Приводимые τ-замкнутые σ-локальные формации конечных групп с заданной структурой подформаций
EDN: OLYGHB
Аннотация
Пусть $\mathfrak{F}$ и $\mathfrak{H}$ – некоторые $\tau$-замкнутые $\sigma$-локальные формации конечных групп. Через $\mathfrak{F}/^\tau_\sigma\mathfrak{H}\cap\mathfrak{F}$ обозначают решетку всех $\tau$-замкнутых $\sigma$-локальных формаций $\mathfrak{X}$ таких, что $\mathfrak{H}\cap\mathfrak{F}\subseteq \mathfrak{X}\subseteq \mathfrak{F}.$ Длину решетки $\mathfrak{F}/^\tau_\sigma\mathfrak{H}\cap\mathfrak{F}$ называют {\it $\mathfrak H^\tau_\sigma$-дефектом}, а при $\mathfrak{H}=(1)$ – формация всех единичных групп, {\it $l^\tau_\sigma$-длиной} формации $\mathfrak{F}$. Изучены общие свойства $\mathfrak H^\tau_\sigma$-дефекта $\tau$-замкнутых $\sigma$-локальных формаций, получено описание структурного строения приводимых $\tau$-замкнутых $\sigma$-локальных формаций, имеющих $\mathfrak H^\tau_\sigma$-дефект $\leq 2$ и $l^\tau_\sigma$-длину $\leq 3$.
Ключевые слова
конечная группа,
подгрупповой функтор,
$\tau$-замкнутая $\sigma$-локальная формация,
критическая $\tau$-замкнутая $\sigma$-локальная формация,
$\mathfrak H^\tau_\sigma$-дефект формации,
$l^\tau_\sigma$-длина формации
При поддержке: The research was carried out within the framework of the State Scientific Research Program “Convergence–2025” with the financial support of the Ministry of Education of the Republic of Belarus (project 20211328)
Об авторах
В. В. Скрундь
Белорусский государственный университет
Беларусь
Беларусь
Минск
И. Н. Сафонова
Белорусский государственный университет
Беларусь
Беларусь
Минск
Список литературы
1. Shemetkov L. A., Skiba A. N. Formations of Algebraic Systems. Moscow, Nauka, 1989. 255 p. (in Russian).
2. Skiba A. N. Algebra of formations. Minsk, Belaruskaya Navuka, 1997. 240 p. (in Russian).
3. 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
4. Skiba A. N. On one generalization of the local formations. Problems of Physics, Mathematics and Technics, 2018, no. 34(1), pp. 79–82.
5. Skiba A. N. On local formations of length 5. Arithmetic and subgroup structure of finite groups: Proceedings of the Gomel Seminar. Minsk, Science and Technics, 1986, pp. 135–149 (in Russian).
6. Skiba A. N.,Targonskii E. A. Classification of local formations of finite groups with nilpotent defect 2. Mathematical Notes, 1987, vol. 41, no. 4, pp. 490–499 (in Russian).
7. Chi Z., Safonov V. G., Skiba A. N. On one application of the theory of n-multiply σ-local formations of finite groups. Problems of Physics, Mathematics and Technics, 2018, no. 2(35), pp. 85–88.
8. Zhang Ch., Skiba A. N. On Σσ-closed classes of finite groups. Ukrainian Mathematical Journal, 2019, vol. 70, no. 2, pp. 1966–1977. https://doi.org/10.1007/s11253-019-01619-6
9. 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
10. Tsarev A. Laws of the lattices of σ-local formations of finite groups. Mediterranean Journal of Mathematics, 2020, vol. 17, no. 3, art. 75. https://doi.org/10.1007/s00009-020-01510-w
11. Safonova I. N., Safonov V. G. On some properties of the lattice of totally σ-local formations of finite groups. Journal of the Belarusian State University. Mathematics and Informatics, 2020, no. 3, pp. 1–14. https://doi.org/10.33581/2520-6508-2020-3-6-16
12. Safonova I. N. On minimal σ-local non-H-formations of finite groups. Problems of Physics, Mathematics and Technics, 2020, no. 4(45), pp. 105–112 (in Russian).
13. Vorob’ev N. N., Stasel’ko I. I., Khodzhagulyev A. O. Separable lattices of multiply σlocal formations. Siberian Mathematical Journal, 2021, vol. 62, no. 4, pp. 586–597. https://doi.org/10.1134/S0037446621040029
14. Safonova I. N. Some properties of n-multiply σ-local formations of finite groups. AsianEuropean Journal of Mathematics, 2022, vol. 15, no. 7, art. 2250138 (12 p.). https://doi.org/10.1142/S1793557122501388
15. Safonova I. N. A criterion for σ-locality of a nonempty formation. Communications in Algebra, 2022, vol. 50, no. 6, pp. 2366–2376. https://doi.org/10.1080/00927872.2021.2006210
16. Safonova I. N. On properties of the lattice of all τ-closed n-multiply σlocal formations. Communications in Algebra, 2023, vol. 51, no. 10, pp. 4454–4461. https://doi.org/10.1080/00927872.2023.2210678
17. Safonova I. N. On critical σ-local formations of finite groups. Trudy Instituta matematiki Natsional’noi akademii nauk Belarusi = Proceedings of the Institute of Mathematics of NAS of Belarus, 2023, vol. 31, no. 2, pp. 63–80 (in Russian).
18. Safonova I. N. On σ-inductive lattices of n-multiply σ-local formations of finite groups. Journal of Algebra and Its Applications, 2024, vol. 23, no. 1, art. 2450017 (13 p.). https://doi.org/10.1142/S0219498824500178
19. 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. https://doi.org/10.32037/agta-2024-005
20. Safonova I. N. On separability of the lattice of τ-closed n-multiply σ-local formations. Communications in Algebra, 2024, vol. 52, no. 2, pp. 3309–3318. https://doi.org/10.1080/00927872.2024.2317458
21. Safonova I. N. On n-multiply σ-locality of a nonempty τ-closed formation of finite groups. Trudy Instituta matematiki Natsional’noi akademii nauk Belarusi = Proceedings of the Institute of Mathematics of NAS of Belarus, 2024, vol. 32, no. 1, pp. 32–38 (in Russian).
22. Safonova I. N., Skrundz V. V. On σ-local formations of finite groups with bounded Hσ-defect. Problems of Physics, Mathematics and Technics, 2025, no. 1(62), pp. 87–101 (in Russian).
23. Safonova I. N., Skrundz V. V. Minimal τ-closed σ-local non-H-formation of finite groups. Doklady Natsional’noi akademii nauk Belarusi = Doklady of the National Academy of Sciences of Belarus, 2025, vol. 69, no. 5, pp. 359–366. https://doi.org/10.29235/1561-8323-2025-69-5-359-366 (in Russian).
24. Shemetkov L. A. Screens of stepped formations. Proceedings of the VI All-Union Symposium on Group Theory. Kyiv, Naukova Dumka, 1980, pp. 37–50 (in Russian).
25. Skiba A. N. On critical formations. Izv. AN BSSR. Ser. phys.-math. sci., 1980, no. 4, pp. 27–33 (in Russian).
26. Birkhoff G. Lattice theory. Moscow, Nauka, 1984. 568 p. (in Russian).
27. Safonov V. G. On multiply local formations of finite groups with bounded nilpotent defect. Voprosy algebry, 1996, no. 9, pp. 161–175 (in Russian).
Рецензия
Для цитирования:
Скрундь В.В., Сафонова И.Н. Приводимые τ-замкнутые σ-локальные формации конечных групп с заданной структурой подформаций. Труды Института математики НАН Беларуси. 2025;33(2):36-53. EDN: OLYGHB
For citation:
Skrundz V.V., Safonova I.N. Reducible τ-closed σ-local formations of finite groups with a given structure of subformations. Proceedings of the Institute of Mathematics of the NAS of Belarus. 2025;33(2):36-53. EDN: OLYGHB
Просмотров:
14
Послать статью по эл. почте 