# Institute of Mathematics

## 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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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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