On Shemetkov’s Question about the $$\mathfrak{F}$$ -Hypercenter

IF 0.6 4区数学 Q3 MATHEMATICS

Mathematical Notes Pub Date : 2024-07-15 DOI:10.1134/s0001434624050134

V. I. Murashka

引用次数: 0

Abstract

The chief factor $H/K$ of a group $G$ is said to be $\mathfrak{F}$-central if

$$(H/K)\rtimes (G/C_G(H/K))\in\mathfrak{F}.$$

The $\mathfrak{F}$-hypercenter of a group $G$ is defined to be a maximal normal subgroup of $G$ such that all $G$-composition factors below it are $\mathfrak{F}$-central in $G$. In 1995, at the Gomel algebraic seminar, L. A. Shemetkov formulated the problem of describing formations of finite groups $\mathfrak{F}$ for which, in any group, the intersection of $\mathfrak{F}$-maximal subgroups coincides with the $\mathfrak{F}$-hypercenter. In the present paper, new properties of such formations are obtained. In particular, a series of hereditary nonsaturated formations of soluble groups is constructed, which answer Shemetkov’s problem.

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关于谢梅特科夫提出的 $$\mathfrak{F}$ - 超中心问题

Abstract 如果 $$(H/K)\rtimes (G/C_G(H/K))\in\mathfrak{F}, 那么一个群 $G$ 的主因子 $H/K$ 被称作是 $\mathfrak{F}$-central 。$$ 一个群 $G$ 的 $\mathfrak{F}$-hypercenter 被定义为 $G$ 的一个最大正则子群，使得它下面的所有 $G$ - 组合因子都是$G$中的$\mathfrak{F}$-中心。1995 年，在戈梅尔代数研讨会上，谢梅特科夫（L. A. Shemetkov）提出了描述有限群 $\mathfrak{F}$的形式的问题，对于这些有限群，在任何群中，$\mathfrak{F}$-最大子群的交集都与$\mathfrak{F}$-超中心重合。在本文中，我们得到了这种形式的新性质。特别是，本文构建了一系列可溶群的遗传非饱和形式，回答了谢梅特科夫的问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematical Notes 数学-数学

CiteScore

0.90

自引率

16.70%

发文量

179

审稿时长

24 months

期刊介绍： Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.