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Keywords:
$p$-nilpotent group; weakly $\mathcal {M}$-supplemented subgroup; finite group
$p$-nilpotent group; weakly $\mathcal {M}$-supplemented subgroup; finite group
Summary:
Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. Subgroup $H$ is said to be weakly $\mathcal {M}$-supplemented in $G$ if there exists a subgroup $B$ of $G$ such that (1) $G=HB$, and (2) if $H_{1}/H_{G}$ is a maximal subgroup of $H/H_{G}$, then $H_{1}B=BH_{1}<G$, where $H_{G}$ is the largest normal subgroup of $G$ contained in $H$. We fix in every noncyclic Sylow subgroup $P$ of $G$ a subgroup $D$ satisfying $1<|D|<|P|$ and study the $p$-nilpotency of $G$ under the assumption that every subgroup $H$ of $P$ with $|H|=|D|$ is weakly $\mathcal {M}$-supplemented in $G$. Some recent results are generalized.
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