On the non-triviality of \(G(D)\) and the existence of maximal subgroups of \(\text{GL}_1(D)\).
type: Item
instance of: scholarly article
title: On the non-triviality of \(G(D)\) and the existence of maximal subgroups of \(\text{GL}_1(D)\).
zbMATH Open document ID: 1099.20027
author: T. Keshavarzipour
author: M. Mahdavi-Hezavehi
published in: Journal of Algebra
publication date: 2005-04-07T00:00:00Z
review text: Let \(F\) be a field, \(D\) a central division \(F\)-algebra of Schur index \(\text{ind}(D)=n\), \(D^*\) the multiplicative group of \(D\), and \(D'\) the commutator subgroup of \(D^*\). Denote by \(G_0(D)\) and \(G(D)\) the quotient groups \(D^*/F^*D'\) and \(D^*/D'RN_{D/F}\), respectively, where \(RN_{D/K}\subseteq D^*\) is the group of reduced norms of \(D/K\). The paper under review proves that \(G_0(D)=\{1\}\), provided that \(D\) is not a quaternion algebra. It shows that if \(F\) contains a primitive \(p\)-th root of unity or \(\text{char}(F)=p\), for a given prime \(p\), and \(\text{ind}(D)\) is a power of \(p\), then \(G(D)\neq\{1\}\) unless \(D\) is a quaternion algebra. As explained by the authors, the motivation for this research comes from the generally open problem of whether \(D^*\) possesses a maximal subgroup with respect to inclusion.
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 20E28
zbMATH DE Number: 2153341
zbMATH Keywords: division rings
zbMATH Keywords: splitting fields
zbMATH Keywords: quaternion algebras
zbMATH Keywords: maximal subgroups
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.11.020
OpenAlex ID: W2026469751
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DOI: 10.1016/J.JALGEBRA.2004.11.020
description: scientific article; zbMATH DE number 2153341