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


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
$G_{i1}\leq K \lt H\leq G_i$, 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 $k$.
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
