On $\mathcal{M}$-normal embedded subgroups and the structure of finite groups

Abstract

Let $G$ be a group and $H$ be a subgroup of $G$. $H$ is said to be $\mathcal{M}$-normal supplemented in $G$ if there exists a normal subgroup $K$ of $G$ such that $G=HK$ and $H_1K<G$ for every maximal subgroup $H_1$ of $H$. Furthermore, $H$ is said to be $\mathcal{M}$-normal embedded in $G$ if there exists a normal subgroup $K$ of $G$ such that $G=HK$ and $H\cap K=1$ or $H\cap K$ is $\mathcal{M}$-normal supplemented in $G$. In this paper, some new criteria for a group to be nilpotent and $p$-supersolvable for some prime $p$ are obtained.