On $\pi\mathfrak{F}$-embedded subgroups of finite groups
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A chief factor $H/K$ of $G$ is called $\mathfrak{F}$-central in $G$ provided $(H/K)\rtimes (G/C_{G}(H/K))\in\mathfrak{F}$. A normal subgroup $N$ of $G$ is said to be $\pi\mathfrak{F}$-hypercentral in $G$ if either $N=1$ or $N\neq1$ 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 $Z_{\pi\mathfrak{F}}(G)$ denotes the $\pi\mathfrak{F}$-hypercentre of $G$, that is, the product of all the normal $\pi\mathfrak{F}$-hypercentral subgroups of $G$. We say that a subgroup $H$ of $G$ is $\pi\mathfrak{F}$-embedded in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is $s$-quasinormal in $G$ and $(H\cap T)H_G/H_G\leq Z_{\pi\mathfrak{F}}(G/H_G)$, where $H_G$ is the maximal normal subgroup of $G$ contained in $H$. In this paper, we use the $\pi\mathfrak{F}$-embedded subgroups to determine the structures of finite groups. In particular, we give some new characterizations of $p$-nilpotency and supersolvability of a group.
Abstract
A chief factor $H/K$ of $G$ is called $\mathfrak{F}$-central in $G$ provided $(H/K)\rtimes (G/C_{G}(H/K))\in\mathfrak{F}$. A normal subgroup $N$ of $G$ is said to be $\pi\mathfrak{F}$-hypercentral in $G$ if either $N=1$ or $N\neq1$ 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 $Z_{\pi\mathfrak{F}}(G)$ denotes the $\pi\mathfrak{F}$-hypercentre of $G$, that is, the product of all the normal $\pi\mathfrak{F}$-hypercentral subgroups of $G$. We say that a subgroup $H$ of $G$ is $\pi\mathfrak{F}$-embedded in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is $s$-quasinormal in $G$ and $(H\cap T)H_G/H_G\leq Z_{\pi\mathfrak{F}}(G/H_G)$, where $H_G$ is the maximal normal subgroup of $G$ contained in $H$. In this paper, we use the $\pi\mathfrak{F}$-embedded subgroups to determine the structures of finite groups. In particular, we give some new characterizations of $p$-nilpotency and supersolvability of a group.
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