Hall classes of groups

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Pub Date : 2024-01-18 DOI:10.1007/s13398-023-01549-w

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Abstract

In 1958, Philip Hall (Ill J Math 2:787–801, 1958) proved that if a group G has a nilpotent normal subgroup N such that \(G/N'\) is nilpotent, then G is nilpotent. The scope of Hall’s nilpotency criterion is not restricted to group theory, and in fact similar statements hold for Lie algebras and more generally for algebraically coherent semiabelian categories (see Chao in Math Z 103:40–42, 1968; Gray in Adv Math 349:911–919, 2019; Stitzinger in Ill J Math 22:499–505, 1978). We say that a group class \({\mathfrak {X}}\) is a Hall class if it contains every group G admitting a nilpotent normal subgroup N such that \(G/N'\) belongs to \({\mathfrak {X}}\) . Thus, Hall’s nilpotency criterion just asserts that nilpotent groups form a Hall class. Many other relevant classes of groups have been proved to be Hall classes; for example, Plotkin (Sov Math Dokl 2:471–474, 1961) and Robinson (Math Z 107:225–231, 1968) proved that locally nilpotent groups and hypercentral groups form Hall classes. Note that these generalizations also hold if groups are replaced by other algebraic structures, for example Lie algebras (see Stitzinger in Ill J Math 22:499–505, 1978). The aim of this paper is to develop a general theory of Hall classes of groups, that could later be reasonably extended to Lie algebras. Among other results, we prove that many natural types of generalized nilpotent groups form Hall classes, and we give examples showing in particular that the class of groups having a finite term in the lower central series is not a Hall class, even if we restrict to the universe of linear groups.

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摘要 1958年，菲利普-霍尔（Ill J Math 2:787-801，1958）证明，如果一个群G有一个零能正常子群N，使得（G/N'\）是零能的，那么G就是零能的。霍尔零势判据的范围并不局限于群论，事实上，类似的说法也适用于李代数和更一般的代数相干半阿贝尔范畴（见Chao在《数学Z》103:40-42，1968；Gray在《Adv Math》349:911-919，2019；Stitzinger在《Ill J Math》22:499-505，1978）。我们说一个群类 \({\mathfrak {X}}\)是一个霍尔类，如果它包含了每一个容许一个零能正子群 N 的群 G，使得 \(G/N'\)属于 \({\mathfrak {X}}\)。因此，霍尔的零能性判据只是断言零能群构成了霍尔类。许多其他相关的群类也被证明是霍尔类；例如，普洛特金（Sov Math Dokl 2:471-474, 1961）和罗宾逊（Math Z 107:225-231, 1968）证明了局部零能群和超中心群构成霍尔类。请注意，如果把群换成其他代数结构，例如李代数，这些概括也是成立的（见 Stitzinger 在 Ill J Math 22:499-505, 1978）。本文的目的是发展一种关于群的霍尔类的一般理论，这种理论以后可以合理地扩展到李代数。在其他结果中，我们证明了许多自然类型的广义零能群构成霍尔类，并举例说明，即使我们把范围局限于线性群，在下中心数列中具有有限项的群类也不是霍尔类。

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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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas

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