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

  Published:2014-08-11
 Printed: Dec 2014
  • Yong Xu,
    School of Mathematics and Statistics, Henan University of Science and Technology (Luoyang), Henan, 471023, China
  • Xinjian Zhang,
    School of Mathematics, Huaiyin Normal University (Huaian), Jiangsu, 223300, China
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Abstract

Let $A$ be a subgroup of a finite group $G$ and $\Sigma : G_0\leq G_1\leq\cdots \leq G_n$ some subgroup series of $G$. Suppose that for each pair $(K,H)$ such that $K$ is a maximal subgroup of $H$ and $G_{i-1}\leq K \lt H\leq G_i$, for some $i$, either $A\cap H = A\cap K$ 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 $\{1\leq G\}$-embedded subgroup $C$ in $G$ such that $G = AT$ and $T\cap A\leq C\leq A$. In this article, some sufficient conditions for a finite group $G$ to be $p$-nilpotent are given whenever all subgroups with order $p^{k}$ of a Sylow $p$-subgroup of $G$ are $m$-embedded for a given positive integer $k$.
Keywords: finite group, $p$-nilpotent group, $m$-embedded subgroup finite group, $p$-nilpotent group, $m$-embedded subgroup
MSC Classifications: 20D10, 20D15 show english descriptions Solvable groups, theory of formations, Schunck classes, Fitting classes, $\pi$-length, ranks [See also 20F17]
Nilpotent groups, $p$-groups
20D10 - Solvable groups, theory of formations, Schunck classes, Fitting classes, $\pi$-length, ranks [See also 20F17]
20D15 - Nilpotent groups, $p$-groups
 

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