Lie properties of the group algebra and the nilpotency class of the group of units
type: Item
instance of: scholarly article
title: Lie properties of the group algebra and the nilpotency class of the group of units
zbMATH Open document ID: 0936.16028
published in: Journal of Algebra
publication date: 2000-05-11T00:00:00Z
review text: Let \(\mathbb{F}\) be a field of characteristic \(p\) and let \(G\) be a group having a nontrivial Sylow \(p\)-subgroup. Then the group of units \(U(\mathbb{F}[G])\) of the group algebra \(\mathbb{F}[G]\) is nilpotent if and only if \(G\) is nilpotent and the commutator subgroup \(G'\) is a group of prime power order [\textit{I. I. Khripta}, Mat. Zametki 11, 191-200 (1972; Zbl 0258.20005)]. This paper is aimed at making a contribution to the determination of the nilpotency class of \(U(\mathbb{F}[G])\). If \(t_L(\mathbb{F}[G])\) and \(t^L(\mathbb{F}[G])\) denote respectively the lower and the upper Lie nilpotency indices of \(\mathbb{F}[G]\), then, in view of the result of \textit{N. Gupta} and \textit{F. Levin} [J. Algebra 81, 225-231 (1983; Zbl 0514.16024)], \(\text{cl}(U(\mathbb{F}[G]))+1\leq t_L(\mathbb{F}[G])\leq t^L(\mathbb{F}[G])\). If \(p>3\), then, by a result of \textit{A. Bhandari} and the reviewer [Bull. Lond. Math. Soc. 24, No. 1, 68-70 (1992; Zbl 0777.20004)], \(t_L(\mathbb{F}[G])=t^L(\mathbb{F}[G])\); using this result \textit{A. Shalev} [Arch. Math. 60, No. 2, 136-145 (1993; Zbl 0818.16025)] determined the Lie nilpotency indices of \(\mathbb{F}[G]\), and consequently \(\text{cl}(U(\mathbb{F}[G]))\), for certain groups. The authors in this paper, extending some of the results of Shalev, determine \(t_L(\mathbb{F}[G])\), \(t^L(\mathbb{F}[G])\), and \(\text{cl}(U(\mathbb{F}[G]))\) for certain metabelian groups. Among other results it is proved that if \(G\) is a nilpotent group with a cyclic commutator subgroup \(G'\) of order \(p^n>2\) and \(\mathbb{F}\) a field of prime characteristic \(p\), then \(U(\mathbb{F}[G])\) is nilpotent of class \(p^n-1\) if \(\text{Syl}_p(G)=G'\), and of class \(p^n\) if \(\text{Syl}_p(G)\neq G'\). The paper concludes with the description of modular group algebras with unit group nilpotent of class 3, thus extending the work of \textit{M. A. Rao} and \textit{R. Sanding} [Can. Math. Bull. 38, No. 1, 112-116 (1995; Zbl 0824.2…
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20F14
zbMATH DE Number: 1271643
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: group of units
zbMATH Keywords: group algebras
zbMATH Keywords: nilpotency classes
zbMATH Keywords: Lie nilpotency indices
zbMATH Keywords: metabelian groups
zbMATH Keywords: nilpotent groups
zbMATH Keywords: modular group algebras
author: Adalbert Bovdi
author: János Kurdics
reviewed by: Inder Bir S. Passi
MaRDI profile type: Publication
full work available at URL: jabr.1998.7617
OpenAlex ID: W2059802592
Wikidata QID: Q115395841
cites work: Lie-Nilpotency Indices of Group Algebras
cites work: Subgroups of Abelian Groups
cites work: Q4338464
cites work: The Centers of a Radical Ring
cites work: On the Lie ideals of a ring
cites work: Q3934684
cites work: Q5182341
cites work: Generalized nilpotency of the multiplicative group of a group ring
cites work: Q5687387
cites work: Engel properties of group algebras. II
cites work: On Lie nilpotent group rings
cites work: Group rings and their augmentation ideals
cites work: Lie dimension subgroups
cites work: The Characterisation of Modular Group Algebras Having Unit Groups of Nilpotency Class 3
cites work: The dimension subgroup problem
cites work: Q4198893
cites work: Q3480234
cites work: The nilpotency class of the unit group of a modular group algebra. I
cites work: The nilpotency class of the unit group of a modular group algebra. III
cites work: Lie ideals in group rings
cites work: Breadth and commutator subgroups of p-groups
DOI: 10.1006/JABR.1998.7617
description: scientific article; zbMATH DE number 1271643