Subgroups of the Fan of Sylow Subgroups and the Supersolvability of a Finite Group

The notion of the fan of a subgroup of a group, which was introduced in 1979 by Z. I. Borevich, is used to prove the supersolvability of finite groups. It is proved that a finite group $$G$$ is supersolvable if and only if any basic subgroup of the fan of every Sylow subgroup either coincides with $$G$$ or can be connected with $$G$$ by a chain of subgroups with prime indices. We also prove the supersolvability of a finite group with supersolvable basic subgroups of the fan of every Sylow subgroup of the group.