Structure of the Macdonald groups in one parameter

Pub Date : 2023-11-08 DOI:10.1515/jgth-2023-0036

Alexander Montoya Ocampo, Fernando Szechtman

{"title":"Structure of the Macdonald groups in one parameter","authors":"Alexander Montoya Ocampo, Fernando Szechtman","doi":"10.1515/jgth-2023-0036","DOIUrl":null,"url":null,"abstract":"Abstract Consider the Macdonald groups <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>G</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>α</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy=\"false\">⟨</m:mo> <m:mrow> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> </m:mrow> <m:mo fence=\"true\" lspace=\"0em\" rspace=\"0em\">∣</m:mo> <m:mrow> <m:mrow> <m:msup> <m:mi>A</m:mi> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> <m:mo stretchy=\"false\">]</m:mo> </m:mrow> </m:msup> <m:mo>=</m:mo> <m:msup> <m:mi>A</m:mi> <m:mi>α</m:mi> </m:msup> </m:mrow> <m:mo rspace=\"0.337em\">,</m:mo> <m:mrow> <m:msup> <m:mi>B</m:mi> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mi>B</m:mi> <m:mo>,</m:mo> <m:mi>A</m:mi> <m:mo stretchy=\"false\">]</m:mo> </m:mrow> </m:msup> <m:mo>=</m:mo> <m:msup> <m:mi>B</m:mi> <m:mi>α</m:mi> </m:msup> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">⟩</m:mo> </m:mrow> </m:mrow> </m:math> G(\\alpha)=\\langle A,B\\mid A^{[A,B]}=A^{\\alpha},\\,B^{[B,A]}=B^{\\alpha}\\rangle , <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant=\"double-struck\">Z</m:mi> </m:mrow> </m:math> \\alpha\\in\\mathbb{Z} . We fill a gap in Macdonald’s proof that <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>G</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>α</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> G(\\alpha) is always nilpotent, and proceed to determine the order, upper and lower central series, nilpotency class, and exponent of <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>G</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>α</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> G(\\alpha) .","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/jgth-2023-0036","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 3

Abstract

Abstract Consider the Macdonald groups G ⁢ ( α ) = ⟨ A , B ∣ A [ A , B ] = A α , B [ B , A ] = B α ⟩ G(\alpha)=\langle A,B\mid A^{[A,B]}=A^{\alpha},\,B^{[B,A]}=B^{\alpha}\rangle , α ∈ Z \alpha\in\mathbb{Z} . We fill a gap in Macdonald’s proof that G ⁢ ( α ) G(\alpha) is always nilpotent, and proceed to determine the order, upper and lower central series, nilpotency class, and exponent of G ⁢ ( α ) G(\alpha) .

查看原文

麦克唐纳群在一个参数中的结构

考虑麦克唐纳群G≠(α)=⟨A,B∣A [A,B]=A α，B [B,A]=B α⟩G(\alpha)= \langle A,B \mid A^{[A,B]}=A^ {\alpha}，\，B^{[B,A]}=B^ {\alpha}\rangle， α∈Z \alpha\in\mathbb{Z}。我们填补了Macdonald证明G(α) G(\alpha)总是幂零的空白，并进一步确定了G(α) G(\alpha)的阶、上、下中心级数、幂零类和指数。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文