Finite subgroups of units of group algebras. I
type: Item
instance of: scholarly article
title: Finite subgroups of units of group algebras. I
zbMATH Open document ID: 0545.16007
DOI: 10.1016/0021-8693(83)90163-1
author: Gerald J. Janusz
published in: Journal of Algebra
publication date: 1983-01-01T00:00:00Z
review text: The following question is considered: If G is a finite group and K an algebraic number field (with ring of integers R), what can be said about the finite subgroups of the unit group of KG which contain G? The main result is proved in section 2, using the existence of an involution for the coefficient ring. Let K have a complex involution *. Define an involution J of KG by \(J(\sum a_ gg)=\sum a^*_ gg,\) and, for a subring A of KG, let \(U_ J(A)=\{x\in A| \quad xJ(x)=1\}.\) Theorem 2. If A is an R-order in KG containing RG, then \(U_ J(A)\) is a finite group containing G. As A ranges over all maximal R-orders containing RG, \(U_ J(A)\) ranges over all maximal finite subgroups of KG containing G. Some interesing consequences are given and the case in which G is a dihedral group is studied in some detail in section 3. This method for the study of finite subgroups of KG arises from a nice result in algebraic number theory.
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16W10
zbMATH DE Number: 3867527
zbMATH Keywords: algebraic number field
zbMATH Keywords: finite subgroups
zbMATH Keywords: unit group
zbMATH Keywords: complex involution
zbMATH Keywords: maximal R-orders
zbMATH Keywords: maximal finite subgroups
reviewed by: Héctor A. Merklen
MaRDI profile type: Publication
full work available at URL: 0021-8693(83)90163-1
OpenAlex ID: W1977943284
cites work: Q4063245
cites work: Q4061092
description: scientific article; zbMATH DE number 3867527