This paper proves: Let $\mathcal {F}$ be a saturated formation containing $\mathcal {U}$ . Suppose that $G$ is a group with a normal subgroup $H$ such that $G/H \in \mathcal {F}$ . (1) If all maximal subgroups of any Sylow subgroup of $F^*(H)$ are $c$ -supple- mented in $G$ , then $G \in \mathcal {F}$ ; (2) If all minimal subgroups and all cyclic subgroups with order 4 of $F^*(H)$ are $c$ -supplemented in $G$ , then $G \in \mathcal {F}$ .

Suppose G is a finite group and H is subgroup of G. H is said to be s-permutably embedded in G if for each prime p dividing |H|, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-permutable subgroup of G; H is called weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup H _se of G contained in H such that G = HT and H ∩ T ≤ H _se . We investigate the influence of weakly s-permutably embedded subgroups on the structure of finite groups. Some recent results are generalized.

Let be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are presented: (1) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every maximal subgroup of all Sylow subgroups of H is either c-normal or S-quasinormally embedded in G. (2) G ∈ if and only if there is a normal subgroup H such that G/H ∈ 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 ∈ if and only if there is a normal subgroup H such that G/H ∈ and every cyclic subgroup of F*(H) of prime order or order 4 is either c-normal or S-quasinormally embedded in G.