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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2012, Volume 13, Issue 1, Pages 18–25 (Mi adm62)

RESEARCH ARTICLE

On $S$-quasinormally embedded subgroups of finite groups

Kh. A. Al-Sharoa, Olga Shemetkovab, Xiaolan Yic

a Al al-Bayt University, St. Al-Zohoor 5–3, Mafraq 25113, Jordan
b Russian Economic University named after G. V. Plekhanov, Stremyanny Per., 36, 117997 Moscow, Russia
c Zhejiang Sci-Tech University, Hangzhou 310018, P. R. China

Abstract: Let $G$ be a finite group. A subgroup $A$ is called: 1) $S$-quasinormal in $G$ if $A$ is permutable with all Sylow subgroups in $G$ 2) $S$-quasinormally embedded in $G$ if every Sylow subgroup of $A$ is a Sylow subgroup of some $S$-quasinormal subgroup of $G$. Let $B_{seG}$ be the subgroup generated by all the subgroups of $B$ which are $S$-quasinormally embedded in $G$. A subgroup $B$ is called $SE$-supplemented in $G$ if there exists a subgroup $T$ such that $G=BT$ and $B\cap T\le B_{seG}$. The main result of the paper is the following.
Theorem. Let $H$ be a normal subgroup in $G$, and $p$ a prime divisor of $|H|$ such that $(p-1,|H|)=1$. Let $P$ be a Sylow $p$-subgroup in $H$. Assume that all maximal subgroups in $P$ are $SE$-supplemented in $G$. Then $H$ is $p$-nilpotent and all its $G$-chief $p$-factors are cyclic.

Keywords: Finite group, $p$-nilpotent, $S$-quasinormal subgroup.

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Bibliographic databases:
MSC: 20D10, 20D20, 20D25
Accepted:31.01.2012
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Citation: Kh. A. Al-Sharo, Olga Shemetkova, Xiaolan Yi, “On $S$-quasinormally embedded subgroups of finite groups”, Algebra Discrete Math., 13:1 (2012), 18–25

Citation in format AMSBIB
\Bibitem{Al-SheXia12} \by Kh.~A.~Al-Sharo, Olga~Shemetkova, Xiaolan~Yi \paper On $S$-quasinormally embedded subgroups of~finite groups \jour Algebra Discrete Math. \yr 2012 \vol 13 \issue 1 \pages 18--25 \mathnet{http://mi.mathnet.ru/adm62} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2963822} \zmath{https://zbmath.org/?q=an:1263.20020}