arXiv subjects

Zhencai Shen

Publications and source records attributed to Zhencai Shen.

6 recordsLinked to original sources

arXiv2024

On generalized covering and avoidance properties of finite groups and saturated fusion systems

A subgroup $A$ of a finite group $G$ is said to be a $CAP$-subgroup of $G$, if for any chief factor $H/K$ of $G$, either $A H= AK$ or $A\cap H = A \cap K$. Let $p$ be a prime, $S$ be a $p$-group and $\mathcal{F}$ be a saturated fusion system over $S$. Then $\mathcal{F}$ is said to be supersolvable, if there exists a series of $S$, namely $1 = S_0 \leq S_1 \leq \cdots \leq S_n = S$, such that $S_{i+1}/S_i$ is cyclic, and $S_i$ is strongly $\mathcal{F}$-closed for any $i=0,1,\cdots,n$. In this paper, we first introduce the concept of strong $p$-$CAP$-subgroups, and investigate the structure of finite groups under the assumptions that some subgroups of $G$ are partial $CAP$-subgroups or strong $(p)$-$CAP$-subgroups of $G$, and obtain some criteria for a group $G$ to be $p$-supersolvable. After that, we investigate the characterizations for supersolvability of $\mathcal{F}_S (G)$ under the assumptions that some subgroups of $G$ are partial $CAP$-subgroups or strong $(p)$-$CAP$-subgroups of $G$, and obtain some criteria for a fusion system $\mathcal{F}_S (G)$ to be supersolvable. The above results improve some known results and develop some new results about $CAP$-subgroups from fusion systems.

math.GR↗

arXiv2024

The influence of weakly $SΦ$-supplemented subgroups on fusion systems of finite groups

Let $G$ be a finite group and $H$ be a subgroup of $G$. Then $H$ is called a weakly $SΦ$-supplemented subgroup of $G$, if there exists a subgroup $T$ of $G$ such that $G =HT$ and $H \cap T \leq Φ(H) H_{sG}$, where $H_{sG}$ denotes the subgroup of $H$ generated by all subgroups of $H$ which are $S$-permutable in $G$. Let $p$ be a prime, $S$ be a $p$-group and $\mathcal{F}$ be a saturated fusion system over $S$. Then $\mathcal{F}$ is said to be supersolvable, if there exists a series of $S$, namely $1 = S_0 \leq S_1 \leq \cdots \leq S_n = S$, such that $S_{i+1}/S_i$ is cyclic, $i=0,1,\cdots, n-1$, $S_i$ is strongly $\mathcal{F}$-closed, $i=0,1,\cdots,n$. In this paper, we investigate the structure of fusion system $\mathcal{F}_S (G)$ under the assumption that certain subgroups of $S$ are weakly $SΦ$-supplemented in $G$, and obtain several new characterizations of supersolvability of $\mathcal{F}_S (G)$.

math.GR↗

arXiv2024

Some results on the norm of finite groups

Let $G$ be a finite group and $N_Ω(G)$ be the intersection of the normalizers of all subgroups belonging to the set $Ω(G),$ where $Ω(G)$ is a set of all subgroups of $G$ which have some theoretical group property. In this paper, we show that $N_Ω(G)= Z_{\infty}(G)$ if $Ω(G)$ is one of the following: (i) the set of all self-normalizing subgroups of $G$; (ii) the set of all subgroups of $G$ satisfying the subnormalizer condition in $G$; (iii) the set of all pronormal subgroups of $G$; (iv) the set of all $\mathscr{H}$-subgroups of $G$; (v) the set of all weakly normal subgroups of $G$; (vi) the set of all $NE$-subgroups of $G$.

math.GR↗

arXiv2024

New characterizations for supersolvability of fusion system and $p$-nilpotency of finite groups

Let $p$ be a prime, $S$ be a $p$-group and $\mathcal{F}$ be a saturated fusion system over $S$. Then $\mathcal{F}$ is said to be supersolvable, if there exists a series of $S$, namely $1 = S_0 \leq S_1 \leq \cdots \leq S_n = S$, such that $S_{i+1}/S_i$ is cyclic, $i=0,1,\cdots, n-1$, $S_i$ is strongly $\mathcal{F}$-closed, $i=0,1,\cdots,n$. In this paper, we investigate the characterizations for supersolvability of $\mathcal{F}_S (G)$ under the assumption that certain subgroups of $G$ satisfy different kinds of generalized normalities in section \ref{1003}. Moreover, we obtain the more advanced and remarkable result of characterizations for generalized saturated fusion system $\mathcal{F}$ in section \ref{Section 4}. Finally, we apply the results in section \ref{1003} and \ref{Section 4} and give characterizations for $p$-nilpotency of finite groups under the assumption that some subgroups of $G$ satisfy different kinds of generalized normalities.

math.GR↗

arXiv2023

On the finite solvable $PNC$-groups

A subgroup $H$ of a finite group $G$ is said to be an NC-subgroup of $G$, if $ H^G N_G (H) =G$, where $H^G$ denotes the normal closure of $H$ in $G$. A finite group $G$ is called a PNC-group, if any subgroup of $G$ is an NC-subgroup of $G$, and $G$ is said to be an ON-group, if for any subgroup $H$ of $G$, either $N_G (H)=H,\,H^G=G$, or $H \unlhd G$. In this paper, we firstly investigate the basic properties of solvable PNC-groups, and then give several sufficient conditions for $G$ to be a solvable PNC-group. In the end of this paper, we discover some characterizations for minimal non-ON-groups, ON-groups, non-abelian simple groups whose second maximal subgroups are solvable PNC-groups and groups whose proper (maximal) subgroups are solvable PNC-groups.

math.GR↗

arXiv2022

A further generalization of the Glauberman-Thompson $p$-nilpotency criterion in fusion systems

Let $p$ be a prime and $\mathcal{F}$ be a saturated fusion system over a finite $p$-group $P$. The fusion system $\mathcal{F}$ is said to be nilpotent if $\mathcal{F}=\mathcal{F}_{P}(P)$. We provide new criteria for a saturated fusion system $\mathcal{F}$ to be nilpotent, which may be viewed as extending the Glauberman-Thompson $p$-nilpotency criterion to fusion systems.

math.GR↗