For an arbitrary partition $\sigma$ of the set $\mathbb{P}$ of all primes, a sufficient condition for the $\sigma$-subnormality of a subgroup in a finite group is given. It is proved that the Kegel—Wielandt $\sigma$-problem has a positive solution in the class of all finite groups all of whose nonabelian composition factors are alternating groups, sporadic groups, or Lie groups of rank 1.
Keywords: finite group, $\sigma$-subnormal subgroup, Kegel—Wielandt $\sigma$-problem, Hall subgroup, complete Hall set
Received July 20, 2023
Revised August 25, 2023
Accepted September 4, 2023
Funding Agency: The work was supported by Belarusian Republican Foundation for Fundamental Research and the Russian Science Foundation (project F23RNF-237).
Sergei Fedorovich Kamornikov, Dr. Phys.-Math. Sci., Prof., F. Skorina Gomel State University, Gomel, 246028 Republic of Belarus, e-mail: sfkamornikov@mail.ru
Valentin Nikolayevich Tyutyanov, Dr. Phys.-Math. Sci., Prof., Gomel Branch of International University “MITSO”, Gomel, 246029 Republic of Belarus, e-mail: vtutanov@gmail.com
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Cite this article as: S.F. Kamornikov, V.N. Tyutyanov. On the Kegel–Wielandt $\sigma$-problem. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 4, pp. 121–129; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 323, Suppl. 1, pp. S113–S120.
