Crossed product conditions for division algebras of prime power degree.
type: Item
instance of: scholarly article
title: Crossed product conditions for division algebras of prime power degree.
zbMATH Open document ID: 1069.16018
author: Dariush Kiani
author: M. Mahdavi-Hezavehi
published in: Journal of Algebra
publication date: 2005-04-07T00:00:00Z
review text: Let \(F\) be a field and \(D\) a central division \(F\)-algebra of degree \(n\). A subgroup \(G\) of the multiplicative group \(D^*\) of \(D\) is said to be irreducible, if \(F[G]=D\). Under the hypothesis that \(n=p^r\), where \(p\) is prime and \(r\) is a positive integer, the paper under review gives several criteria for \(D\) to be a crossed product. The authors show that \(D\) has this property if and only if \(D^*\) possesses an irreducible solvable subgroup. They prove that \(D\) is a crossed product, in case \(D^*\) has an irreducible finite subgroup. When \(p>2\) or \(\text{char}(F)=0\), it is established that \(D\) is such a product if and only if \(D^*\) contains an irreducible subgroup satisfying a group identity. The proof of the third result relies on the Tits alternative, namely the fact that a finitely generated linear group contains a noncyclic free subgroup, unless it is solvable-by-finite (see [\textit{J. Tits}, J. Algebra 20, 250-270 (1972; Zbl 0236.20032)]).
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 16U60
zbMATH DE Number: 2153361
zbMATH Keywords: irreducible subgroups
zbMATH Keywords: crossed products
zbMATH Keywords: Tits alternative
zbMATH Keywords: central division algebras
zbMATH Keywords: group identities
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.04.022
OpenAlex ID: W1963978679
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DOI: 10.1016/J.JALGEBRA.2004.04.022
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description: scientific article; zbMATH DE number 2153361