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On s-semipermutable or s-quasinormally Embedded Subgroups of Finite Groups

Published online by Cambridge University Press:  20 November 2018

Qingjun Kong  and
Xiuyun Guo
Qingjun Kong
Affiliation:
Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, People’s Republic of China e-mail: shkqj2929@163.com
Xiuyun Guo
Affiliation:
Department of Mathematics, Shanghai University, Shanghai 200444, People’s Republic of China
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Abstract

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Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. $H$ is said to be $s$-semipermutable in $G$ if $H{{G}_{p}}\,=\,{{G}_{p}}H$ for any Sylow $p$-subgroup ${{G}_{p}}$ of $G$ with $\left( p,\,\left| H \right| \right)\,=\,1$; $H$ is said to be $s$-quasinormally embedded in $G$ if for each prime $p$ dividing the order of $H$, a Sylow $p$-subgroup of $H$ is also a Sylow $p$-subgroup of some $s$-quasinormal subgroup of $G$. In every non-cyclic Sylow subgroup $P$ of $G$ we fix some subgroup $D$ satisfying $1\,<\,\left| D \right|\,<\,\left| P \right|$ and study the structure of $G$ under the assumption that every subgroup $H$ of $P$ with $\left| H \right|\,=\,\left| D \right|$ is either $s$-semipermutable or $s$-quasinormally embedded in $G$. Some recent results are generalized and unified.

Keywords

20D1020D20s-semipermutable subgroups-quasinormally embedded subgroupsaturated formation
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 4 , 01 December 2015 , pp. 799 - 807
Copyright
Copyright © Canadian Mathematical Society 2015

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