Über die Projektivitätengruppen von Semikörperebenen gerader Ordnung. (About groups of projectivities of semifield planes of even order)
type: Item
instance of: scholarly article
title: Über die Projektivitätengruppen von Semikörperebenen gerader Ordnung. (About groups of projectivities of semifield planes of even order)
zbMATH Open document ID: 0678.51006
DOI: 10.1007/BF01222267
author: Dirk Kilmer
published in: Journal of Geometry
publication date: 1989-01-01T00:00:00Z
review text: By Th. Grundhöfer, the group of projectivities \(\Pi\) of any non- desarguesian (projective) translation plane P of order n contains the alternating group on \(n+1\) letters. Motivated by examples of Grundhöfer, Longwitz and Herzer, \textit{Th. Grundhöfer} [Result. Math. 6, 163-182 (1983; Zbl 0532.51002)] was led to conjecture that \(\Pi\) equals the symmetric group if the order of P is odd, and the alternating group if it is even. In the present note the author determines the group of projectivities \(\Pi\) of translation planes over some proper semifields (i.e. distributive quasifields) S of even order. He shows that, according to Grundhöfer's conjecture, \(\Pi\) consists only of even permutations, provided S is commutative or if S is of Knuth type II, III, or IV and \(| S| >16.\) On the other hand, proving the conjecture above wrong, he gives some examples of semifields of even order the group of projectivities of planes over which equals the symmetric group. In particular, the groups of projectivities of the two proper semifield-planes of order 16 contain odd permutations.
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 4111440
zbMATH Keywords: finite projective planes
zbMATH Keywords: group of projectivities
zbMATH Keywords: translation planes
zbMATH Keywords: semifields
zbMATH Keywords: Knuth type
reviewed by: Franz B. Kalhoff
MaRDI profile type: Publication
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cites work: Die Gruppe \(\pi(g)\) in den endlichen Andre-Ebenen gerader Ordnung
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full work available at URL: https://doi.org/10.1007/bf01222267
OpenAlex ID: W2914343020
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description: scientific article; zbMATH DE number 4111440