Finite Groups with some s -Permutably Embedded and Weakly s -Permutable Subgroups

Fenfang Xie[1]; Jinjin Wang[1]; Jiayi Xia[1]; Guo Zhong[1]

  • [1] School of Mathematical Sciences, Guangxi Teachers Education University, Nanning, Guangxi, 530023, P. R. China

Confluentes Mathematici (2013)

  • Volume: 5, Issue: 1, page 93-100
  • ISSN: 1793-7434

Abstract

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Let G be a finite group, p the smallest prime dividing the order of G and P a Sylow p -subgroup of G with the smallest generator number d . There is a set ℳ d ( P ) = { P 1 , P 2 , ⋯ , P d } of maximal subgroups of P such that ⋂ i = 1 d P i = Φ ( P ) . In the present paper, we investigate the structure of a finite group under the assumption that every member of ℳ d ( P ) is either s -permutably embedded or weakly s -permutable in G to give criteria for a group to be p -supersolvable or p -nilpotent.

How to cite

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Xie, Fenfang, et al. "Finite Groups with some $s$-Permutably Embedded and Weakly $s$-Permutable Subgroups." Confluentes Mathematici 5.1 (2013): 93-100. <http://eudml.org/doc/275595>.

@article{Xie2013,
abstract = {Let $G$ be a finite group, $p$ the smallest prime dividing the order of $G$ and $P$ a Sylow $p$-subgroup of $G$ with the smallest generator number $d$. There is a set $\mathcal\{M\}_d(P) = \lbrace P_1, P_2, \cdots , P_d\rbrace $ of maximal subgroups of $P$ such that $\bigcap ^d _\{i=1\}P_i=\Phi (P)$. In the present paper, we investigate the structure of a finite group under the assumption that every member of $\mathcal\{M\}_d(P)$ is either $s$-permutably embedded or weakly $s$-permutable in $G$ to give criteria for a group to be $p$-supersolvable or $p$-nilpotent.},
affiliation = {School of Mathematical Sciences, Guangxi Teachers Education University, Nanning, Guangxi, 530023, P. R. China; School of Mathematical Sciences, Guangxi Teachers Education University, Nanning, Guangxi, 530023, P. R. China; School of Mathematical Sciences, Guangxi Teachers Education University, Nanning, Guangxi, 530023, P. R. China; School of Mathematical Sciences, Guangxi Teachers Education University, Nanning, Guangxi, 530023, P. R. China},
author = {Xie, Fenfang, Wang, Jinjin, Xia, Jiayi, Zhong, Guo},
journal = {Confluentes Mathematici},
keywords = {weakly $s$-permutable subgoups; $s$-permutably embedded subgroups; $p$-nilpotent groups; finite groups; weakly -permutable subgoups; -permutably embedded subgroups; -nilpotent groups; maximal subgroups; Sylow subgroups; -nilpotency; -supersolvability},
language = {eng},
number = {1},
pages = {93-100},
publisher = {Institut Camille Jordan},
title = {Finite Groups with some $s$-Permutably Embedded and Weakly $s$-Permutable Subgroups},
url = {http://eudml.org/doc/275595},
volume = {5},
year = {2013},
}

TY - JOUR
AU - Xie, Fenfang
AU - Wang, Jinjin
AU - Xia, Jiayi
AU - Zhong, Guo
TI - Finite Groups with some $s$-Permutably Embedded and Weakly $s$-Permutable Subgroups
JO - Confluentes Mathematici
PY - 2013
PB - Institut Camille Jordan
VL - 5
IS - 1
SP - 93
EP - 100
AB - Let $G$ be a finite group, $p$ the smallest prime dividing the order of $G$ and $P$ a Sylow $p$-subgroup of $G$ with the smallest generator number $d$. There is a set $\mathcal{M}_d(P) = \lbrace P_1, P_2, \cdots , P_d\rbrace $ of maximal subgroups of $P$ such that $\bigcap ^d _{i=1}P_i=\Phi (P)$. In the present paper, we investigate the structure of a finite group under the assumption that every member of $\mathcal{M}_d(P)$ is either $s$-permutably embedded or weakly $s$-permutable in $G$ to give criteria for a group to be $p$-supersolvable or $p$-nilpotent.
LA - eng
KW - weakly $s$-permutable subgoups; $s$-permutably embedded subgroups; $p$-nilpotent groups; finite groups; weakly -permutable subgoups; -permutably embedded subgroups; -nilpotent groups; maximal subgroups; Sylow subgroups; -nilpotency; -supersolvability
UR - http://eudml.org/doc/275595
ER -

References

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