Geodesic
Subgroups of the Fan of Sylow Subgroups and the Supersolvability of a Finite Group
Matematičeskie zametki, Tome 110 (2021) no. 2, pp. 192-203

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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, basic subgroup of a fan, contranormalizer of a subgroup
Mots-clés : fan, supersolvable group. 
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     author = {T. I. Vasilyeva},
     title = {Subgroups of the {Fan} of {Sylow} {Subgroups} and the {Supersolvability} of a {Finite} {Group}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {192--203},
     publisher = {mathdoc},
     volume = {110},
     number = {2},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2021_110_2_a2/}
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T. I. Vasilyeva. Subgroups of the Fan of Sylow Subgroups and the Supersolvability of a Finite Group. Matematičeskie zametki, Tome 110 (2021) no. 2, pp. 192-203. http://geodesic.mathdoc.fr/item/MZM_2021_110_2_a2/