Supersoluble crossed product criterion for division algebras.
type: Item
instance of: scholarly article
title: Supersoluble crossed product criterion for division algebras.
zbMATH Open document ID: 1070.16021
DOI: 10.1007/BF02786697
author: R. Ebrahimian
author: M. Mahdavi-Hezavehi
author: Dariush Kiani
published in: Israel Journal of Mathematics
publication date: 2005-05-03T00:00:00Z
review text: Let \(D\) be a finite-dimensional noncommutative division algebra over its centre \(F\), and let \(D^*\) be the multiplicative group of \(D\). A subgroup \(G\) of \(D^*\) is called generating, if \(F[G]=D\). The paper under review shows that \(D\) is presentable as a supersoluble (respectively, nilpotent, Abelian or cyclic) crossed product, i.e. \(D\) contains as a maximal subfield a Galois extension of \(F\) with a supersoluble (nilpotent, Abelian, cyclic) Galois group, if and only if, there exists a generating subgroup \(G\) of \(D^*\), which possesses an Abelian normal subgroup \(A\), such that the quotient group \(G/A\) is supersoluble (respectively, nilpotent, Abelian or cyclic). In particular, it proves that the existence of a locally nilpotent generating subgroup of \(D^*\) of \(D\) ensures the presentability of \(D\) as a nilpotent crossed product. Also, the authors show that their main result need not be valid, if \(D\) is merely a finite-dimensional central simple \(F\)-algebra.
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16S35
zbMATH DE Number: 2164547
zbMATH Keywords: supersoluble crossed products
zbMATH Keywords: generating subgroups of multiplicative groups of division algebras
zbMATH Keywords: nilpotent crossed products
zbMATH Keywords: Abelian crossed products
zbMATH Keywords: cyclic crossed products
MaRDI profile type: Publication
cites work: Cyclicity conditions for division algebras of prime degree
cites work: Q4039952
cites work: Q3737662
cites work: Soluble subgroups of the multiplicative group of a field
full work available at URL: https://doi.org/10.1007/bf02786697
OpenAlex ID: W2014007868
description: scientific article; zbMATH DE number 2164547