Normalizing isomorphisms between Burnside rings
type: Item
instance of: scholarly article
title: Normalizing isomorphisms between Burnside rings
zbMATH Open document ID: 1055.19001
author: Luis Valero-Elizondo
author: Alberto G. Raggi-Cárdenas
published in: Journal of Algebra
publication date: 2004-10-01T00:00:00Z
review text: Let \(G\) be a finite group and \(B(G)\) its Burnside ring. Given two finite groups \(G\) and \(G'\) one asks when are their corresponding Burnside rings isomorphic. There are infinitely many examples of nonisomorphic groups with isomorphic Burnside rings. However, what can be said about \(G\) and \(G'\) if \(B(G)\cong B(G')\)? The main theorem is the following: Let \(G\) and \(G'\) be finite groups with isomorphic Burnside rings. Then, there exists a normalized isomorphism \(\theta:B(G)\to B(G')\), i.e. \(\theta\) is a ring isomorphism such that \(\theta(G/1)=G'/1\). As an application the authors have the following corollary: Let \(G\) and \(G'\) be finite groups such that their Burnside rings are isomorphic. Then, there is a one-to-one correspondence between the conjugacy classes of soluble subgroups of \(G\) and \(G'\) which preserves order of subgroups and cardinality of the conjugacy class.
Mathematics Subject Classification ID: 19A22
zbMATH DE Number: 2105243
zbMATH Keywords: Burnside ring
zbMATH Keywords: table of marks
zbMATH Keywords: normalized isomorphism
reviewed by: Daniel Juan Pineda
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.10.031
OpenAlex ID: W2081052081
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DOI: 10.1016/J.JALGEBRA.2003.10.031
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description: scientific article; zbMATH DE number 2105243