On the ${mathcal F}{Phi}$-Hypercentre of Finite Groups

Canadian mathematical bulletin

Long Miao¹  Juping Tang¹ 

[1] School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, People's Republic of China

关键词: ${mathcal F}{Phi}$-hypercentre;    Sylow subgroups;    $mathcal M$-supplemented subgroups;    formation;   

DOI  :  10.4153/CMB-2014-021-9

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Let $G$ be a finite group, $mathcal F$ a class of groups.Then $Z_{{mathcal F}{Phi}}(G)$ is the ${mathcal F}{Phi}$-hypercentreof $G$ which is the product of all normal subgroups of $G$ whosenon-Frattini $G$-chief factors are $mathcal F$-central in $G$. Asubgroup $H$ is called $mathcal M$-supplemented in a finite group$G$, if there exists a subgroup $B$ of $G$ such that $G=HB$ and$H_1B$ is a proper subgroup of $G$ for any maximal subgroup $H_1$of $H$. The main purpose of this paper is to prove: Let $E$ be anormal subgroup of a group $G$. Suppose that every noncyclicSylowsubgroup $P$ of $F^{*}(E)$ has a subgroup $D$ such that$1lt |D|lt |P|$ and every subgroup $H$ of $P$ with order $|H|=|D|$is$mathcal M$-supplemented in $G$, then $Eleq Z_{{mathcalU}{Phi}}(G)$.

【 授权许可】

Unknown