{"title":"Automorphic word maps and the Amit–Ashurst conjecture","authors":"Harish Kishnani, Amit Kulshrestha","doi":"10.1515/jgth-2023-0151","DOIUrl":null,"url":null,"abstract":"In this article, we address the Amit–Ashurst conjecture on lower bounds of a probability distribution associated to a word on a finite nilpotent group. We obtain an extension of a result of [R. D. Camina, A. Iñiguez and A. Thillaisundaram, Word problems for finite nilpotent groups, <jats:italic>Arch. Math. (Basel)</jats:italic> 115 (2020), 6, 599–609] by providing improved bounds for the case of finite nilpotent groups of class 2 for words in an arbitrary number of variables, and also settle the conjecture in some cases. We achieve this by obtaining words that are identically distributed on a group to a given word. In doing so, we also obtain an improvement of a result of [A. Iñiguez and J. Sangroniz, Words and characters in finite 𝑝-groups, <jats:italic>J. Algebra</jats:italic> 485 (2017), 230–246] using elementary techniques.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/jgth-2023-0151","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this article, we address the Amit–Ashurst conjecture on lower bounds of a probability distribution associated to a word on a finite nilpotent group. We obtain an extension of a result of [R. D. Camina, A. Iñiguez and A. Thillaisundaram, Word problems for finite nilpotent groups, Arch. Math. (Basel) 115 (2020), 6, 599–609] by providing improved bounds for the case of finite nilpotent groups of class 2 for words in an arbitrary number of variables, and also settle the conjecture in some cases. We achieve this by obtaining words that are identically distributed on a group to a given word. In doing so, we also obtain an improvement of a result of [A. Iñiguez and J. Sangroniz, Words and characters in finite 𝑝-groups, J. Algebra 485 (2017), 230–246] using elementary techniques.
在这篇文章中，我们讨论了阿米特-阿舒斯特猜想中与有限零能群上的一个词相关的概率分布的下界问题。我们得到了[R.D. Camina, A. Iñiguez and A. Thillaisundaram, Word problems for finite nilpotent groups, Arch.Math. (Basel) 115 (2020), 6, 599-609] 为任意变量个数的单词提供了改进的 2 类有限零potent 群的约束，并在某些情况下解决了猜想。我们通过获得与给定词在一个群上同分布的词来实现这一目标。在此过程中，我们还得到了对 [A. Iñiguez and J. M. B.] 结果的改进。Iñiguez and J. Sangroniz, Words and characters in finite 𝑝-groups, J. Algebra 485 (2017), 230-246] 的一个结果的改进。
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