J. Guo, W. Guo, D.O. Revin, V.N. Tyutyanov. On the Baer–Suzuki width of some radical classes ... P. 96-105

Let $\sigma=\{\sigma_i\mid i\in I\}$ be a fixed partition of the set of all primes into pairwise disjoint nonempty subsets $\sigma_i$. A finite group is called $\sigma$-nilpotent if it has a normal $\sigma_i$-Hall subgroup for any $i\in I$. Any finite group possesses a $\sigma$-nilpotent radical, which is the largest normal $\sigma$-nilpotent subgroup. In this note, it is proved that there exists an integer $m=m(\sigma)$ such that the $\sigma$-nilpotent radical of any finite group coincides with the set of elements $x$ such that any $m$ conjugates of $x$ generate a $\sigma$-nilpotent subgroup. Other possible analogs of the classical Baer-Suzuki theorem are discussed.

Keywords: Baer-Suzuki width, $\sigma$-nilpotent group, $\sigma$-solvable group, complete class of groups

Received April 10, 2022

Revised April 20, 2022

Accepted April 25, 2022

Funding Agency: J. Guo and W. Guo were supported by the National Natural Science Foundation of China (project nos. 11961017 and 12171126). D.O.Revin and V.N.Tyutyanov were supported by the joint grant of the Russian Foundation for Basic Research (project no. 20-51-00007) and the Belarusian Republican Foundation for Fundamental Research (project no. F20R-291). D. O. Revin was also supported by the Program for Fundamental Research of the Russian Academy of Sciences (project no. FWNF-2022-0002).

Jin Guo, PhD, Prof. (assoc.), School of Science, Hainan University, Haikou, Hainan, 570228, P.R. China, e-mail: guojinecho@163.com

Wenbin Guo, PhD, Dr. Phys.-Math. Sci., Prof., School of Science, Hainan University, Haikou, Hainan, 570228, P.R. China, Department of Mathematics, University of Science and Technology of China, Hefei 230026, P. R. China, e-mail: wbguo@ustc.edu.cn

Danila Olegovich Revin, Dr. Phys.-Math. Sci., Prof., Sobolev Institute of Mathematics of the Siberia Branch of the Russian Academy of Sciences, Novosibirsk, 630090 Russia; Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: revin@math.nsc.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: J. Guo, W. Guo, D.O. Revin, V.N. Tyutyanov. On the Baer–Suzuki width of some radical classes. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 2, pp. 96–105; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2022, Vol. 317, Suppl. 1, pp. S90–S97.