A criterion for infinite non-Dickson nearfields of dimension two
type: Item
instance of: scholarly article
title: A criterion for infinite non-Dickson nearfields of dimension two
zbMATH Open document ID: 0696.12021
DOI: 10.1007/BF03322613
author: Theo Grundhöfer
author: H. Zassenhaus
published in: Results in Mathematics
publication date: 1989-01-01T00:00:00Z
review text: Infinite nearfields which are not Dickson nearfields were constructed for the first time by H. Zassenhaus in 1987. However, there is a gap in the proof of the assertion that some of these nearfields are non-Dickson nearfields. In this paper a theorem is proved which states that if a nearfield N contains a field k in its center and the following three properties are satisfied: (i) N has dimension 2 over k, (ii) the factor group \(N^*/k^*\) contains no element of order 2, (iii) the characteristic of N is distinct from 2, then N is not a Dickson nearfield. This criterion is applied to some of the nearfields constructed by Zassenhaus, thus proving that these infinite nearfields are indeed non- Dickson nearfields.
reviewed by: W. Kerby
Mathematics Subject Classification ID: 12K05
zbMATH DE Number: 4139892
zbMATH Keywords: Dickson nearfields
zbMATH Keywords: infinite nearfields
MaRDI profile type: Publication
cites work: Q4108256
cites work: Transitive linear groups and nearfields with solubility conditions
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cites work: Q3671335
cites work: On Frobenius groups. II: Universal completion of nearfields of finite degree over a field of reference
description: scientific article; zbMATH DE number 4139892