On the compatibility of planar and nonplanar involutions in translation planes
type: Item
instance of: scholarly article
title: On the compatibility of planar and nonplanar involutions in translation planes
zbMATH Open document ID: 0538.51004
DOI: 10.1007/BF02941863
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1982-01-01T00:00:00Z
review text: It is well known that in translation planes of odd order \(p^ r\) affine elations and Baer p-collineations cannot exist simultaneously. In translation planes of even order, elations and Baer involutions can simultaneously exist. Most of such known planes are derivable. The author considers derivable translation planes of even order admitting both elations and Baer involutions that fix the derivable net. He discusses in detail two incompatibility conjectures: I. A derivable translation plane cannot simultaneously admit two nontrivial elations with the same axis and two nontrivial Baer involutions with the same Baer axis that fix a derivable net. II. Let \(\pi\) be a derivable translation plane of even order admitting an elation group of order \(\geq 4\) that fixes a derivable net. Then the full collineation group of the derived plane \({\bar \pi}\) is the group inherited from \(\pi\) (that is, leaves the associated derivable net invariant).
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 51A35
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 3855860
zbMATH Keywords: Baer involutions
zbMATH Keywords: derivable net
zbMATH Keywords: collineation group
author: Norman L. Johnson
MaRDI profile type: Publication
cites work: Q3251484
cites work: Baer p-elements in translation planes
cites work: Subplanes of Partial Spreads in Translation Planes
cites work: Baer involutions in semifield planes of even order
cites work: On subgroups with trivial normalizer intersection
cites work: On shears of translation planes
cites work: Q4126984
cites work: A Classification of Semi-Translation Planes
cites work: On nonstrict semi-translation planes of Lenz-Barlotti class I-1
cites work: The translation planes of order 16 that admit nonsolvable collineation groups
cites work: The geometry of \(GL(2,q)\) in translation planes of even order \(q^ 2\)
cites work: Translation planes of dimension two and characteristic two
cites work: On Elations of Derived Semifield Planes
cites work: Generalization of Hall planes of odd order
cites work: The existence of Fano subplanes in generalized Hall planes
description: scientific article; zbMATH DE number 3855860