On the converse of Hall's theorem

In this paper, we mainly investigate the converse of a well-known theorem proved by P. Hall, and present detailed characterizations under the various assumptions of the existence of some families of Hall subgroups. In particular, we prove that if $p\neq 3$ and a finite group $G$ has a Hall $\{p,q\}$-subgroup for every prime $q\neq p$, then $G$ is $p$-soluble

On a problem from the Kourovka Notebook

In this manuscript, a solution to Problem 18.91(b) in the Kourovka Notebook is given by proving the following theorem. Let $P$ be a Sylow $p$-subgroup of a group $G$ with $|P| = p^n$. Suppose that there is an integer $k$ such that $1 < k < n$ and every subgroup of $P$ of order $p^k$ is $S$-propermutable in $G$, and also, in the case that $p=2$, $k = 1$ and $P$ is non-abelian, every cyclic subgroup of $P$ of order $4$ is $S$-propermutable in $G$. Then $G$ is $p$-nilpotent

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

The Decomposition of Permutation Module for Infinite Chevalley Groups

Let ${\bf G}$ be a connected reductive group defined over $\mathbb{F}_q$, the finite field with $q$ elements. Let ${\bf B}$ be an Borel subgroup defined over $\mathbb{F}_q$. In this paper, we completely determine the composition factors of the induced module \mathbb{M}(\op{tr})=\Bbbk{\bf G}\otimes_{\Bbbk{\bf B}}\op{tr} (\op{tr} is the trivial ${\bf B}$-module) for any field $\Bbbk$.Comment: Accepted by Science China Mathematic

On the π\piF\mathfrak{F}-norm and the H\mathfrak{H}-F\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 $\pi$ denote a set of primes and let $\mathfrak{G}_\pi$ denote the class of all finite $\pi$-groups. We call the subgroup $\mathcal{N}_{\mathfrak{G}_\pi,\mathfrak{F}}(G)$ of $G$ the $\pi\mathfrak{F}$-norm of $G$. A normal subgroup $N$ of $G$ is called $\pi\mathfrak{F}$-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 $\pi$ is $\mathfrak{F}$-central in $G$. Let $Z_{\pi\mathfrak{F}}(G)$ denote the $\pi\mathfrak{F}$-hypercentre of $G$, that is, the product of all $\pi\mathfrak{F}$-hypercentral normal subgroups of $G$. In this paper, we study the properties of the $\mathfrak{H}$-$\mathfrak{F}$-norm, especially of the $\pi\mathfrak{F}$-norm of a finite group $G$. In particular, we investigate the relationship between the $\pi'\mathfrak{F}$-norm and the $\pi\mathfrak{F}$-hypercentre of $G$

On Π\Pi-supplemented subgroups of a finite group

A subgroup $H$ of a finite group $G$ is said to satisfy $\Pi$-property in $G$ if for every chief factor $L/K$ of $G$, $|G/K:N_{G/K}(HK/K\cap L/K)|$ is a $\pi(HK/K\cap L/K)$-number. A subgroup $H$ of $G$ is called to be $\Pi$-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 $\Pi$-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 $\Pi$-supplemented in $G$. The main result we proved improves a large number of earlier results.Comment: arXiv admin note: text overlap with arXiv:1301.636

The Permutation Module on Flag Varieties in Cross Characteristic

Let ${\bf G}$ be a connected reductive group over $\bar{\mathbb{F}}_q$, the algebraically closure of $\mathbb{F}_q$ (the finite field with $q=p^e$ elements), with the standard Frobenius map $F$. Let ${\bf B}$ be an $F$-stable Borel subgroup. Let $\Bbbk$ be a field of characteristic $r\neq p$. In this paper, we completely determine the composition factors of the induced module $Ind_{B}^{G}{tr}=\Bbbk{G}\otimes_{\Bbbk{\bf B}}$ tr (here $\Bbbk{H}$ is the group algebra of the group ${H}$, and tr is the trivial $B$-module). In particular, we find a new family of infinite dimensional irreducible abstract representations of $G$.Comment: Accepted by Mathematische Zeitschrif

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

On HC-subgroups of a finite group

A subgroup $H$ of a finite group $G$ 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 $H^g \cap N_T(H)\leq H$ for all $g\in G$. In this paper, we investigate the structure of a finite group $G$ under the assumption that certain subgroups of $G$ of arbitrary prime power order are $\mathscr{H}C$-subgroups of $G$

On weakly Fs\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