Narrow Results

Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be an ${\mathscr{H}}C$-subgroup of $G$ if there exists a normal subgroup $T$ of $G$ such that $G=HT$ and $H^g\cap N_T(H)\le H$ for all $g\in G.$ We say that $H$ is weakly ${\mathscr{H}}C$-embedded in $G$ if there exists a normal subgroup $T$ of $G$ such that $H^G=HT$ and $H^g\cap N_T(H)\le H$ for all $g\in G$. In this paper, we investigate the structure of the finite group $G$ under the assumption that some subgroups of prime power order are weakly ${\mathscr{H}}C$-embedded in $G$. Our results improve and generalize several recent results in the literature.

A subgroup $H$ of a group $G$ is said to be an $\mathcal{\mathscr{H}}C$-subgroup of $G$, if there exists a normal subgroup $K$ of $G$ such that $G=HK$ and $H^g\cap N_K(H)\le H$, for all $g\in G$. In this paper, we investigate the structure of groups based on the assumption that every subgroup of $P\cap G^{{{𝒩}}_p}$ of order $p$ or 4 (if $p=2$) is an $\mathcal{\mathscr{H}}C$-subgroup of $N_G(P)$, here $P$ is a Sylow $p$-subgroup of $G$. Some results for a group to be $p$-nilpotent and supersolvable are obtained and many known results are generalized.

In this paper, a new characterization of $p$-hypercyclical embeddability of a normal subgroup of a finite group is obtained based on the notion of ${\mathscr{H}}C$-subgroups and some known results are generalized and extended.

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.

A subgroup $H$ of a group $G$ is called an $\mathcal{\mathscr{H}}C$-subgroup of $G$ if there exists a normal subgroup $T$ of $G$ such that $G=HT$ and $H^g\cap N_T(H)\le H$ for all $g\in G$. In this paper, we obtain new criteria for a normal subgroup to be contained in the $p𝔉$-hypercenter of a finite group by assuming that some of its subgroups are $\mathcal{\mathscr{H}}C$-subgroups. Our results generalize and uniform many known results.