On $c$-supplemented maximal and minimal subgroups of Sylow subgroups of finite groups

This paper proves: Let ${\cal F}$ be a saturated formation containing ${\cal U}$. Suppose that $G$ is a group with a normal subgroup $H$ such that $G/H \in {\cal F}$.

(1) If all maximal subgroups of any Sylow subgroup of $F^*(H)$ are $c$-supple- mented in $G$, then $G \in {\cal F}$;

(2) If all minimal subgroups and all cyclic subgroups with order 4 of $F^*(H)$ are $c$-supplemented in $G$, then $G \in {\cal F}$.