S.F. Kamornikov, V.N. Tyutyanov. On the Kegel–Wielandt $\sigma$-problem ... P. 121-129

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

REFERENCES

1.   Kegel O.H. Sylow-Gruppen und Subnormalteiler endlicher Gruppen. Math. Z., 1962, vol. 78, pp. 205–221. doi: 10.1007/BF01195169

2.   Wielandt H. Zusammengesetzte Gruppen: Hölders Programm heute. In: The Santa Cruz conf. on finite groups (Santa Cruz, 1979), Providence RI: Amer. Math. Soc., 1980, Ser. Proc. Sympos. Pure Math., vol. 37, pp. 161–173. doi: 10.1090/pspum/037

3.   Kleidman P.B. A proof of the Kegel–Wielandt conjecture on subnormal subgroups. Ann. Math. (2), 1991, vol. 133, no. 2, pp. 369–428. doi: 10.2307/294

4.   Guralnick R., Kleidman P.B., Lyons R. Sylow p-subgroups and subnormal subgroups of finite groups. Proc. London Math. Soc. (3), 1993, vol. 66, no. 1, pp. 129–151. doi: 10.1112/plms/s3-66.1.129

5.   The Kourovka notebook. Unsolved problems in group theory, 20th ed., eds. V.D. Mazurov, E.I. Khukhro, Novosibirsk: Inst. Math. SO RAN Publ., 2022, 269 p. Available at: https://kourovka-notebook.org/ .

6.    Skiba A.N. On σ-subnormal and $\sigma$-permutable subgroups of finite groups. J. Algebra, 2015, vol. 436, pp. 1–16. doi: 10.1016/j.jalgebra.2015.04.010

7.   Kamornikov S.F., Tyutyanov V.N. On $\sigma$-subnormal subgroups of finite groups. Siberian Math. J., 2020, vol. 61, no. 2, pp. 266–270. doi: 10.1134/S0037446620020093

8.    Ballester-Bolinches A., Kamornikov S.F., Tyutyanov V.N. On the Kegel-Wielandt $\sigma$-problem for binary partitions. Annali di Matematica Pura ed Applicata, 2022, vol. 201, no. 1, pp. 443–451. doi: 10.1007/s10231-021-01123-4

9.   Kamornikov S.F., Tyutyanov V.N. On some aspects of the Kegel-Wielandt $\sigma$-problem. Russian Math. (Iz. VUZ), 2022, vol. 66, no. 2, pp. 15–23. doi: 10.3103/S1066369X22020050

10.   Kamornikov S.F., Tyutyanov V.N. On $\sigma$-subnormal subgroups of finite 3′-groups. Ukr. Math. J., 2020, vol. 72, no. 6, pp. 935–941. doi: 10.1007/s11253-020-01833-7

11.   Kamornikov S.F., Tyutyanov V.N. On the Kegel–Wielandt $\sigma$-problem. Math. Notes, 2021, vol. 109, no. 4, pp. 580–584. doi: 10.1134/S0001434621030263

12.   Doerk K., Hawkes T. Finite soluble groups. Berlin; NY: Walter de Gruyter, 1992, 891 p. doi: 10.1515/9783110870138

13.   Huppert B. Endliche Gruppen I. Berlin: Springer-Verlag, 1967, 796 p. doi: 10.1007/978-3-642-64981-3

14.   Bray J.N., Holt D.F., Roney-Dougal C.M. The maximal subgroups of the low-dimensional finite classical groups, Cambridge: Cambridge Univ. Press, 2013, London Math. Soc. Lect. Note Ser., vol. 407, 438 p. doi: 10.1017/CBO9781139192576

15.   Vdovin E.P., Revin D.O. Theorems of Sylow type. Russian Math. Surveys, 2011, vol. 66, no. 5, pp. 829–870. doi: 10.1070/RM2011v066n05ABEH004762

16.   Kazarin L.S. On the product of finite groups. Dokl. Akad. Nauk SSSR, 1983, vol. 269, no. 3, pp. 528–531.

17.   Conway J.N., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A. Atlas of finite groups. Oxford: Oxford Univ. Press, 1985, 252 p. ISBN: 0-19-853199-0 .

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.