Narrow Results

Let G be a finite group. A subgroup H of G is called an ℋ-subgroup of G if N_G(H) ∩ H^g ≤ H for all g ∈ G; G is said to be an ℋ_p-group if every cyclic subgroup of G of prime order or order 4 is an ℋ-subgroup of G. In this paper, the structure of the finite groups all of whose maximal subgroups of even order are ℋ_p-subgroups have been characterized.

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.

Let $G$ be a finite group and $H$ a subgroup of $G$. We say that $H$ is an ${\mathscr{H}}$-subgroup in $G$ if $N_G(H)\cap H\le H^g$ for all $g\in G$; $H$ is called weakly ${\mathscr{H}}$-subgroup in $G$ if it has a normal subgroup $K$ such that $G=HK$ and $H\cap K$ is an ${\mathscr{H}}$-subgroup in $G$. In this paper, we present some sufficient conditions for a group to be $p$-nilpotent under the assumption that certain subgroups of fixed prime power orders are weakly ${\mathscr{H}}$-subgroups in $G$. The main results improve and extend new and recent results in the literature.

A subgroup $H$ of a finite group $G$ is said to be an $\mathcal{\mathscr{H}}$-subgroup in $G$ if $N_G(H)\cap H^g\le H$ for all $g\in G$; and $H$ is said to be a weakly $\mathcal{\mathscr{H}}$-subgroup in $G$ if there is a normal subgroup $T$ of $G$ such that $G=HT$ and $H\cap T$ is an $\mathcal{\mathscr{H}}$-subgroup of $G$. In this paper, we give a positive answer to a problem posed by Li and Qiao [On weakly $\mathcal{\mathscr{H}}$-subgroups and $p$-nilpotency of finite groups, J. Algebra Appl.16 (2017) 1750042].

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.

The norm $N(G)$ of a group $G$ is the intersection of the normalizers of all subgroups in $G$. In this paper, the norm is generalized by studying on Sylow subgroups and ${\mathscr{H}}$-subgroups in finite groups which is denoted by $C(G)$ and $A(G)$, respectively. It is proved that the generalized norms $A(G)$ and $C(G)$ are all equal to the hypercenter of $G$.

Let be a saturated formation containing , and G be a finite group. Li etc. proposed a problem: whether there is a normality such that the following two statements are equivalent: (i) . (ii) There exists a normal subgroup H of G such that and for each Sylow subgroup P of F*(H), every member in some satisfies the above normality in G. In this paper, we find a normality satisfying the above problem. Moreover, by using the concept of ℋ-subgroups, we obtain other results about the influence of the members of some fixed on the structure of G.