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 ${\mathcal{M}}_{d}\left(P\right)=\{{P}_{1},{P}_{2},\cdots ,{P}_{d}\}$ of maximal subgroups of $P$ such that ${\bigcap}_{i=1}^{d}{P}_{i}=\Phi \left(P\right)$. In the present paper, we investigate the structure of a finite group under the assumption that every member of ${\mathcal{M}}_{d}\left(P\right)$ is either $s$-permutably embedded or weakly $s$-permutable in $G$ to give criteria for a group to be $p$-supersolvable or $p$-nilpotent.
Keywords: weakly $s$-permutable subgoups; $s$-permutably embedded subgroups; $p$-nilpotent groups
Fenfang Xie ^{1}; Jinjin Wang ^{1}; Jiayi Xia ^{1}; Guo Zhong ^{1}
@article{CML_2013__5_1_93_0, author = {Fenfang Xie and Jinjin Wang and Jiayi Xia and Guo Zhong}, title = {Finite {Groups} with some $s${-Permutably} {Embedded} and {Weakly} $s${-Permutable} {Subgroups}}, journal = {Confluentes Mathematici}, pages = {93--100}, publisher = {Institut Camille Jordan}, volume = {5}, number = {1}, year = {2013}, doi = {10.5802/cml.4}, mrnumber = {3143613}, language = {en}, url = {https://cml.centre-mersenne.org/articles/10.5802/cml.4/} }
TY - JOUR AU - Fenfang Xie AU - Jinjin Wang AU - Jiayi Xia AU - Guo Zhong TI - Finite Groups with some $s$-Permutably Embedded and Weakly $s$-Permutable Subgroups JO - Confluentes Mathematici PY - 2013 SP - 93 EP - 100 VL - 5 IS - 1 PB - Institut Camille Jordan UR - https://cml.centre-mersenne.org/articles/10.5802/cml.4/ DO - 10.5802/cml.4 LA - en ID - CML_2013__5_1_93_0 ER -
%0 Journal Article %A Fenfang Xie %A Jinjin Wang %A Jiayi Xia %A Guo Zhong %T Finite Groups with some $s$-Permutably Embedded and Weakly $s$-Permutable Subgroups %J Confluentes Mathematici %D 2013 %P 93-100 %V 5 %N 1 %I Institut Camille Jordan %U https://cml.centre-mersenne.org/articles/10.5802/cml.4/ %R 10.5802/cml.4 %G en %F CML_2013__5_1_93_0
Fenfang Xie; Jinjin Wang; Jiayi Xia; Guo Zhong. Finite Groups with some $s$-Permutably Embedded and Weakly $s$-Permutable Subgroups. Confluentes Mathematici, Volume 5 (2013) no. 1, pp. 93-100. doi : 10.5802/cml.4. https://cml.centre-mersenne.org/articles/10.5802/cml.4/
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