A structure theorem for KT-nearfields
type: Item
instance of: scholarly article
title: A structure theorem for KT-nearfields
zbMATH Open document ID: 0604.12019
DOI: 10.1007/BF02188193
author: W. Kerby
published in: Aequationes Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137163
review text: All known examples of sharply 3-transitive permutation groups G contain an abelian subgroup fixing one point and acting regularly on the remaining points. Hence G can be written as \(G=\{(x\to a+mx)\), \((x\to a+m \sigma (b+x)) |\) a,b,m\(\in F\), \(m\neq 0\}\) for a suitable KT-near- field (F,\(\sigma)\), i.e. a near-field F (with distributive law \(x(y+z)=xy+xz)\) with an automorphism \(\sigma\) of the multiplicative group of F satisfying the functional equation \(\sigma (1+\sigma (x))=1-\sigma (1+x).\) The author proves the following result: for every KT-near-field (F,\(\sigma)\) with char(F)\(\neq 2\), the kernel \(K=\{k\in F |\) \((x+y)k=xk+yk\) for every x,y\(\in F\}\) of F is a commutative field, and \(\sigma (k)=k^{-1}\) for every \(k\in K^*.\) This generalizes results of \textit{W. Benz} and \textit{S. Elliger} [ibid. 1, 267-274 (1968; Zbl 0165.172)] and \textit{E. Hartmann} [Mitt. Math. Ges. Hamb. 10, 757-762 (1980; Zbl 0562.12021)]. If char(F)\(\neq 0\) and char(F)\(\not\equiv 1\) (mod 4), then the prime field of F is contained in the center of F; examples shown that the restrictions on the characteristic of F cannot be relaxed.
Mathematics Subject Classification ID: 12K05
Mathematics Subject Classification ID: 20B22
Mathematics Subject Classification ID: 20B20
zbMATH DE Number: 3975054
zbMATH Keywords: KT-field
zbMATH Keywords: sharply 3-transitive permutation groups
zbMATH Keywords: KT-near-field
zbMATH Keywords: kernel
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3975054