Czechoslovak Mathematical Journal, first online, pp. 1-7


Finite $p$-nilpotent groups with some subgroups weakly $\mathcal{M}$-supplemented

Liushuan Dong

Received June 3, 2018.   Published online November 19, 2019.

Abstract:  Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. Subgroup $H$ is said to be weakly $\mathcal{M}$-supplemented in $G$ if there exists a subgroup $B$ of $G$ such that (1) $G=HB$, and (2) if $H_1/H_G$ is a maximal subgroup of $H/H_G$, then $H_1B=BH_1<G$, where $H_G$ is the largest normal subgroup of $G$ contained in $H$. We fix in every noncyclic Sylow subgroup $P$ of $G$ a subgroup $D$ satisfying $1<|D|<|P|$ and study the $p$-nilpotency of $G$ under the assumption that every subgroup $H$ of $P$ with $|H|=|D|$ is weakly $\mathcal{M}$-supplemented in $G$. Some recent results are generalized.
Keywords:  $p$-nilpotent group; weakly $\mathcal{M}$-supplemented subgroup; finite group
Classification MSC:  20D10, 20D20
DOI:  10.21136/CMJ.2019.0273-18

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References:
[1] D. Gorenstein: Finite Groups. Chelsea Publishing Company, New York (1980). MR 0569209 | Zbl 0463.20012
[2] B. Huppert: Endliche Gruppen I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 134, Springer, Berlin (1967). (In German.) DOI 10.1007/978-3-642-64981-3 | MR 0224703 | Zbl 0217.07201
[3] Y. Li, Y. Wang, H. Wei: The influence of $\pi$-quasinormality of some subgroups of a finite group. Arch. Math. 81 (2003), 245-252. DOI 10.1007/s00013-003-0829-6 | MR 2013253 | Zbl 1053.20017
[4] L. Miao: On weakly $\mathcal M$-supplemented subgroups of Sylow $p$-subgroups of finite groups. Glasg. Math. J. 53 (2011), 401-410. DOI 10.1017/S0017089511000036 | MR 2783169 | Zbl 1218.20017
[5] L. Miao, W. Lempken: On $\mathcal M$-supplemented subgroups of finite groups. J. Group Theory 12 (2009), 271-287. DOI 10.1515/JGT.2008.077 | MR 2502219 | Zbl 1202.20021
[6] L. Miao, W. Lempken: On weakly $\mathcal M$-supplemented primary subgroups of finite groups. Turk. J. Math. 34 (2010), 489-500. DOI 10.3906/mat-0901-32 | MR 2721962 | Zbl 1210.20022
[7] D. J. S. Robinson: A Course in the Theory of Groups. Graduate Texts in Mathematics 80, Springer, New York (1982). DOI 10.1007/978-1-4419-8594-1 | MR 1261639 | Zbl 0483.20001
[8] H. Wei, Y. Wang: On $c^*$-normality and its properties. J. Group Theory 10 (2007), 211-223. DOI 10.1515/JGT.2007.017 | MR 2302616 | Zbl 1125.20011

Affiliations:   Liushuan Dong, College of Information and Business, Zhongyuan University of Technology, No. 41 Zhongyuan Road, Zhengzhou 450007, P. R. China, e-mail: dk091234@163.com


 
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