Die Gruppe der speziellen gebrochen-affinen Transformationen eines KT- Fastkörpers. (The group of special fractional affine transformations of a KT-nearfield)
type: Item
instance of: scholarly article
title: Die Gruppe der speziellen gebrochen-affinen Transformationen eines KT- Fastkörpers. (The group of special fractional affine transformations of a KT-nearfield)
zbMATH Open document ID: 0705.20002
DOI: 10.1007/BF01189140
published in: Archiv der Mathematik
publication date: 1990-01-01T00:00:00Z
review text: The sharply 3-transitive groups with stabilizers containing a regular abelian normal subgroup can be described as the ``fractional linear'' groups T(N,\(\epsilon\)), where N is a nearfield with an inversion-like mapping \(\epsilon\), cf. \textit{H. Wähling} [Theorie der Fastkörper (Thales Verlag, Essen 1987; Zbl 0669.12014)] or \textit{W. Kerby} [On infinite sharply multiply transitive groups (Vandenhoeck \& Ruprecht, Göttingen 1974; Zbl 0291.20039)]. Let \(| N| >3\). The authors show that T(N,\(\epsilon\)) has a simple normal subgroup S(N,\(\epsilon\)), which is the smallest normal subgroup of T(N,\(\epsilon\)). Furthermore, if \(S(N_ 1,\epsilon_ 1)\) and \(S(N_ 2,\epsilon_ 2)\) are permutation- isomorphic, then \((N_ i,\epsilon_ i)\) are closely related (but not isomorphic in general). If char \(N\neq 2\) and if S(N,\(\epsilon\)) is permutation-isomorphic to PSL(2,F) for some field F, then N is a Dickson nearfield constructed from F, and \(\epsilon\) is the inversion of F.
Mathematics Subject Classification ID: 20B22
Mathematics Subject Classification ID: 12K05
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E32
zbMATH DE Number: 4156740
zbMATH Keywords: fractional linear groups
zbMATH Keywords: sharply 3-transitive groups
zbMATH Keywords: regular abelian normal subgroup
zbMATH Keywords: nearfield
zbMATH Keywords: simple normal subgroup
zbMATH Keywords: permutation- isomorphic
zbMATH Keywords: Dickson nearfield
author: Heinz Waehling
author: Heinrich Wefelscheid
MaRDI profile type: Publication
cites work: Über die Funktionalgleichung \(f(1+x)+ f(1+f(x)) = 1\). (On the functional equation \(f(1+x)+ f(1+f(x)) = 1\))
cites work: Endliche Gruppen I
cites work: Über die Einfachheit der speziellen projektiven Gruppen
cites work: Zusammenhänge zwischen Fastbereichen, scharf zweifach transitiven Permutationsgruppen und 2-Strukturen mit Rechtecksaxiom
cites work: Q4041787
cites work: Über eine scharf 3-fach transitiven Gruppen zugeordnete algebraische Struktur
cites work: ZT-subgroups of sharply 3-transitive groups
Wikidata QID: Q59383320
full work available at URL: https://doi.org/10.1007/bf01189140
OpenAlex ID: W2011032748
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description: scientific article; zbMATH DE number 4156740