S.F. Kamornikov, V.N. Tyutyanov. On two problems from “The Kourovka Notebook” ... P. 98-102

We solve Problems 19.87 and 19.88 formulated by A.N. Skiba in "The Kourovka Notebook". It is proved that if, for every Sylow subgroup $P$ of a finite group $G$ and every maximal subgroup $V$ of $P$, there is a $\sigma$-soluble ($\sigma$-nilpotent) subgroup $T$ such that $VT=G$, then $G$ is $\sigma$-soluble ($\sigma$-nilpotent, respectively).

Keywords: finite group, $\sigma$-soluble group, $\sigma$-nilpotent group, partition of the set of all prime numbers, Sylow subgroup, maximal subgroup

Received January 17, 2021

Revised February 10, 2021

Accepted February 18, 2021

Funding Agency: This work was supported by the Russian Foundation for Basic Research and the Belarusian Republican Foundation for Fundamental Research (project no. F20R-291).

Sergei Fedorovich Kamornikov, Dr. Phys.-Math. Sci., Prof., Francisk Skorina Gomel State University, 246019, Gomel, 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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2.   Skiba A.N. On σ-subnormal and σ-permutable subgroups of finite groups . J. Algebra, 2015, vol. 436, pp. 1–16. doi: 10.1016/j.jalgebra.2015.04.010 

3.   Guralnick R.M. Subgroups of prime power index in a simple group. J. Algebra, 1983, vol. 81, no. 2, pp. 304–311. doi: 10.1016/0021-8693(83)90190-4 

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Cite this article as: S.F. Kamornikov, V.N. Tyutyanov. On two problems from “The Kourovka Notebook”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 1, pp. 98–102.