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1. CMB 2014 (vol 57 pp. 884)
$m$-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 |