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Keywords:
finite group; $p$-supersoluble group; $p$-nilpotent group; the $\Pi $-property
finite group; $p$-supersoluble group; $p$-nilpotent group; the $\Pi $-property
Summary:
Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies the $\Pi $-property in $G$ if for any chief factor $L / K$ of $G$, $ |G/K : N_{G/K}(HK/K\cap L/K )| $ is a $\pi (HK/K\cap L/K)$-number. We obtain some criteria for the $p$-supersolubility or $p$-nilpotency of a finite group and extend some known results by concerning some subgroups that satisfy the $\Pi $-property.
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