A translation plane of order 49 with non-solvable collineation group
type: Item
instance of: scholarly article
title: A translation plane of order 49 with non-solvable collineation group
zbMATH Open document ID: 0552.51005
DOI: 10.1007/BF01223529
author: Gabor Korchmáros
published in: Journal of Geometry
publication date: 1985-01-01T00:00:00Z
review text: The author constructs a translation plane of order 49 such that the translation complement S has a normal subgroup F of order 24 and S mod F is isomorphic to the symmetric group \(S_ 5\). There is a normal subgroup of index 2 which induces the same orbit decomposition on \(1_{\infty}\) as does SL(2,5) on the Desarguesian affine plane of order 49, but this plane is not Desarguesian. The plane is a rank 3 plane with two orbits on \(1_{\infty}\) (under S) of respective lengths 30 and 20. The construction goes back to an idea of Bruen involving what he calls chains of circles. (The line at infinity of a Desarguesian affine plane of order \(q^ 2\) becomes a Miquelian inversive plane in which the ''circles'' are the derivation sets. The circles in Bruen's chains are not disjoint.) The plane of this paper appears not to belong to any class of known translation planes; in any case, no known translation plane has been given explicitly with a proof that the translation complement has the properties of this particular one. We have here an interesting exceptional case.
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 3879615
zbMATH Keywords: collineation group
zbMATH Keywords: translation plane
zbMATH Keywords: chains of circles
zbMATH Keywords: exceptional case
Wikidata QID: Q59249319
reviewed by: Theodore G. Ostrom
MaRDI profile type: Publication
cites work: Q3314469
cites work: A translation plane of order 81 and its full collineation group
cites work: Q5606475
cites work: A translation plane of order \(11^ 2\)
cites work: Q3945140
cites work: Q3891315
cites work: Finite translation planes
cites work: Q3695923
cites work: Translation Planes of Order 112
cites work: A new class of finite translation planes obtained from the exceptional near-fields
description: scientific article; zbMATH DE number 3879615