arXiv subjects
Publications and source records attributed to Hangyang Meng.
A second maximal subgroup of a finite group $G$ is a subgroup $H$ that is maximal in every maximal overgroup of $H$ in $G$. We prove that every maximal subgroup of a prime-index subgroup of a finite group is a second maximal subgroup.
A positive integer $m$ is termed a \emph{Hall number} if every finite group $G$ whose order is precisely divisible by $m$ possesses a Hall subgroup of order $m$. Seeking generalizations of Sylow's theorem and Hall's theorem for finite solvable groups, Jiping Zhang asked for a full classification of Hall numbers, a problem recently solved by Guo, Hu and Li. Inspired by Zhang's problem, Guohua Qian put forward an analogous problem on a full classification of Hall complement numbers. Recall that a positive integer $m$ is called a \emph{Hall complement number} provided that every finite group $G$ with $m$ precisely dividing $|G|$ admits a Hall subgroup of order $|G|/m$. In the present paper, we prove that every Hall complement number is either $1$ or of the form $4k+2$ for some non-negative integer $k$, thus answering Qian's problem.
Let $G$ be a finite group. For a prime $p$ and an integer $e \geq 0$, we denote by $\Gamma_{p,e}(G)$ the set of all pairs $(H, \varphi)$, where $H$ is a $p$-subgroup of $G$ of order greater than $p^e$ and $\varphi$ is a complex irreducible character of $H$. In this paper, we investigate the connected components of the poset $\Gamma_{p,e}(G)$. For the case $e = 0$, we prove that $\Gamma_{p,0}(G)$ is disconnected if and only if either $G$ has a strongly $p$-embedded subgroup, or every Sylow $p$-subgroup of $G$ contains a unique subgroup of order $p$. Furthermore, for $e = 1$ and $G$ a $p$-group, we show that the number of connected components of $\Gamma_{p,1}(G)$ equals the order of the intersection of all subgroups of $G$ of order $p^2$.
Let $G$ be a finite group and $p$ be a prime. We denote by $C_p(G)$ the poset of all cosets of $p$-subgroups of $G$. We characterize the homotopy type of the geometric realization $|\Delta C_p(G)|$ for $p$-closed groups $G$, which is motivated by K.S.Brown's Question. We will further demonstrate that $\chi(C_{p}(G)) \equiv |G|_{p'} (\text{mod} p)$ for any finite group $G$ and any prime $p$.
In this paper, we investigate the difference between weak second maximal subgroups and second maximal subgroups. A sufficient and necessary condition is also given to describe such class of groups whose weak second maximal subgroups coincide with its second maximal subgroups(called WSM-groups) under the solvable case. As an application, we show that every non-vanishing element of a solvable WSM-group lies in its Fitting subgroup.