Units of group rings of solvable and Frobenius groups over large rings of cyclotomic integers
type: Item
instance of: scholarly article
title: Units of group rings of solvable and Frobenius groups over large rings of cyclotomic integers
zbMATH Open document ID: 0796.16025
author: Sudarshan K. Sehgal
author: Jürgen Ritter
published in: Journal of Algebra
publication date: 1994-09-29T00:00:00Z
review text: In recent years the problem of finding units which generate a subgroup of finite index in \(U(\mathbb{Z} G)\), the group of units of an integral group ring, has seen a lot of progress, a great deal of which is due to the authors of the paper under review. In the present work they consider group rings of finite groups \(G\) over the ring \(R[\zeta]\), where \(\zeta\) denotes a primitive root of unity of order \(| G|\), provided that \(G\) satisfies a restriction first introduced by \textit{G. Janusz} [Proc. Am. Math. Soc. 17, 520--523 (1966; Zbl 0151.02203)]. This condition is verified for both solvable and Frobenius groups. In this case, a finite set of units generating a subgroup of finite index is given. This set includes the so-called Bass cyclic units of \(G\) with respect to \(R\) and a family of units introduced here for the first time. The paper is of great interest to anyone working in the area. In this context, also the paper by \textit{E. Jespers} and \textit{G. Leal} [Manuscr. Math. 78, No. 3, 303--315 (1993; Zbl 0802.16025)] should be consulted.
reviewed by: C. Polcino Milies
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 218572
zbMATH Keywords: unit group
zbMATH Keywords: solvable groups
zbMATH Keywords: subgroup of finite index
zbMATH Keywords: group of units
zbMATH Keywords: integral group ring
zbMATH Keywords: group rings of finite groups
zbMATH Keywords: Frobenius groups
zbMATH Keywords: Bass cyclic units
MaRDI profile type: Publication
full work available at URL: jabr.1993.1126
OpenAlex ID: W1966502994
DOI: 10.1006/JABR.1993.1126
description: scientific article; zbMATH DE number 218572