $AH = AK$. Then $A$ is said to be $Sigma$-embedded in $G$; $A$is said to be $m$-embedded in $G$ if $G$ has a subnormal subgroup$T$ and a ${1leq G}$-embedded subgroup $C$ in $G$ such that $G =AT$ and $Tcap Aleq Cleq A$. In this article, some sufficientconditions for a finite group $G$ to be $p$-nilpotent are givenwhenever all subgroups with order $p^{k}$ of a Sylow $p$-subgroup of$G$ are $m$-embedded for a given positive integer $k$.

【 授权许可】

Unknown