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On the ℱϕ-Hypercentre of Finite Groups

Published online by Cambridge University Press:  20 November 2018

Juping Tang  and
Long Miao
Juping Tang
Affiliation:
School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, People’s Republic of China e-mail: lmiao@yzu.edu.cntangjuping 205@126.com
Long Miao*
Affiliation:
School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, People’s Republic of China e-mail: lmiao@yzu.edu.cntangjuping 205@126.com
*
L. Miao is the corresponding author
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Abstract

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

Keywords

20D1020D20ℱϕ-hypercentreSylow subgroupsM-supplemented subgroupsformation.
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 3 , 01 September 2014 , pp. 648 - 657
Copyright
Copyright © Canadian Mathematical Society 2014

Footnotes

This research is supported by NSFC (Grant #11271016), the Postgraduate Innovation Project of Jiangsu Province (No. CXZZ13–0890), and the Natural Science Fund for Colleges and Universities in Anhui Province (Grant #KJ2013B138).

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