Finite groups with some weakly s-permutably embedded and weakly s-supplemented subgroups

Zhong, Guo; Ma, XuanLong; Lin, Shixun; Xia, Jiayi; Jin, Jianxing

doi:10.2140/involve.2015.8.253

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Finite groups with some weakly $s$-permutably embedded and weakly $s$-supplemented subgroups

Guo Zhong, XuanLong Ma, Shixun Lin, Jiayi Xia and Jianxing Jin

Vol. 8 (2015), No. 2, 253–262

DOI: 10.2140/involve.2015.8.253

Abstract

Let $G$ be a finite group. A subgroup $H$ of $G$ is called weakly $s$ -permutably embedded in $G$ if there is a subnormal subgroup $T$ of $G$ and an $s$ -permutably embedded subgroup $H_{s e}$ of $G$ contained in $H$ such that $G = H T$ and $H \cap T \leq H_{s e}$ . The subgroup $H$ is called weakly $s$ -supplemented in $G$ if $G$ has a subgroup $K$ such that $H K = G$ and $H \cap K \leq H_{s G}$ , where $H_{s G}$ is the largest $s$ -permutable subgroup of $G$ contained in $H$ . In this paper, we investigate the influence of weakly $s$ -permutably embedded and weakly $s$ -supplemented subgroups on the structure of finite groups. Some recent results are generalized.

Keywords

weakly $s$-permutably embedded subgroups, weakly $s$-supplemented subgroups, $p$-nilpotent groups

Mathematical Subject Classification 2010

Primary: 20D10

Secondary: 20D20

Milestones

Received: 10 January 2013

Revised: 6 August 2013

Accepted: 7 August 2013

Published: 3 March 2015

Communicated by Joseph A. Gallian

Authors

Guo Zhong
School of Mathematical Sciences Guangxi Teachers Education University Nanning, Guangxi 530023 China

XuanLong Ma
School of Mathematical Sciences Beijing Normal University Beijing 100875 China

Shixun Lin
College of Mathematics and Statistics Zhaotong University Zhaotong, Yunnan 657000 China

Jiayi Xia
School of Mathematical Sciences Guangxi Teachers Education University Nanning, Guangxi 530023 China

Jianxing Jin
School of Mathematical Sciences Guangxi Teachers Education University Nanning, Guangxi 530023 China

Vol. 8, No. 2, 2015