UDK 512.542, 519.6
MSC: 20D10, 20B40
DOI: 10.21538/0134-4889-2025-31-1-154-165
The research was supported by the Russian Science Foundation and by the Belarusian Republican Foundation for Fundamental Research (project no. Φ23PHΦ-237)
For a wide family of formations $\mathfrak{F}$ (which includes Baer-local formations) of finite groups it is proved that the $ \mathfrak{F}$-hypercenter of a permutation finite group of degree $n$ can be computed in polynomial time in $n$. In particular, the algorithms for computing the $\mathfrak{F}$-hypercenter for the following classes of groups are suggested: hereditary local formations with the Shemetkov property, rank formations, formations of all quasinilpotent, Sylow tower of type $\varphi$, $p$-nilpotent, supersoluble, $w$-supersoluble and $SC$-groups. For some of these formations $\mathfrak{F}$ algorithms for the computation of the intersection of all maximal $\mathfrak{F}$-subgroups of a finite group are suggested.
Keywords: Finite group, $\mathfrak{F}$-hypercenter, Baer-local formation, permutation group computation, polynomial time algorithm
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Received October 11, 2024
Revised January 8, 2025
Accepted January 13, 2025
Funding Agency: The research was supported by the Russian Science Foundation and by the Belarusian Republican Foundation for Fundamental Research (project no. Φ23PHΦ-237).
Viachaslau Igaravich Murashka, Cand. Sci. (Phys.-Math.), Francisk Skorina Gomel State University, Gomel, 246019 Belarus, e-mail: mvimath@yandex.ru
Cite this article as: V.I. Murashka. Formations of finite groups in polynomial time II: the $\mathfrak{F}$-hypercenter and its generalizations.Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 1, pp. 154–165.
Русский
В.И. Мурашко. Распознавание формаций конечных групп за полиномиальное время II: $\mathfrak{F}$-гиперцентр и его обобщения
Для широкого семейства формаций $\mathfrak{F}$ (включающего в себя композиционные формации) конечных групп доказано, что $\mathfrak{F}$-гиперцентр конечной группы перестановок степени $n$ может быть вычислен за полиномиальное время от $n$. В частности, предложены алгоритмы вычисления $\mathfrak{F}$-гиперцентра для следующих классов групп: наследственные локальные формации с условием Шеметкова, ранговые формации, формации всех квазинильпотентных, $\varphi$-дисперсивных, $p$-нильпотентных, сверхразрешимых, $w$-сверхразрешимых и $SC$-групп. Для некоторых из этих формаций $\mathfrak{F}$ предложены алгоритмы вычисления пересечения всех максимальных $\mathfrak{F}$-подгрупп конечной группы.
Ключевые слова: Конечная группа, $\mathfrak{F}$-гиперцентр, композиционная формация, вычисления в группах перестановок, полиномиальный алгоритм
