Narrow Results

Let $G$ be a finite group and $H$ a subgroup of $G$. We say that $H$ is an ${\mathscr{H}}$-subgroup of $G$ if $N_G(H)\cap H^g\le H$ for all $g\in G$. We say that $H$ is weakly ${\mathscr{H}}{𝒞}$-embedded in $G$ if $G$ has a normal subgroup $T$ such that $H^G=HT$ and $N_T(H)\cap H^g\le H$ for all $g\in G,$ where $H^G$ is the normal closure of $H$ in $G$. For each prime $p$ dividing the order of $G,$ let $P$ be a Sylow $p$-subgroup of $G$. We fix a subgroup of $P$ of order $d$ with $1<d<\left|P\right|$ and study the structure of $G$ under the assumption that every subgroup of $P$ of order $p^{n}d$$(n=0,1)$ is weakly ${\mathscr{H}}{𝒞}$-embedded in $G$. Our results improve and generalize several recent results in the literature.

Let $G$ be a finite group and $H$ a subgroup of $G$. We say that $H$ is an ${\mathscr{H}}$-subgroup of $G$ if $N_G(H)\cap H^g\le H$ for all $g\in G$; $H$ is called weakly ${\mathscr{H}}C$-embedded in $G$ if $G$ has a normal subgroup $T$ such that $H^G=HT$ and $N_T(H)\cap H^g\le H$ for all $g\in G,$ where $H^G$ is the normal closure of $H$ in $G$. In this paper, we study the $p$-nilpotence of a group $G$ in which every subgroup of order $d$ of a Sylow $p$-subgroup $P$ with $1<d<\left|P\right|$ is weakly ${\mathscr{H}}C$-embedded in $G$. Many recent results in the literature related to $p$-nilpotence of $G$ are generalized.