Abstract:
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.
Keywords:finite group, $\mathbb{P}$-subnormal subgroup, Sylow subgroup, fan, basic subgroup of a fan, contranormalizer of a subgroup, supersolvable group.
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