Let G be a finite group, and let F be a formation of finite group. We say that a subgroup H of G is p F -normal in G if there exists a normal subgroup T of G such that HT is a permutable Hall subgroup of G and G G H H T H / ) ( is contained in the F-hypercenter ) / ( G F H G Z of G H G / . In this note, we get some results about the p F -normal subgroups and then use them to study the structure...

Let $mathfrak{F}$ be a formation and $G$ a finite group. A subgroup $H$ of $G$ is said to be weakly $mathfrak{F}_{s}$-quasinormal in $G$ if $G$ has an $S$-quasinormal subgroup $T$ such that $HT$ is $S$-quasinormal in $G$ and $(Hcap T)H_{G}/H_{G}leq Z_{mathfrak{F}}(G/H_{G})$, where $Z_{mathfrak{F}}(G/H_{G})$ denotes the $mathfrak{F}$-hypercenter of $G/H_{G}$. In this paper, we study the structur...

let $mathfrak{f}$ be a formation and $g$ a finite group. a subgroup $h$ of $g$ is said to be weakly $mathfrak{f}_{s}$-quasinormal in $g$ if $g$ has an $s$-quasinormal subgroup $t$ such that $ht$ is $s$-quasinormal in $g$ and $(hcap t)h_{g}/h_{g}leq z_{mathfrak{f}}(g/h_{g})$, where $z_{mathfrak{f}}(g/h_{g})$ denotes the $mathfrak{f}$-hypercenter of $g/h_{g}$. in this paper, we study the structur...