On the Fφ-Hypercentre of Finite Groups

Tang, Juping ; Miao, Long

Canadian mathematical bulletin, Tome 57 (2014) no. 3, pp. 648-657

Résumé

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 $E\,\le \,{{Z}_{\mathcal{U}\Phi }}\left( G \right)$ .

Détail

DOI : 10.4153/CMB-2014-021-9

Mots-clés : 20D10, 20D20, Fφ-hypercentre, Sylow subgroups, M-supplemented subgroups, formation.

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