{"title":"An algorithm to recognize echelon subgroups of a free group","authors":"Dario Ascari","doi":"10.1142/s021819672450019x","DOIUrl":null,"url":null,"abstract":"<p>We provide an algorithm that, given a finite set of generators for a subgroup <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>H</mi></math></span><span></span> of a finitely generated free group <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>F</mi></math></span><span></span>, determines whether <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>H</mi></math></span><span></span> is echelon or not and, in case of affirmative answer, also computes a basis with respect to which <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>H</mi></math></span><span></span> is in echelon form. This gives an answer to a question of Rosenmann. We also prove, by means of a counterexample, that intersection of two echelon subgroups needs not to be echelon, answering another question of Rosenmann.</p>","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Algebra and Computation","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s021819672450019x","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We provide an algorithm that, given a finite set of generators for a subgroup of a finitely generated free group , determines whether is echelon or not and, in case of affirmative answer, also computes a basis with respect to which is in echelon form. This gives an answer to a question of Rosenmann. We also prove, by means of a counterexample, that intersection of two echelon subgroups needs not to be echelon, answering another question of Rosenmann.
期刊介绍：
The International Journal of Algebra and Computation publishes high quality original research papers in combinatorial, algorithmic and computational aspects of algebra (including combinatorial and geometric group theory and semigroup theory, algorithmic aspects of universal algebra, computational and algorithmic commutative algebra, probabilistic models related to algebraic structures, random algebraic structures), and gives a preference to papers in the areas of mathematics represented by the editorial board.
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