{"title":"The Isomorphism Problem of Normalized Unit Groups of Group Algebras of a Class of Finite 2-groups","authors":"Yu Lei Wang, He Guo Liu","doi":"10.1007/s10114-023-2261-0","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>p</i> be a prime and <span>\\({\\mathbb{F}_p}\\)</span> be a finite field of <i>p</i> elements. Let <span>\\({\\mathbb{F}_p}G\\)</span> denote the group algebra of the finite <i>p</i>-group <i>G</i> over the field <span>\\({\\mathbb{F}_p}\\)</span> and <span>\\(V({\\mathbb{F}_p}G)\\)</span> denote the group of normalized units in <span>\\({\\mathbb{F}_p}G\\)</span>. Suppose that <i>G</i> and <i>H</i> are finite <i>p</i>-groups given by a central extension of the form </p><div><div><span>$$1 \\to {\\mathbb{Z}_{{p^m}}} \\to G \\to {\\mathbb{Z}_p} \\times \\cdots \\times {\\mathbb{Z}_p} \\to 1$$</span></div></div><p> and <span>\\({G^\\prime } \\cong {\\mathbb{Z}_p},\\,\\,m \\ge 1\\)</span>. Then <span>\\(V({\\mathbb{F}_p}G) \\cong V({\\mathbb{F}_p}H)\\)</span> if and only if <i>G</i> ≅ <i>H</i>. Balogh and Bovdi only solved the isomorphism problem when <i>p</i> is odd. In this paper, the case <i>p</i> = 2 is determined.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10114-023-2261-0.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Sinica-English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10114-023-2261-0","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let p be a prime and \({\mathbb{F}_p}\) be a finite field of p elements. Let \({\mathbb{F}_p}G\) denote the group algebra of the finite p-group G over the field \({\mathbb{F}_p}\) and \(V({\mathbb{F}_p}G)\) denote the group of normalized units in \({\mathbb{F}_p}G\). Suppose that G and H are finite p-groups given by a central extension of the form
and \({G^\prime } \cong {\mathbb{Z}_p},\,\,m \ge 1\). Then \(V({\mathbb{F}_p}G) \cong V({\mathbb{F}_p}H)\) if and only if G ≅ H. Balogh and Bovdi only solved the isomorphism problem when p is odd. In this paper, the case p = 2 is determined.
期刊介绍:
Acta Mathematica Sinica, established by the Chinese Mathematical Society in 1936, is the first and the best mathematical journal in China. In 1985, Acta Mathematica Sinica is divided into English Series and Chinese Series. The English Series is a monthly journal, publishing significant research papers from all branches of pure and applied mathematics. It provides authoritative reviews of current developments in mathematical research. Contributions are invited from researchers from all over the world.