Units in group rings: Splittings and the isomorphism problem
type: Item
instance of: scholarly article
title: Units in group rings: Splittings and the isomorphism problem
zbMATH Open document ID: 0588.16004
DOI: 10.1016/0021-8693(85)90018-3
author: Klaus W. Roggenkamp
author: Leonard L. Scott
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: In 1975, \textit{R. K. Dennis} posed the question of when the natural injection \(G\to V(RG)\) of the theory of group rings is the section map of a splitting, so that \(V(RG)\) is of the form \(N\leftthreetimes G\) [see Ring theory II, Proc. 2nd Okla. Conf. 1975, 103-130 (1977; Zbl 0357.16003)]. For the most important case \(R={\mathbb{Z}}\), several results, especially those of \textit{G. H. Cliff}, \textit{S. K. Sehgal} and \textit{A. R. Weiss} [see J. Algebra 73, 167-185 (1981; Zbl 0484.16004)] and of the authors [see J. Pure Appl. Algebra 27, 299-314 (1983; Zbl 0509.16006)] suggested that the answer is affirmative mainly for metabelian groups G of the form \(1\to A\to G\to G/A\to 1,\) where G/A is of odd order. The present paper strengthens this feeling by showing the following Theorem. If G is non-abelian, simple and not isomorphic to a PSL(n,q) \((n>1\), q a prime power), then \(G\to V({\mathbb{Z}}G)\) is the section of a splitting. They also extend the result of Cliff-Sehgal-Weiss to the modular case (cf. Theorem III in the paper). For this case, they suppose that A and G/A are abelian p-groups, A being elementary, and that \(R={\mathbb{Z}}/p{\mathbb{Z}}\). Also, in order to prove the existence of the splitting, they have to introduce another definition which produces a group which is different from V(RG) in this case. The proof of the theorem stated above, which is the main result in this paper, follows from an adapted version of the congruence subgroup theorem of \textit{H. Bass}, \textit{J. Milnor} and \textit{J. P. Serre} [see Inst. Haut. Etud. Sci., Publ. Math. 33, 59-137 (1967; Zbl 0174.052)]. The main observation is that, fixing a maximal order \(\Lambda\) in \({\mathbb{Q}}G\) containing \({\mathbb{Z}}G\), any splitting epimorphism \(V({\mathbb{Z}}G)\to G\) induces an epimorphism SL(\({\mathbb{Z}}G)\to G\) whose kernel contains a congruence subgroup of \(\Lambda\). Using this, the authors are able to show that, under the stated assumption, G has a PSL(…
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16H05
zbMATH DE Number: 3944055
zbMATH Keywords: group rings
zbMATH Keywords: section map of a splitting
zbMATH Keywords: metabelian groups
zbMATH Keywords: congruence subgroup theorem
zbMATH Keywords: maximal order
zbMATH Keywords: isomorphism problem
reviewed by: Héctor A. Merklen
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90018-3
OpenAlex ID: W1979229498
cites work: Circle Groups of Nilpotent Rings
cites work: Q5562621
cites work: Solution of the congruence subgroup problem for \(\text{SL}_ n\) \((n\geq 3)\) and \(\text{Sp}_{2n}\) \((n\geq 2)\)
cites work: Units of integral group rings of metabelian groups
cites work: Deux groupes finis distincts ayant la même algèbre de groupe sur tout corps
cites work: Q4129579
cites work: Finite Abelian groups with isomorphic group algebras
cites work: Q3921361
cites work: Endliche Gruppen I
cites work: THE GROUPS OF UNITS OF THE INTEGRAL GROUP RINGS OF FINITE METABELIAN AND FINITE NILPOTENT GROUPS
cites work: On the Circle Group of a Nilpotent Ring
cites work: Q4177670
cites work: On the units of the integral group ring of a dihedral group
cites work: Isomorphism of modular group algebras
cites work: Isomorphic groups and group rings
cites work: Units in integral group rings
cites work: On the isomorphism problem for integral group-rings of circle-groups
cites work: UNITS IN INTEGRAL METABELIAN GROUPRINGS I, JACKSON'S UNIT THEOREM REVISITED
cites work: Q3898627
cites work: Group rings of circle and unit groups
cites work: Q4198893
description: scientific article; zbMATH DE number 3944055