{"title":"On generalized concise words","authors":"Costantino Delizia, Michele Gaeta, Carmine Monetta","doi":"10.1515/jgth-2024-0148","DOIUrl":null,"url":null,"abstract":"The study of verbal subgroups within a group is well known for being an effective tool to obtain structural information about a group. Therefore, conditions that allow the classification of words in a free group are of paramount importance. One of the most studied problems is to establish which words are concise, where a word 𝑤 is said to be concise if the verbal subgroup <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>w</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2024-0148_ineq_0001.png\"/> <jats:tex-math>w(G)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is finite in each group 𝐺 in which 𝑤 takes only a finite number of values. The purpose of this article is to present some results, in which a hierarchy among words is introduced, generalizing the concept of concise word.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"37 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Group Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/jgth-2024-0148","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The study of verbal subgroups within a group is well known for being an effective tool to obtain structural information about a group. Therefore, conditions that allow the classification of words in a free group are of paramount importance. One of the most studied problems is to establish which words are concise, where a word 𝑤 is said to be concise if the verbal subgroup w(G)w(G) is finite in each group 𝐺 in which 𝑤 takes only a finite number of values. The purpose of this article is to present some results, in which a hierarchy among words is introduced, generalizing the concept of concise word.
众所周知，研究一个词组中的言语亚群是获取词组结构信息的有效工具。因此，对自由组中的词语进行分类的条件至关重要。研究最多的问题之一是确定哪些词是简明词，如果在每个𝑤取值有限的群𝐺中，言语子群 w ( G ) w(G) 都是有限的，则称𝑤为简明词。本文的目的是介绍一些结果，其中引入了词的层次结构，概括了简洁词的概念。
期刊介绍：
The Journal of Group Theory is devoted to the publication of original research articles in all aspects of group theory. Articles concerning applications of group theory and articles from research areas which have a significant impact on group theory will also be considered.
Topics:
Group Theory-
Representation Theory of Groups-
Computational Aspects of Group Theory-
Combinatorics and Graph Theory-
Algebra and Number Theory
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