Finite groups with $X$-quasipermutable subgroups of prime power order

Document Type: Research Paper

Authors

Department of Mathematics, Zhejiang Sci-Tech University, 310018, Hangzhou, P. R. China.

Abstract

Let $H$, $L$ and $X$ be subgroups of a finite group $G$. Then $H$ is said to be $X$-permutable with $L$ if for some
$xin X$ we have $AL^{x}=L^{x}A$. We say that $H$ is  emph{$X$-quasipermutable } (emph{$X_{S}$-quasipermutable}, respectively) in $G$ provided $G$ has a subgroup $B$ such that $G=N_{G}(H)B$ and $H$ $X$-permutes  with $B$ and with all subgroups (with all Sylow subgroups, respectively) $V$ of $B$ such that $(|H|, |V|)=1$. In this paper, we analyze the influence of $X$-quasipermutable and $X_{S}$-quasipermutable subgroups on the structure of $G$. Some known results are generalized.

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History

• Receive Date: 01 December 2014
• Revise Date: 01 February 2015
• Accept Date: 03 February 2015