Minimal τ-closed σ-local non-Hformation of finite groups
https://doi.org/10.29235/1561-8323-2025-69-5-359-366
Abstract
A description of minimal τ-closed σ-local non-H-formations is obtained for an arbitrary σ-local formation H of classical type, i. e., a σ-local formation that has such a σ-local definition, all non-abelian values of which are σ-local formations. This result provides a solution to L. A. Shemetkov’s problem (1980) on describing critical formations in the class of τ-closed σ-local formations, where σ is some partition of the set of all prime numbers, τ is an arbitrary subgroup functor.
(Communicated by Academician Nikolay A. Izobov)
Keywords
finite group,
formation σ-function,
σ-local formation,
subgroup functor,
τ-closed formation,
critical σ-local formation,
formation of classical type
Supporting agencies: Research was supported by the Ministry of Education of the Republic of Belarus (project no. 20211328)
About the Authors
I. N. Safonova
Belarusian State University
Belarus
Belarus
Safonova Inna N. – Ph. D. (Physics and Mathematics), Associate Professor.
4, Nezavisimosti Ave., 220030, Minsk
V. V. Skrundz
Belarusian State University
Belarus
Belarus
Skrundz Valentina V. – Postgraduate Student.
4, Nezavisimosti Ave., 220030, Minsk
References
1. Shemetkov L. A. Screens of step formations. Trudy VI Vsesoyuznogo simpoziuma po teorii grupp [Proceedings of the VI All-Union Symposium on Group Theory]. Kyiv, 1980, рр. 37–50 (in Russian).
2. Skiba A. N. On critical formations. Izvestiya AN BSSR, 1980, no. 4, рр. 27–33 (in Russian).
3. Skiba A. N. On critical formations. Beskonechnye gruppy i primykayushchie algebraicheskie struktury [Infinite Groups and Adjoining Algebraic Structures]. Kyiv, 1993, рр. 258–268 (in Russian).
4. Shemetkov L. A., Skiba A. N. Formations of Algebraic Systems. Moscow, 1989. 257 р. (in Russian).
5. Skiba A. N. Algebra of Formations. Minsk, 1997. 239 p. (in Russian).
6. Safonova I. N. On minimal σ-local non-H-formations of finite groups. Problemy fiziki, matematiki i tekhniki = Problems of Physics, Mathematics and Technology, 2020, no. 4 (45), pp. 105–112 (in Russian).
7. Safonova I. N. On critical σ-local formations of finite groups. Trudy Instituta Matematiki = Proceedings of the Institute of Mathematics, 2023, vol. 31, no. 2, pp. 63–80 (in Russian).
8. Safonova I. N., Skrund V. V. On the theory of Hστ -critical formations of finite groups. Mezhdunarodnaya konferent siya “Algebra i dinamicheskie sistemy”, posvyashchennaya 90-letiyu so dnya rozhdeniya V. A. Belonogova, Nal’chik, 3–7 iyunya 2025 g. [International Conference “Algebra and Dynamical Systems” dedicated to the 90th anniversary of V. A. Belonogov, Nalchik, 3–7 June 2025]. Nalchik, 2025, pp. 137–140 (in Russian).
9. Safonova I. N., Skrundz V. V. Оn Hστ -critical formations of finite groups. Problemy fiziki, matematiki i tekhniki = Problems of Physics, Mathematics and Technology, 2025, no. 3 (64), рр. 99–111 (in Russian).
10. 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
11. Skiba A. N. On one generalization of the local formations. Problemy fiziki, matematiki i tekhniki = Problems of Physics, Mathematics and Technology, 2018, no. 1 (34), pp. 79–82 (in Russian).
12. Chi Z., Safonov V. G., Skiba A. N. On n-multiply σ-local formations of finite groups. Communications in Algebra, 2019, vol. 47, no. 3, pp. 957–968. https://doi.org/10.1080/00927872.2018.1498875
13. Safonova I. N. Some properties of n-multiply σ-local formations of finite groups. Asian-European Journal of Mathematics, 2022, vol. 15, no. 7, art. 2250138 (12 p). https://doi.org/10.1142/s1793557122501388
14. Safonova I. N. On the τ-closedness of n-multiply σ-local formation. Advances in Group Theory and Applications, 2024, vol. 18, pp. 123–136.
15. Safonova I. N. Оn n-multiple σ-locality of a non-empty τ-closed formation of finite groups. Trudy Instituta Matematiki NAN Belarusi = Proceedings of the Institute of Mathematics of the NAS of Belarus, 2024, vol. 32, no. 1, pp. 32–38 (in Russian).
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