Mod \(2\) group algebras with metabelian unit groups
type: Item
instance of: scholarly article
title: Mod \(2\) group algebras with metabelian unit groups
zbMATH Open document ID: 0926.16027
DOI: 10.1016/S0022-4049(97)00105-9
author: Donald B. Coleman
author: Robert Sandling
published in: Journal of Pure and Applied Algebra
publication date: 1999-11-29T00:00:00Z
review text: The authors classify the finite groups \(G\) such that the group of units of the group ring \(\mathbb{F} G\) is metabelian in case the characteristic of \(\mathbb{F}\) is 2. \textit{A. Shalev} gave [in J. Pure Appl. Algebra 72, No. 3, 295-302 (1991; Zbl 0735.16020)] the odd characteristic case. The authors obtain that the unit group of \(\mathbb{F} G\) is metabelian in case the characteristic of \(\mathbb{F}\) is 2 in exactly the following cases: Either \(G\) is nilpotent of class less or equal to 2 and the derived subgroup of \(G\) is a subgroup of a Klein four group, or \(\mathbb{F}\) is of cardinality 2 and \(G\) is a split extension of an elementary Abelian 3-group and a cyclic group of order 2 which acts by inversion. The authors prove first that for a 2-group \(G\) the unit group of \(\mathbb{F} G\) is not metabelian for large classes of groups \(G\). They essentially look at double commutators of certain elements in the group of units. This leaves as only possibility the case mentioned in the statement. The case of a non-2 group is less labourous.
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 1308221
zbMATH Keywords: finite groups
zbMATH Keywords: groups of units
zbMATH Keywords: group rings
reviewed by: Alexander Zimmermann
describes a project that uses: Cayley
MaRDI profile type: Publication
cites work: On the Solvability of Unit Groups of Group Algebras
cites work: Q3333220
cites work: Q4226368
cites work: Endliche Gruppen I
cites work: The groups of order 128
cites work: Q3338422
cites work: Q3728186
cites work: The Characterisation of Modular Group Algebras Having Unit Groups of Nilpotency Class 3
cites work: Presentations for Unit Groups of Modular Group Algebras of Groups of Order 16
cites work: Meta-abelian unit groups of group algebras are usually abelian
full work available at URL: s0022-4049(97)00105-9
OpenAlex ID: W2085551906
description: scientific article; zbMATH DE number 1308221