On $Π$-supplemented subgroups of a finite group

A subgroup $H$ of a finite group $G$ is said to satisfy $Π$-property in $G$ if for every chief factor $L/K$ of $G$, $|G/K:N_{G/K}(HK/K\cap L/K)|$ is a $π(HK/K\cap L/K)$-number. A subgroup $H$ of $G$ is called to be $Π$-supplemented in $G$ if there exists a subgroup $T$ of $G$ such that $G=HT$ and $H\cap T\leq I\leq H$, where $I$ satisfies $Π$-property in $G$. In this paper, we investigate the structure of a finite group $G$ under the assumption that some primary subgroups of $G$ are $Π$-supplemented in $G$. The main result we proved improves a large number of earlier results.

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