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Wenbin Guo

Publications and source records attributed to Wenbin Guo.

46 records · Page 3Linked to original sources
arXiv2014

Confirmation for Wielandt's conjecture

Let $π$ be a set of primes. By H.Wielandt definition, {\it Sylow $π$-theorem} holds for a finite group $G$ if all maximal $π$-subgroups of $G$ are conjugate. In the paper, the following statement is proven. Assume that $π$ is a union of disjoint subsets $σ$ and $τ$ and a finite group $G$ possesses a $π$-Hall subgroup which is a direct product of a $σ$-subgroup and a $τ$-subgroup. Furthermore, assume that both the Sylow $σ$-theorem and $τ$-theorem hold for $G$. Then the Sylow $π$-theorem holds for $G$. This result confirms a conjecture posed by H.\,Wielandt in~1959.

math.GR↗
arXiv2014

On weakly $\frak{F}_{s}$-quasinormal subgroups of finite groups

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 $(H\cap T)H_{G}/H_{G}\leq Z_{\mathfrak{F}}(G/H_{G})$, where $Z_{\mathfrak{F}}(G/H_{G})$ denotes the $\mathfrak{F}$-hypercenter of $G/H_{G}$. In this paper, we study the structure of finite groups by using the concept of weakly $\mathfrak{F}_{s}$-quasinormal subgroups.

math.GR↗
arXiv2014

Finite groups in which SS-permutability is a transitive relation

A subgroup $H$ of a finite group $G$ is said to be SS-permutable in $G$ if $H$ has a supplement $K$ in $G$ such that $H$ permutes with every Sylow subgroup of $K$. A finite group $G$ is called an SST-group if SS-permutability is a transitive relation on the set of all subgroups of $G$. The structure of SST-groups is investigated in this paper.

math.GR↗
arXiv2014

On the $π$$\mathfrak{F}$-norm and the $\mathfrak{H}$-$\mathfrak{F}$-norm of a finite group

Let $\mathfrak{H}$ be a Fitting class and $\mathfrak{F}$ a formation. We call a subgroup $\mathcal{N}_{\mathfrak{H},\mathfrak{F}}(G)$ of a finite group $G$ the $\mathfrak{H}$-$\mathfrak{F}$-norm of $G$ if $\mathcal{N}_{\mathfrak{H},\mathfrak{F}}(G)$ is the intersection of the normalizers of the products of the $\mathfrak{F}$-residuals of all subgroups of $G$ and the $\mathfrak{H}$-radical of $G$. Let $π$ denote a set of primes and let $\mathfrak{G}_π$ denote the class of all finite $π$-groups. We call the subgroup $\mathcal{N}_{\mathfrak{G}_π,\mathfrak{F}}(G)$ of $G$ the $π\mathfrak{F}$-norm of $G$. A normal subgroup $N$ of $G$ is called $π\mathfrak{F}$-hypercentral in $G$ if either $N=1$ or $N>1$ and every $G$-chief factor below $N$ of order divisible by at least one prime in $π$ is $\mathfrak{F}$-central in $G$. Let $Z_{π\mathfrak{F}}(G)$ denote the $π\mathfrak{F}$-hypercentre of $G$, that is, the product of all $π\mathfrak{F}$-hypercentral normal subgroups of $G$. In this paper, we study the properties of the $\mathfrak{H}$-$\mathfrak{F}$-norm, especially of the $π\mathfrak{F}$-norm of a finite group $G$. In particular, we investigate the relationship between the $π'\mathfrak{F}$-norm and the $π\mathfrak{F}$-hypercentre of $G$.

math.GR↗
arXiv2014

The influence of $\mathfrak{F_{\mathrm s}}$-quasinormality of subgroups on the structure of finite groups

Let $\frak{F}$ be a class of finite groups. A subgroup $H$ of a finite group $G$ is said to be $\mathfrak{F_{\mathrm s}}$-quasinormal in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is $s$-permutable in $G$ and $(H\cap T)H_G/H_G$ is contained in the $\frak{F}$-hypercenter $Z_\infty^\mathfrak{F}(G/H_G)$ of $G/H_G$. In this paper, we investigate further the influence of $\mathfrak{F_{\mathrm s}}$-quasinormality of some subgroups on the structure of finite groups. New characterization of some classes of finite groups are obtained.

math.GR↗
arXiv2014

On $Π$-supplemented subgroups of a finite group

A subgroup $H$ of a finite group $G$ is said to satisfy $Π$-property in $G$ if for every chief factor $L/K$ of $G$, $|G/K:N_{G/K}(HK/K\cap L/K)|$ is a $π(HK/K\cap L/K)$-number. A subgroup $H$ of $G$ is called to be $Π$-supplemented in $G$ if there exists a subgroup $T$ of $G$ such that $G=HT$ and $H\cap T\leq I\leq H$, where $I$ satisfies $Π$-property in $G$. In this paper, we investigate the structure of a finite group $G$ under the assumption that some primary subgroups of $G$ are $Π$-supplemented in $G$. The main result we proved improves a large number of earlier results.

math.GR↗
arXiv2013

On weakly S-embedded subgroups and weakly $τ$-embedded subgroups

Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be weakly S-embedded in $G$ if there exists $K\unlhd G$ such that $HK$ is S-quasinormal in $G$ and $H\cap K\leq H_{seG}$, where $H_{seG}$ is the subgroup generated by all those subgroups of $H$ which are S-quasinormally embedded in $G$. We say that $H$ is weakly $τ$-embedded in $G$ if there exists $K\unlhd G$ such that $HK$ is S-quasinormal in $G$ and $H\cap K\leq H_{τG}$, where $H_{τG}$ is the subgroup generated by all those subgroups of $H$ which are $τ$-quasinormal in $G$. In this paper, we study the properties of the weakly S-embedded subgroups and the weakly $τ$-embedded subgroups, and use them to determine the structure of finite groups.

math.GR↗
arXiv2012

On the intersection of the $\cal F$-maximal subgroups and the generalized ${\cal F}$-hypercentre of a finite group

Let $\cal F$ be a class of groups. A chief factor $H/K$ of a group $G$ is called \emph{${\cal F}$-central in $G$} provided $(H/K)\rtimes (G/C_{G}(H/K)) \in {\cal F}$. We write $Z_{π{\cal F}}(G)$ to denote the product of all normal subgroups of $G$ whose $G$-chief factors of order divisible by at least one prime in $π$ are $\cal F$-central. We call $Z_{π{\cal F}}(G)$ the $π{\cal F}$-hypercentre of $G$. A subgroup $U$ of a group $G$ is called \emph{$\cal F$-maximal} in $G$ provided that (a) $U\in {\cal F}$, and (b) if $U\leq V\leq G$ and $V\in {\cal F}$, then $U=V$. In this paper we study the properties of the intersection of all $\cal F$-maximal subgroups of a finite group. In particular, we analyze the condition under which $Z_{π{\cal F}}(G)$ coincides with the intersection of all $\cal F$-maximal subgroups of $G$.

math.GR↗