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

  Published:2014-04-28
 Printed: Sep 2014
  • Juping Tang,
    School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, People's Republic of China
  • Long Miao,
    School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, People's Republic of China
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Abstract

Let $G$ be a finite group, $\mathcal F$ a class of groups. Then $Z_{{\mathcal F}{\Phi}}(G)$ is the ${\mathcal F}{\Phi}$-hypercentre of $G$ which is the product of all normal subgroups of $G$ whose non-Frattini $G$-chief factors are $\mathcal F$-central in $G$. A subgroup $H$ 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 a normal subgroup of a group $G$. Suppose that every noncyclic Sylow subgroup $P$ of $F^{*}(E)$ has a subgroup $D$ such that $1\lt |D|\lt |P|$ and every subgroup $H$ of $P$ with order $|H|=|D|$ is $\mathcal M$-supplemented in $G$, then $E\leq Z_{{\mathcal U}{\Phi}}(G)$.
Keywords: ${\mathcal F}{\Phi}$-hypercentre, Sylow subgroups, $\mathcal M$-supplemented subgroups, formation ${\mathcal F}{\Phi}$-hypercentre, Sylow subgroups, $\mathcal M$-supplemented subgroups, formation
MSC Classifications: 20D10, 20D20 show english descriptions Solvable groups, theory of formations, Schunck classes, Fitting classes, $\pi$-length, ranks [See also 20F17]
Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure
20D10 - Solvable groups, theory of formations, Schunck classes, Fitting classes, $\pi$-length, ranks [See also 20F17]
20D20 - Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure
 

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