Near nearfield planes
type: Item
instance of: scholarly article
title: Near nearfield planes
zbMATH Open document ID: 0764.51009
DOI: 10.1007/BF00181294
author: Yutaka Hiramine
author: Norman L. Johnson
published in: Geometriae Dedicata
publication date: 1992-09-27T00:00:00Z
review text: If \((F,+,\cdot)\) is a planar nearfield then we know that the affine plane over \(F\) admits the two groups \(G_ 1,G_ 2\) of affine perspectivities: \(G_ 1\) (resp. \(G_ 2)\) consists of all homologies with the \(x\)-axis \(X\) \((y\)-axis \(Y)\) as axis and the \(y\)-axis \(Y\) \((x\)-axis \(X)\) as direction and these two groups are acting transitively on the set \(Y\backslash X\) (resp. \(X\backslash Y)\). Both groups, \(G_ 1\) and \(G_ 2\) are isomorphic to the multiplicative group \((F^*,\cdot)\) of the nearfield. The authors call a translation plane \(\pi\) of finite odd order \(k\) a near nearfield plane if this plane admits at least two affine homology groups \(H_ 1\) and \(H_ 2\) each of order \((k-1)/2\). By this definition any affine plane over a finite nearfield \(F\) belongs to this class if \((F^*,\cdot)\) contains a subgroup of index 2. This is the case for all finite Dickson nearfields (including fields) of odd order and for the exceptional nearfields \(F\) of type (III) \((| F|=7^ 2)\) and type (IV) \((| F|=23^ 2)\) [cf. \textit{H. Wähling}, Theorem der Fastkörper (1987; Zbl 0669.12014), p. 373]. The authors show firstly that each near nearfield plane \(\pi\) admits always such homology groups \(H_ 1\) and \(H_ 2\) where the axis of \(H_ 2\) is the direction of \(H_ 1\) and vice versa. Then they show that, besides these mentioned nearfield examples, the only near nearfield planes are the exceptional Lüneburg plane of order \(7^ 2\) and the generalized André planes [for def. cf. \textit{H. Lüneburg}, Translation planes (1980; Zbl 0446.51003)].
reviewed by: Helmut Karzel
Mathematics Subject Classification ID: 51E15
Mathematics Subject Classification ID: 51A40
zbMATH DE Number: 64285
zbMATH Keywords: translation plane
zbMATH Keywords: near nearfield planes
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00181294
OpenAlex ID: W1971558142
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description: scientific article; zbMATH DE number 64285