# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 136, 2016, pp. 175–189**

**DOI: 10.4171/RSMUP/136-12**

Published online: 2016-12-22

On $\pi\mathfrak{F}$-supplemented subgroups of a finite group

Li Zhang^{[1]}, Xiaoyu Chen

^{[2]}and Wenbin Guo

^{[3]}(1) University of Scinece and Technology of China, Hefei, China

(2) Nanjing Normal University, China

(3) University of Scinece and Technology of China, Hefei, China

Let $\mathfrak{F}$ be a class of groups and $G$ a finite group. A chief factor $H/K$ of $G$ is called *$\mathfrak{F}$-central in * $G$ provided $(H/K)\rtimes (G/C_{G}(H/K))\in\mathfrak{F}$. A normal subgroup $N$ of $G$ is said to be *$\pi\mathfrak{F}$-hypercentral in* $G$ if every chief factor of $G$ below $N$ of order divisible by at least one prime in $\pi$ is $\mathfrak{F}$-central in $G$. The $\pi\mathfrak{F}$-hypercentre of $G$ is the product of all the normal $\pi\mathfrak{F}$-hypercentral subgroups of $G$. In this paper, we study the structure of finite groups by using the notion of $\pi\mathfrak{F}$-hypercentre. New characterizations of some classes of finite groups are obtained.

*Keywords: *$\mathfrak{F}$-hypercentre, $\pi\mathfrak{F}$-hypercentre, Sylow subgroups, $n$-maximal subgroups

Zhang Li, Chen Xiaoyu, Guo Wenbin: On $\pi\mathfrak{F}$-supplemented subgroups of a finite group. *Rend. Sem. Mat. Univ. Padova* 136 (2016), 175-189. doi: 10.4171/RSMUP/136-12