arXiv subjects
Publications and source records attributed to Zhenfeng Wu.
A normal subgroup $E$ of a group $G$ is said to be hypercyclically embedded in $G$ if either $E=1$ or $E\neq 1$ and every chief factor of $G$ below $E$ is cyclic. In this article, we present some new characterizations of a normal subgroup to be hypercyclically embedded. Some recent results in this literature are generalized and unified.
Let $π$ be a proper subset of the set of all primes. Denote by $r$ the smallest prime which does not belong to $π$ and set $m = r$ if $r = 2$ or $3$ and $m = r-1$ if $r \geqslant 5$. We study the following conjecture: a conjugacy class $D$ of a finite group $G$ is contained in the $π$-radical $\mathrm{O}_π(G)$ of $G$ if and only if every $m$ elements of $D$ generate a $π$-subgroup. We confirm this conjecture for each group $G$ whose nonabelian composition factors are isomorphic to sporadic or alternating groups.
Let $π$ be a set of primes such that $|π|\geqslant 2$ and $π$ differs from the set of all primes. Denote by $r$ the smallest prime which does not belong to $π$ and set $m=r$ if $r=2,3$ and $m=r-1$ if $r\geqslant 5$. We study the following conjecture: a conjugacy class $D$ of a finite group $G$ is contained in $Oπ(G)$ if and only if every $m$ elements of $D$ generate a $π$-subgroup. We confirm this conjecture for each group $G$ whose nonabelian composition factors are isomorphic to alternating, linear and unitary simple groups.
Let $G$ be a finite group and $σ=\{σ_{i}|i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, that is, $\mathbb{P}=\bigcup_{i\in I}σ_{i}$ and $σ_{i}\cap σ_{j}=\emptyset$ for all $i\neq j$. A chief factor $H/K$ of $G$ is said to be $σ$-central in $G$, if the semidirect product $(H/K)\rtimes(G/C_G(H/K))$ is a $σ_i$-group for some $i\in I$. The group $G$ is said to be $σ$-nilpotent if either $G=1$ or every chief factor of $G$ is $σ$-central. In this paper, we study the properties of a finite group $G=AB$, factorized by two $σ$-nilpotent subgroups $A$ and $B$, and also generalize some known results.
Let $σ=\{σ_{i}|i\in I\}$ be some partition of the set $\mathbb{P}$ of all primes, that is, $\mathbb{P}=\bigcup_{i\in I}σ_{i}$ and $σ_{i}\cap σ_{j}=\emptyset$ for all $i\neq j$. Let $G$ be a finite group. A set $\mathcal {H}$ of subgroups of $G$ is said to be a complete Hall $σ$-set of $G$ if every non-identity member of $\mathcal {H}$ is a Hall $σ_{i}$-subgroup of $G$ and $\mathcal {H}$ contains exactly one Hall $σ_{i}$-subgroup of $G$ for every $σ_{i}\in σ(G)$. $G$ is said to be a $σ$-group if it possesses a complete Hall $σ$-set. A $σ$-group $G$ is said to be $σ$-dispersive provided $G$ has a normal series $1 = G_1<G_2<\cdots< G_t< G_{t+1} = G$ and a complete Hall $σ$-set $\{H_{1}, H_{2}, \cdots, H_{t}\}$ such that $G_iH_i = G_{i+1}$ for all $i= 1,2,\ldots t$. In this paper, we give a characterizations of $σ$-dispersive group, which give a positive answer to an open problem of Skiba in the paper.
Let G be a finite group and σ = {σ_i, i \in I} be a partition of the set of all primes \mathbb{P}. A set \mathcal{H} of subgroups of G with 1 \in \mathcal{H} is said to be a complete Hall σ-set of G if every non-identity member of \mathcal{H} is a Hall σ_i-subgroup of G. A subgroup H of G is said to be σ-permutable if G possesses a complete Hall σ-set \mathcal{H} such that HA^x = A^xH for all A \in \mathcal{H} and all x \in G. We say that a subgroup H of G is weakly σ-permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H \cap T \leq H_σG. where H_σG is the subgroup of H generated by all those subgroups of H which are σ-permutable in G. By using this new notion, we establish some new criterias for a group G to be a σ-soluble and supersoluble, and also we give the conditions under which a normal subgroup of G is hypercyclically embedded.