# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 136, 2016, pp. 175–189**

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

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

Li Zhang^{[1]}, Xiaoyu Chen

^{[2]}and Wenbin Guo

^{[3]}(1) School of Mathematical Sciences, University of Scinece and Technology of China, 230026, Hefei, Anhui, China

(2) School of Mathematical Sciences, Nanjing Normal University, 210023, Nanjing, China

(3) School of Mathematical Sciences, University of Scinece and Technology of China, 230026, Hefei, Anhui, 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