New criteria for $p$-nilpotency of finite groups

Abstract

A subgroup $H$ of a group $G$ is said to be weakly $s$-supplementedly embedded in $G$ if there exists a subgroup $T$ of $G$ such that $G=HT$ and $H\cap T\leq H_{se}\leq H$, where $H_{se}$ is an S-permutably embedded subgroup of $G$. In this paper, we investigate the structure of $G$ under the assumption that some subgroups of prime-power order are weakly S-supplementedly embedded in $G$, and some new criteria for $p$-nilpotency are obtained.