Partially $S$-embedded minimal subgroups of finite groups

Abstract

Suppose that $H$ is a subgroup of $G$‎, ‎then $H$ is said to be‎ ‎$s$-permutable in $G$‎, ‎if $H$ permutes with every Sylow subgroup of‎ ‎$G$‎. ‎If $HP=PH$ hold for every Sylow subgroup $P$ of $G$ with $(|P|‎, ‎|H|)=1$)‎, ‎then $H$ is called an $s$-semipermutable subgroup of $G$‎. ‎In this paper‎, ‎we say that $H$ is partially $S$-embedded in $G$ if‎ ‎$G$ has a normal subgroup $T$ such that $HT$ is $s$-permutable in‎ ‎$G$ and $Hcap Tleq H_{overline{s}G}$‎, ‎where $H_{overline{s}G}$‎ ‎is generated by all $s$-semipermutable subgroups of $G$ contained in‎ ‎$H$‎. ‎We investigate the influence of some partially $S$-embedded‎ ‎minimal subgroups on the nilpotency and supersolubility of a finite‎ ‎group $G$‎. ‎A series of known results in the literature are unified‎ ‎and generalized.‎