On Frobenius groups. II: Universal completion of nearfields of finite degree over a field of reference
type: Item
instance of: scholarly article
title: On Frobenius groups. II: Universal completion of nearfields of finite degree over a field of reference
zbMATH Open document ID: 0649.12018
DOI: 10.1007/BF03323278
published in: Results in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: In the 1930's the author characterizing the complements in Frobenius groups [see Abh. Math. Semin. Hamb. Univ. 11, 187-220 (1935; Zbl 0011.10302)] put a milestone in the basement of the modern finite group theory. The suitable language in this paper included near-fields: fields were characterized as commutative complete near-fields, and all finite non-Dickson complete near-fields, 7 in number, were found. (A near-field is said to be complete if the left multiplication is defined for each element as a left multiplier.) In this paper the author constructs a new class of complete non-Dickson near-fields which are universal in some sense. The starting point is some subgroup \(G_0\) of \(\mathrm{GL}(n,k)\) which is fixed-point-free in the sense that only the identity element of \(G_0\) has eigenvalue 1. There is also an application to the problem of Biliotti about Dembowski's generalization of Hughes planes. [For part I see ibid. 5, 132--145 (1985; Zbl 0591.20017).]
Mathematics Subject Classification ID: 12K05
Mathematics Subject Classification ID: 51J20
Mathematics Subject Classification ID: 20B05
Mathematics Subject Classification ID: 20D40
zbMATH DE Number: 4059543
zbMATH Keywords: complements in Frobenius groups
zbMATH Keywords: non-Dickson near-fields
author: H. Zassenhaus
reviewed by: Evgenii I. Khukhro
MaRDI profile type: Publication
cites work: Q4195527
cites work: FINITE GROUPS WITH FIXED-POINT-FREE AUTOMORPHISMS OF PRIME ORDER
cites work: On a Normal Form of the Orthogonal Transformation I
full work available at URL: https://doi.org/10.1007/bf03323278
OpenAlex ID: W2018380917
description: scientific article; zbMATH DE number 4059543