The derived length of Lie soluble group rings. I
type: Item
instance of: scholarly article
title: The derived length of Lie soluble group rings. I
zbMATH Open document ID: 0765.16007
DOI: 10.1016/0022-4049(92)90111-R
author: Aner Shalev
published in: Journal of Pure and Applied Algebra
publication date: 1993-01-16T00:00:00Z
review text: Let the group algebra \(KG\) of a group \(G\) over a field \(K\) be considered as a Lie algebra under commutation. The precise conditions under which \(KG\) is Lie soluble are known [\textit{I. B. S. Passi}, \textit{D. S. Passman} and \textit{S. K. Sehgal}, Can. J. Math. 25, 748-757 (1973; Zbl 0266.16011)]. In characteristic zero this is the case if and only if \(G\) is Abelian. This raises the problem of relating the structure of \(G\) to \(d\ell(KG)\) the derived length of \(KG\). The present paper is a contribution in this direction. The main result (Theorem A) states that if \(K\) is a field of characteristic \(p>0\) and \(G\) is a non-abelian group, then \(d\ell(KG)\geq\lceil\log_2(p+1)\rceil\), where \(\lceil r\rceil\) denotes the upper integral part of a real number \(r\). Furthermore, it is proved (Proposition B) that equality holds if the commutator subgroup of \(G\) is central of order \(p\).
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 20C07
zbMATH DE Number: 89554
zbMATH Keywords: group algebras
zbMATH Keywords: Lie algebras
zbMATH Keywords: Lie soluble group rings
zbMATH Keywords: derived lengths
zbMATH Keywords: commutator subgroup
reviewed by: Inder Bir S. Passi
MaRDI profile type: Publication
Wikidata QID: Q115364427
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description: scientific article; zbMATH DE number 89554