论不在切尔马克-德尔加多网格中的子群较少的群

Q3 Mathematics Arnold Mathematical Journal Pub Date : 2023-09-20 DOI:10.1007/s40598-023-00237-2

David Burrell, William Cocke, Ryan McCulloch

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引用次数: 0

摘要

我们研究了有限群有多少子群不在其 Chermak-Delgado 网格中的问题。有限群的 Chermak-Delgado 网格是子群的自偶网格，具有许多耐人寻味的性质。Fasolă 和 Tărnăuceanu (Bull Aust Math Soc 107(3):451-455, 2023) 询问了有多少子群不在 Chermak-Delgado 网格中，并对所有有两个或两个以下子群不在 Chermak-Delgado 网格中的群进行了分类。我们扩展了他们的工作，将不在 Chermak-Delgado 网格中的所有子群少于五个的群进行了分类。此外，我们还证明了不在 Chermak-Delgado 网格中的子群少于五个的群是零能群。在这方面，我们还证明了唯一一个在切尔马克-戴尔加多网格中有五个或更少子群的非零能群是（S_3\）。

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On Groups with Few Subgroups not in the Chermak–Delgado Lattice

We investigate the question of how many subgroups of a finite group are not in its Chermak–Delgado lattice. The Chermak–Delgado lattice for a finite group is a self-dual lattice of subgroups with many intriguing properties. Fasolă and Tărnăuceanu (Bull Aust Math Soc 107(3):451–455, 2023) asked how many subgroups are not in the Chermak–Delgado lattice and classified all groups with two or less subgroups not in the Chermak–Delgado lattice. We extend their work by classifying all groups with less than five subgroups not in the Chermak–Delgado lattice. In addition, we show that a group with less than five subgroups not in the Chermak–Delgado lattice is nilpotent. In this vein, we also show that the only non-nilpotent group with five or fewer subgroups in the Chermak–Delgado lattice is \(S_3\).

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Arnold Mathematical Journal Mathematics-Mathematics (all)

CiteScore

1.50

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0.00%

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期刊介绍： The Arnold Mathematical Journal publishes interesting and understandable results in all areas of mathematics. The name of the journal is not only a dedication to the memory of Vladimir Arnold (1937 – 2010), one of the most influential mathematicians of the 20th century, but also a declaration that the journal should serve to maintain and promote the scientific style characteristic for Arnold''s best mathematical works. Features of AMJ publications include: Popularity. The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions necessary for the understanding must be provided but also informal motivations even if the latter are well-known to the experts in the field. Interdisciplinary and multidisciplinary mathematics. AMJ publishes research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, for this reason, this kind of research is often under-represented in specialized mathematical journals. AMJ will try to compensate for this.Problems, objectives, work in progress. Most scholarly publications present results of a research project in their “final'' form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned, but the very process of mathematical discovery remains hidden. Following Arnold, publications in AMJ will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. AMJ publishes well-motivated research problems on a regular basis. Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold''s principle, a general formulation is less desirable than the simplest partial case that is still unknown.Being interesting. The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author''s responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author''s understanding of the overall picture is presented; however, these parts must be clearly indicated.

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