On the ℱϕ-Hypercentre of Finite Groups
On the ℱϕ-Hypercentre of Finite Groups
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DOI：
10.4153/cmb-2014-021-9
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发表时间：
2014-09
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影响因子：
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通讯作者：
Juping Tang;L. Miao
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文献类型：
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作者：
Juping Tang;L. Miao
设$G$是一个有限群，$数学{F}$是一类群。则${Z}_{\Mathcal{F}\Phi}}\Left(G\Right)$是$G$的$\Mathcal{F}\Phi$-超中心，它是$G$的所有正规子群的乘积，其非Frattini$G$-主因子是$G$中的$\Mathcal{F}$-中心.如果存在$G$的子群$B$，使得对于$H$的任何极大子群${H}_{1}}，$G$是$G$的真子群，则称$H$子群为$\数学{M}$-在有限群$G$中补充。本文的主要目的是证明：设$E$是群$G$的正规子群。假设每个非循环Sylow子群$P\，\Text{of}，{{F}^{*}}\Left(E\Right)$都有一个子群$D$，使得$1，\，\Left|D\Right|，<；左|P\右|$和每个子群$H\，文本{of}，P$顺序为$\Left|H\Right|\，=\，\Left|D\Right|$是$\Mathcal{M}$-用$G$补充，然后是$E\，\le\，{{Z}_{\Mathcal{U}\Phi}}\Left(G\Right)$。
Abstract Let $G$ be a finite group and let $\mathcal{F}$ be a class of groups. Then ${{Z}_{\mathcal{F}\Phi }}\left( G \right)$ 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 $G\,=\,HB\,\text{and}\,{{H}_{1}}B$ is a proper subgroup of $G$ for any maximal subgroup ${{H}_{1}}$ of $H$ . The main purpose of this paper is to prove the following: Let $E$ be a normal subgroup of a group $G$ . Suppose that every noncyclic Sylow subgroup $P\,\text{of}\,{{F}^{*}}\left( E \right)$ has a subgroup $D$ such that $1\,<\,\left| D \right|\,<\left| P \right|$ and every subgroup $H\,\text{of}\,P$ with order $\left| H \right|\,=\,\left| D \right|$ is $\mathcal{M}$ -supplemented in $G$ , then $E\,\le \,{{Z}_{\mathcal{U}\Phi }}\left( G \right)$ .