Gauss units in integral group rings
type: Item
instance of: scholarly article
title: Gauss units in integral group rings
zbMATH Open document ID: 0917.16012
author: Olaf Neisse
author: Sudarshan K. Sehgal
published in: Journal of Algebra
publication date: 1998-12-07T00:00:00Z
review text: In a number of papers the generation of the unit group \(U(RG)\) of an integral group ring \(RG\) of a finite group \(G\) over some ring \(R\) of algebraic integers has been studied, and often a finite set of explicit units could be given that generates \(U(RG)\) up to a finite index. In this paper \(R\) is the ring of integers in \(\mathbb{Q}(\sqrt{-p}:p\mid m)\) where \(m\in\mathbb{N}\) only allows prime divisors \(p\equiv 3\bmod 4\), but is itself not a power of 3, and where \(G\) is generated by \(a,b\) subject to the relations \(a^m=1=b^n\) \(a^b=a^{-1}\), \(\gcd(n,m)=1\), \(n\equiv 0\bmod 4\). Such \(G\) have fixed point free representations and so Bass units and bicyclic units will not suffice for getting a finite index in \(U(RG)\). The new units introduced here, which replace the bicyclic ones and guarantee a finite index, are so-called Gauß units and look like \(1+x^+yx^-\), \(1+x^-yx^+\) with \(y\in RG\) and \(x^\pm=(\widehat g-p^k)\cdot\sum_{j=0}^{p^k-1}\sqrt{-p} g^{j^2}\pm g^{p^{j^2}}\) for \(g\in G\) of order \(p^k\).
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 1218676
zbMATH Keywords: unit groups
zbMATH Keywords: integral group rings
zbMATH Keywords: finite groups
zbMATH Keywords: fixed point free representations
zbMATH Keywords: Bass units
zbMATH Keywords: bicyclic units
zbMATH Keywords: finite index
zbMATH Keywords: Gauß units
reviewed by: Jürgen Ritter
MaRDI profile type: Publication
full work available at URL: jabr.1997.7379
OpenAlex ID: W1999443644
cites work: Generators of large subgroups of units of integral group rings of nilpotent groups
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cites work: Q4298518
cites work: Construction of Units in Integral Group Rings of Finite Nilpotent Groups
cites work: Q4271282
cites work: Q5546515
DOI: 10.1006/JABR.1997.7379
description: scientific article; zbMATH DE number 1218676