vestifmИзвестия Национальной академии наук Беларуси. Серия физико-математических наукProceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series1561-24302524-2415The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-2430-2024-60-4-271-279vestifm-807Research ArticleМАТЕМАТИКАMATHEMATICSК вопросам Шеметкова и Аграваля об обобщениях гиперцентра конечных группTo the questions of Shemetkov and Agrawal about the generalizations of the hypercenter of finite groupshttps://orcid.org/0000-0002-0769-2519МурашкоВ. И.MurashkaV. I.

Мурашко Вячеслав Игоревич – кандидат физико- математических наук, доцент, доцент кафедры алгеб ры и геометрии, ведущий научный сотрудник НИС

ул. Советская, 104, 246028, Гом ел

Viachaslau I. Murashka – Ph. D. (Physics and Ma thematics), Associate Professor, Associate Professor of the Department of Algebra and Geometry, Leading Researcher of Research Sector

104, Sovetskaya Str., 246028, Gomel

mvimath@yandex.ruГомельский государственный университет имени Франциска СкориныFrancisk Skorina Gomel State University202408012025604271279Copyright © Мурашко В.И., 20252025Мурашко В.И.Murashka V.I.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestifm.belnauka.by/jour/article/view/807

Формация F называется формацией Бэра – Шеметкова в классе групп X, если в любой конечной X-группе пересечение F-максимальных подгрупп совпадает с F-гиперцентром. Доказано, что для непустой наследственной насыщенной формации F существует наибольшая по включению наследственная насыщенная формация BSF, в которой F является формацией Бэра – Шеметкова. Установлена связь данного результата с решениями вопросов Аграваля (1976) и Шеметкова (1995). Для класса U всех сверхразрешимых групп описан класс BSU и приведен алгоритм распознавания принадлежности группы данному классу.

A formation F is called a Baer – Shemetkov formation in a class X of groups if in any finite X-group the intersection of all F-maximal subgroups coincides with the F-hypercenter. It is proved that for a non-empty hereditary saturated formation F there exists the greatest by inclusion hereditary saturated formation BSF such that F is a Baer – Shemetkov formation in BSF. The connection of this result with the solution of Agrawal’s (1976) and Shemetkov’s (1995) questions is discussed. For the class U of all supersolvable groups the class BSU is described and the algorithm for its recognition is presented.

конечная группасверхразрешимая группаF-гиперцентробобщенный гиперцентрнаследственная насыщенная формацияформация Бэра – Шеметковаfinite groupsupersolvable groupF-hypercentergeneralized hypercenterhereditary saturated formationBaer – Shemetkov formationРабота выполнена при финансовой поддержке Белорусского республиканского фонда фундаментальных исследований (проект № Ф23PHФM-63).The work was financially supported by the Belarusian Republican Foundation for Fundamental Research (project no. Ф23PHФM-63).ReferencesBaer R. Supersolvable Immersion. Canadian Journal of Mathematics, 1959, vol. 11, pp. 353–369. https://doi.org/10.4153/cjm-1959-036-2Baer R. Supersolvable Immersion. Canadian Journal of Mathematics, 1959, vol. 11, pp. 353–369. https://doi.org/10.4153/cjm-1959-036-2Huppert B. Zur Theorie der Formationen. Archiv der Mathematik, 1969, vol. 19, no. 6, pp. 561–574. https://doi.org/10.1007/BF01899382Huppert B. Zur Theorie der Formationen. Archiv der Mathematik, 1969, vol. 19, no. 6, pp. 561–574. https://doi.org/10.1007/BF01899382Šemetkov L. A. Graduated formations of groups. Mathematics of the USSR-Sbornik, 1974, vol. 23, no. 4, pp. 593–611. https://doi.org/10.1070/sm1974v023n04abeh002184Šemetkov L. A. Graduated formations of groups. Mathematics of the USSR-Sbornik, 1974, vol. 23, no. 4, pp. 593–611. https://doi.org/10.1070/sm1974v023n04abeh002184Shemetkov L. A., Skiba A. N. Formations of Algebraic Systems. Moscow, Nauka Publ., 1989. 256 p. (in Russian).Shemetkov L. A., Skiba A. N. Formations of Algebraic Systems. Moscow, Nauka Publ., 1989. 256 p. (in Russian).Hall P. On the System Normalizers of a Solvable Group. Proceedings of the London Mathematical Society, 1938, vol. 43, no. 1, pp. 507–528. https://doi.org/10.1112/plms/s2-43.6.507Hall P. On the System Normalizers of a Solvable Group. Proceedings of the London Mathematical Society, 1938, vol. 43, no. 1, pp. 507–528. https://doi.org/10.1112/plms/s2-43.6.507Baer R. Group elements of prime power index. Transactions of the American Mathematical Society, 1953, vol. 75, pp. 20–47. https://doi.org/10.1090/s0002-9947-1953-0055340-0Baer R. Group elements of prime power index. Transactions of the American Mathematical Society, 1953, vol. 75, pp. 20–47. https://doi.org/10.1090/s0002-9947-1953-0055340-0Skiba A. N. On the F-hypercentre and the intersection of all F-maximal subgroups of a finite group. Journal of Pure and Applied Algebra, 2012, vol. 216, no. 4, pp. 789–799. https://doi.org/10.1016/j.jpaa.2011.10.006Skiba A. N. On the F-hypercentre and the intersection of all F-maximal subgroups of a finite group. Journal of Pure and Applied Algebra, 2012, vol. 216, no. 4, pp. 789–799. https://doi.org/10.1016/j.jpaa.2011.10.006Murashka V. I. On the F-hypercenter and the intersection of F-maximal subgroups of a finite group. Journal of Group Theory, 2018, vol. 21, no. 3, pp. 463–473. https://doi.org/10.1515/jgth-2017-0043Murashka V. I. On the F-hypercenter and the intersection of F-maximal subgroups of a finite group. Journal of Group Theory, 2018, vol. 21, no. 3, pp. 463–473. https://doi.org/10.1515/jgth-2017-0043Murashka V. I. On Shemetkov’s Question about the F-Hypercenter. Mathematical Notes, 2024, vol. 115, no. 5, pp. 779–788. https://doi.org/10.1134/s0001434624050134Murashka V. I. On Shemetkov’s Question about the F-Hypercenter. Mathematical Notes, 2024, vol. 115, no. 5, pp. 779–788. https://doi.org/10.1134/s0001434624050134Beidleman J. C., Heineken H. A note of intersection of maximal F-subgroups. Journal of Algebra, 2010, vol. 333, no. 1, pp. 120–127. https://doi.org/10.1016/j.jalgebra.2010.10.017Beidleman J. C., Heineken H. A note of intersection of maximal F-subgroups. Journal of Algebra, 2010, vol. 333, no. 1, pp. 120–127. https://doi.org/10.1016/j.jalgebra.2010.10.017Mazurov V., Khukhro E. I. (eds.). Unsolved Problems in Group Theory. The Kourovka Notebook, no. 20. Novosibirsk, 2022. Available at: https://arxiv.org/pdf/1401.0300 (accessed at 16 August 2024).Mazurov V., Khukhro E. I. (eds.). Unsolved Problems in Group Theory. The Kourovka Notebook, no. 20. Novosibirsk, 2022. Available at: https://arxiv.org/pdf/1401.0300 (accessed at 16 August 2024).Vasil’ev A. F., Murashka V. I. Arithmetic graphs and classes of finite groups. Siberian Mathematical Journal, 2019, vol. 60, no. 1, pp. 41–55. https://doi.org/10.1134/s0037446619010051Vasil’ev A. F., Murashka V. I. Arithmetic graphs and classes of finite groups. Siberian Mathematical Journal, 2019, vol. 60, no. 1, pp. 41–55. https://doi.org/10.1134/s0037446619010051Agrawal R. K. Generalized Center and Hypercenter of a Finite Group. Proceedings of the American Mathematical Society, 1976, vol. 58, no. 1, pp. 13–21. https://doi.org/10.1090/S0002-9939-1976-0409651-8Agrawal R. K. Generalized Center and Hypercenter of a Finite Group. Proceedings of the American Mathematical Society, 1976, vol. 58, no. 1, pp. 13–21. https://doi.org/10.1090/S0002-9939-1976-0409651-8Weinstein M. (ed.). Between Nilpotent and Solvable. Passaic, NJ, Polygonal Publishing House, 1982. 231 p.Weinstein M. (ed.). Between Nilpotent and Solvable. Passaic, NJ, Polygonal Publishing House, 1982. 231 p.Monakhov V. S. Finite groups with a given set of Schmidt subgroups. Mathematical Notes, 1995, vol. 58, no. 5, pp. 1183–1186. https://doi.org/10.1007/bf02305002Monakhov V. S. Finite groups with a given set of Schmidt subgroups. Mathematical Notes, 1995, vol. 58, no. 5, pp. 1183–1186. https://doi.org/10.1007/bf02305002Vasil’ev A. F., Vasil’eva T. I., Tyutyanov V. N., On the finite groups of supersolvable type. Siberian Mathematical Journal, 2010, vol. 51, no. 6, pp. 1004–1012. https://doi.org/10.1007/s11202-010-0099-zVasil’ev A. F., Vasil’eva T. I., Tyutyanov V. N., On the finite groups of supersolvable type. Siberian Mathematical Journal, 2010, vol. 51, no. 6, pp. 1004–1012. https://doi.org/10.1007/s11202-010-0099-zMonakhov V. S., Kniahina V. N. Finite groups with ℙ-subnormal subgroups. Ricerche di Matematica, 2013, vol. 62, pp. 307–322. https://doi.org/10.1007/s11587-013-0153-9Monakhov V. S., Kniahina V. N. Finite groups with ℙ-subnormal subgroups. Ricerche di Matematica, 2013, vol. 62, pp. 307–322. https://doi.org/10.1007/s11587-013-0153-9Seress Á. Permutation Group Algorithms. Cambridge University Press, 2003. 264 p. https://doi.org/10.1017/CBO9780511546549Seress Á. Permutation Group Algorithms. Cambridge University Press, 2003. 264 p. https://doi.org/10.1017/CBO9780511546549Höfling B. Computing projectors, injectors, residuals and radicals of finite solvable groups. Journal of Symbolic Computation, 2001, vol. 32, no. 5, pp. 499–511. https://doi.org/10.1006/jsco.2001.0477Höfling B. Computing projectors, injectors, residuals and radicals of finite solvable groups. Journal of Symbolic Computation, 2001, vol. 32, no. 5, pp. 499–511. https://doi.org/10.1006/jsco.2001.0477Eick B., Wright C. R. Computing subgroups by exhibition in finite solvable groups. Journal of Symbolic Computation, 2002, vol. 33, no. 2, pp. 129–143. https://doi.org/10.1006/jsco.2000.0503Eick B., Wright C. R. Computing subgroups by exhibition in finite solvable groups. Journal of Symbolic Computation, 2002, vol. 33, no. 2, pp. 129–143. https://doi.org/10.1006/jsco.2000.0503Murashka V. I. Formations of finite groups in polynomial time: F-residuals and F-subnormality. Journal of Symbolic Computation, 2024, vol. 122, art. ID 102271. https://doi.org/10.1016/j.jsc.2023.102271Murashka V. I. Formations of finite groups in polynomial time: F-residuals and F-subnormality. Journal of Symbolic Computation, 2024, vol. 122, art. ID 102271. https://doi.org/10.1016/j.jsc.2023.102271GAP – Groups, Algorithms, Programming – a System for Computational Discrete Algebra. GAP. Available at: https:// www.gap-system.org (accessed at 16 August 2024).GAP – Groups, Algorithms, Programming – a System for Computational Discrete Algebra. GAP. Available at: https:// www.gap-system.org (accessed at 16 August 2024).Doerk K., Hawkes T. O. Finite Solvable Groups. De Gruyter Expositions in Mathematics, vol. 4. Berlin, New York, De Gruyter, 1992. 891 p. https://doi.org/10.1515/9783110870138Doerk K., Hawkes T. O. Finite Solvable Groups. De Gruyter Expositions in Mathematics, vol. 4. Berlin, New York, De Gruyter, 1992. 891 p. https://doi.org/10.1515/9783110870138Eick B., Wright C. R. GAP package. FORMAT 1.4.4 Computing with formations of finite solvable groups. GAP. 2024. Available at: https://www.gap-system.org/Packages/format.html (accessed at 16 August 2024).Eick B., Wright C. R. GAP package. FORMAT 1.4.4 Computing with formations of finite solvable groups. GAP. 2024. Available at: https://www.gap-system.org/Packages/format.html (accessed at 16 August 2024).Guo W. Structure Theory for Canonical Classes of Finite Groups. Berlin, Heidelberg, Springer-Verlag, 2015. 359 p. https://doi.org/10.1007/978-3-662-45747-4Guo W. Structure Theory for Canonical Classes of Finite Groups. Berlin, Heidelberg, Springer-Verlag, 2015. 359 p. https://doi.org/10.1007/978-3-662-45747-4Kantor W. M. Sylow’s theorem in polynomial time. Journal of Computer and System Sciences, 1985, vol. 30, no. 3, pp. 359–394. https://doi.org/10.1016/0022-0000(85)90052-2Kantor W. M. Sylow’s theorem in polynomial time. Journal of Computer and System Sciences, 1985, vol. 30, no. 3, pp. 359–394. https://doi.org/10.1016/0022-0000(85)90052-2Babai L. On the length of subgroup chains in the symmetric group. Communications in Algebra, 1986, vol. 14, no. 9, pp. 1729–1736. https://doi.org/10.1080/00927878608823393Babai L. On the length of subgroup chains in the symmetric group. Communications in Algebra, 1986, vol. 14, no. 9, pp. 1729–1736. https://doi.org/10.1080/00927878608823393

The authors declare that there are no conflicts of interest present.