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Search: All articles in the CMB digital archive with keyword $m$-embedded subgroup

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Xu, Yong; Zhang, Xinjian
$m$-embedded Subgroups and $p$-nilpotency of Finite Groups
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
Categories:20D10, 20D15

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