- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnlilne Submission ㆍMy Manuscript - For Reviewers - For Editors
 On $\pi\mathfrak{F}$-embedded subgroups of finite groups Bull. Korean Math. Soc. 2016 Vol. 53, No. 1, 91-102 https://doi.org/10.4134/BKMS.2016.53.1.91Printed January 31, 2016 Wenbin Guo, Haifeng Yu, and Li Zhang University of Science and Technology of China, Hefei University, University of Science and Technology of China Abstract : A chief factor $H/K$ of $G$ is called $\mathfrak{F}$-central in $G$ provided $(H/K)\rtimes (G/C_{G}(H/K))\in\mathfrak{F}$. A normal subgroup $N$ of $G$ is said to be $\pi\mathfrak{F}$-hypercentral in $G$ if either $N=1$ or $N\neq1$ and every chief factor of $G$ below $N$ of order divisible by at least one prime in $\pi$ is $\mathfrak{F}$-central in $G$. The symbol $Z_{\pi\mathfrak{F}}(G)$ denotes the $\pi\mathfrak{F}$-hypercentre of $G$, that is, the product of all the normal $\pi\mathfrak{F}$-hypercentral subgroups of $G$. We say that a subgroup $H$ of $G$ is $\pi\mathfrak{F}$-embedded in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is $s$-quasinormal in $G$ and $(H\cap T)H_G/H_G\leq Z_{\pi\mathfrak{F}}(G/H_G)$, where $H_G$ is the maximal normal subgroup of $G$ contained in $H$. In this paper, we use the $\pi\mathfrak{F}$-embedded subgroups to determine the structures of finite groups. In particular, we give some new characterizations of $p$-nilpotency and supersolvability of a group. Keywords : $\pi\mathfrak{F}$-hypercenter, $\pi\mathfrak{F}$-embedded subgroup, Sylow subgroup, $n$-maximal subgroup MSC numbers : 20D10, 20D15, 20D20 Downloads: Full-text PDF