Products of finite groups with conditions of formation subnormality on subgroups of factors

The connection between the solution of Shemetkov’s problem on Z-saturated formations and the study of classes of groups defined by systems of formation subnormal subgroups was established. A criterion for a superradical formation to be solvably saturated was found. The superradicality of hereditary S -formations was proved. All hereditary formations F of solvable groups that contain every group representable as a product of its subgroups, all H-subgroups of which are F-subnormal (or K-F-subnormal), where H is a saturated homomorph, were described.

1. Shemetkov L. A., Skiba A. N. Formations of Algebraic Systems. Moscow, Nauka Publ., 1989. 256 p. (in Russian).

2. Doerk K., Hawkes T. Finite Soluble Groups. Berlin, New York, Walter de Gruyter, 1992. 891 p.

3. Kamornikov S. F., Sel’kin V. M. Subgroup Functors and Classes of Finite Groups. Minsk, Belaruskaya navuka Publ., 2003. 254 p. (in Russian).

4. Ballester-Bolinches A., Ezquerro L. M. Classes of Finite Groups. Springer, Dordrecht, 2006. 385 p.

5. Shemetkov L. A. Formations of Finite Groups. Moscow, Nauka Publ., 1978. 272 p. (in Russian).

6. Murashka V. I. On Questions Posed by Shemetkov, Ballester-Bolinches, and Perez-Ramos in Finite Group Theory. Mathematical Notes, 2022, vol. 112, no. 6, pp. 932–939. https://doi.org/10.1134/s000143462211027x

7. Shemetkov L. A. Frattini extensions of finite groups and formations. Communications in Algebra, 1997, vol. 25, no. 3, pp. 955–964. https://doi.org/10.1080/00927879708825900

8. Ballester-Bolinches A., Perez-Ramos M. D. On a question of L. A. Shemetkov. Communications in Algebra, 1999, vol. 27, no. 11, pp. 5615–5618. https://doi.org/10.1080/00927879908826777

9. Aivazidis S., Safonova I. N., Skiba A. N. Subnormality and residuals for saturated formations: A generalization of Schenkman’s theorem. Journal of Group Theory, 2021, vol. 24, iss. 4, pp. 807–818. https://doi.org/10.1515/jgth-2020-0149

10. Vasil’ev A. F., Vasil’eva T. I. On Finite Groups with Generalized Subnormal Sylow Subgroups. Problemy fiziki, matematiki i tekhniki = Problems of Physics, Mathematics and Technics, 2011, no. 4, pp. 86–91 (in Russian).

11. Murashka V. I. Classes of finite groups with generalized subnormal cyclic primary subgroups. Siberian Mathematical Journal, 2014, vol. 55, pp. 1105–1115. https://doi.org/10.1134/s0037446614060135

12. Yi X., Kamornikov S. F. On hereditary superradical formations. Siberian Mathematical Journal, 2016, vol. 57, pp. 260–264. https://doi.org/10.1134/s0037446616020087

13. Ballester-Bolinches A., Kamornikov S. F., Tyutyanov V. N. On a problem of L. A. Shemetkov on superradical formations of finite groups. Journal of Algebra, 2014, vol. 403, pp. 69–76. https://doi.org/10.1016/j.jalgebra.2014.01.004

14. Vasil’ev A. F., Vasil’eva T. I., Vegera A. S. Finite groups with generalized subnormal embedding of Sylow subgroups. Siberian Mathematical Journal, 2016, vol. 57, pp. 200–212. https://doi.org/10.1134/s0037446616020038

15. Murashka V. I. Finite groups with given sets of F-subnormal subgroups. Asian-European Journal of Mathematics, 2020, vol. 13, no. 4, art. ID 2050073. https://doi.org/10.1142/s1793557120500734

16. Murashka V. I., Vasil’ev A. F. On the σ-nilpotent hypercenter of finite groups. Journal of Group Theory, 2022, vol. 25, no. 6, pp. 1083–1098. https://doi.org/10.1515/jgth-2021-0138

17. Murashka V. I. Regular saturated formations of finite soluble groups. Journal of Group Theory, 2025, vol. 28, no. 6, pp. 1307–1324. https://doi.org/10.1515/jgth-2024-0143

18. Mazurov V. D., Khukhro E. I. (eds.). Unsolved Problems in Group Theory. The Kourovka Notebook no. 21. Novosibirsk, 2026. Available at: https://arxiv.org/pdf/1401.0300 (accessed at 2 June 2026).

19. Semenchuk V. N., Vasil’ev A. F. Characterization of Local Formations F by Given Properties of Minimal Non- Fgroups. Issledovanie normal’nogo i podgruppovogo stroeniya konechnykh grupp: Trudy Gomel’skogo seminara [Investigation of Normal and Subgroup Structure of Finite Groups: Proceedings of the Gomel Seminar]. Minsk, 1984, pp. 175–181 (in Russian).

20. Vasil’ev A. F. On a Question of the Theory of Formations of Finite Groups. Vestsі Akademіі navuk Belarusi. Seryya fіzіka-matematychnykh navuk = Proceedings of the Academy of Sciences of Belarus. Physics and Mathematics series, 1994, no. 3, pp. 121–122 (in Russian).

21. Kamornikov S. F. On two problems by L. A. Shemetkov. Siberian Mathematical Journal, 1994, vol. 35, pp. 713–721. https://doi.org/10.1007/bf02106614

22. Semenchuk V. N. Solvable F-radical formations. Math Notes, 1996, vol. 59, no. 2, pp. 185–188. https://doi.org/10.1007/bf02310958

23. Murashka V. I. Arithmetic Graphs and Factorizable Finite Groups. Trudy Instituta Matematiki i Mekhaniki UrO RAN [Proceedings of Krasovskii Institute of Mathematics and Mechanics UB RAS], 2023, vol. 29, no. 4, pp. 181–192 (in Russian). https://doi.org/10.21538/0134-4889-2023-29-4-181-192

24. Guo W. Structure Theory for Canonical Classes of Finite Groups. Berlin, Heidelberg, Springer-Verlag, 2015. 258 p. https://doi.org/10.1007/978-3-662-45747-4

25. Vasil’ev A. F., Murashka V. I. Formations and Products of F(G)-Subnormal Subgroups of Finite Solvable Groups. Mathematical Notes, 2020, vol. 107, no. 3, pp. 413–424. https://doi.org/10.1134/s0001434620030050

26. Vasil’ev A. F., Murashka V. I. Arithmetic graphs and classes of finite groups. Siberian Mathematical Journal, 2019. vol. 60, pp. 41–55. https://doi.org/10.1134/s0037446619010051

27. Murashka V. I. On the lattice of hereditary Shemetkov formations of finite groups in a class X. Mathematical Notes, 2025, vol. 118, no. 6, pp. 1272–1280. https://doi.org/10.1134/s0001434625605787

28. Skiba A. N. On σ-subnormal and σ-permutable subgroups of finite groups. Journal of Algebra, 2015, vol. 436, pp. 1–16. https://doi.org/10.1016/j.jalgebra.2015.04.010

29. Skiba A. N. On some results in the theory of finite partially soluble groups. Communications in Mathematics and Statistics, 2016, vol. 4, no. 3, pp. 281–309. https://doi.org/10.1007/s40304-016-0088-z