Let $\cal F$ be a saturated formation containing the class of supersolvable groups and let $G$ be a finite group. The following theorems are presented: (1) $G\in \cal F$ if and only if there is a normal subgroup $H$ such that $G/H\in \cal F$ and every maximal subgroup of all Sylow subgroups of $H$ is either $c$-normal or $S$-quasinormally embedded in $G$. (2) $G\in \cal F$ if and only if there is a normal subgroup $H$ such that $G/H\in \cal F$ and every maximal subgroup of all Sylow subgroups of $F^*(H)$, the generalized Fitting subgroup of $H$, is either $c$-normal or $S$-quasinormally embedded in $G$. (3) $G\in \cal F$ if and only if there is a normal subgroup $H$ such that $G/H\in \cal F$ and every cyclic subgroup of $F^*(H)$ of prime order or order 4 is either $c$-normal or $S$-quasinormally embedded in $G$.
A subgroup $H$ of a group $G$ is said to be complemented in $G$ if there exists a subgroup $K$ of $G$ such that $G=HK$ and $H\cap K=1$. In this paper we determine the structure of finite groups with some complemented primary subgroups, and obtain some new results about $p$-nilpotent groups.
A subgroup $H$ of a finite group $G$ is weakly-supplemented in $G$ if there exists a proper subgroup $K$ of $G$ such that $G=HK$. In the paper it is proved that a finite group $G$ is $p$-nilpotent provided $p$ is the smallest prime number dividing the order of $G$ and every minimal subgroup of $P\cap G'$ is weakly-supplemented in $N_{G}(P),$ where $P$ is a Sylow $p$-subgroup of $G$. As applications, some interesting results with weakly-supplemented minimal subgroups of $P\cap G'$ are obtained.