mathnasТруды Института математики НАН БеларусиProceedings of the Institute of Mathematics of the NAS of Belarus1812-5093Институт математики НАН БеларусиOLYGHBmathnas-97Research ArticleАЛГЕБРА И ТЕОРИЯ ЧИСЕЛALGEBRA AND NUMBER THEORYПриводимые τ-замкнутые σ-локальные формации конечных групп с заданной структурой подформацийReducible τ-closed σ-local formations of finite groups with a given structure of subformationsСкрундьВ. В.SkrundzV. V.

Минск

vallik@mail.ruСафоноваИ. Н.SafonovaI. N.

Минск

Minsk

in.safonova@mail.ruБелорусский государственный университетBelarusian State University2025050120263323653Copyright © Скрундь В.В., Сафонова И.Н., 20262026Скрундь В.В., Сафонова И.Н.Skrundz V.V., Safonova I.N.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://mathnas.ejournal.by/jour/article/view/97

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 $\leq 3$.

Let $\mathfrak{F}$ and $\mathfrak{H}$ be some $\tau$-closed $\sigma$-local formations of finite groups. By $\mathfrak{F}/^\tau_\sigma\mathfrak{H}\cap\mathfrak{F}$ we denote the lattice of all $\tau$-closed $\sigma$-local formations $\mathfrak{X}$ such that $\mathfrak{H}\cap\mathfrak{F}\subseteq \mathfrak{X}\subseteq \mathfrak{F}.$ The length of the lattice $\mathfrak{F}/^\tau_\sigma\mathfrak{H}\cap\mathfrak{F}$ is called the {\it $\mathfrak H^\tau_\sigma$-defect}, and for $\mathfrak{H}=(1)$ it is the formation of all identity groups, {\it $l^\tau_\sigma$-length} of $\mathfrak{F}$. The general properties of the $\mathfrak H^\tau_\sigma$-defect of $\tau$-closed $\sigma$-local formations are studied, and a description of the structural structure of reducible $\tau$-closed $\sigma$-local formations with $\mathfrak H^\tau_\sigma$-defect $\leq 2$ and $l^\tau_\sigma$-length $\leq 3$ is obtained.

конечная группаподгрупповой функтор$\tau$-замкнутая $\sigma$-локальная формациякритическая $\tau$-замкнутая $\sigma$-локальная формация$\mathfrak H^\tau_\sigma$-дефект формации$l^\tau_\sigma$-длина формацииfinite groupsubgroup functor$\tau$-closed $\sigma$-local formationcritical $\tau$-closed $\sigma$-local formation$\mathfrak H^\tau_\sigma$-defect of $\tau$-closed $\sigma$-local formationThe 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)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)ReferencesShemetkov L. A., Skiba A. N. Formations of Algebraic Systems. Moscow, Nauka, 1989. 255 p. (in Russian).Shemetkov L. A., Skiba A. N. Formations of Algebraic Systems. 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The authors declare that there are no conflicts of interest present.