On $pimathfrak{F}$-embedded subgroups of finite groups

摘要

A chief factor $H/K$ of $G$ is called $mathfrak{F}$-central in $G$ provided $(H/K)times (G/C_{G}(H/K))inmathfrak{F}$. A normal subgroup $N$ of $G$ is said to be $pimathfrak{F}$-hypercentral in $G$ if either $N=1$ or $Neq1$ and every chief factor of $G$ below $N$ of order divisible by at least one prime in $pi$ is $mathfrak{F}$-central in $G$. The symbol $Z_{pimathfrak{F}}(G)$ denotes the $pimathfrak{F}$-hypercentre of $G$, that is, the product of all the normal $pimathfrak{F}$-hypercentral subgroups of $G$. We say that a subgroup $H$ of $G$ is $pimathfrak{F}$-embedded in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is $s$-quasinormal in $G$ and $(Hcap T)H_G/H_Gleq Z_{pimathfrak{F}}(G/H_G)$, where $H_G$ is the maximal normal subgroup of $G$ contained in $H$. In this paper, we use the $pimathfrak{F}$-embedded subgroups to determine the structures of finite groups. In particular, we give some new characterizations of $p$-nilpotency and supersolvability of a group.