Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-20T04:51:15.151Z Has data issue: false hasContentIssue false
  • English
  • Français

Congruence-Preserving Isomorphisms of the Translation Group associated with a Translation Plane

Published online by Cambridge University Press:  20 November 2018

F. Radó
F. Radó*
Affiliation:
University of Waterloo, Waterloo, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let II, II′ be projective translation planes, their sets of points, l∞, l∞′ the improper lines, and T, T′ the corresponding translation groups. T is an Abelian group, simply transitive on . The set of the subgroups Ts = {τ|τ ∈ T, cen τ = S} for all S ∈ l∞ is called the congruence of II (cen τ = centre of τ). An injective map , where , is said to be a collineation of when and three points in are collinear if and only if their images are collinear; the set of these φ is denoted by and for we write

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 2 , April 1971 , pp. 214 - 221
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Aczél, J., Collineations on three and on four lines of projective planes over fields, Mathematica 8 (81) (1966), 7–13.Google Scholar
2. Aczél, J. and Benz, W., Kollineationen auf Drei- und Vierecken in der Desargues s chen projektiven Ebene und Àquivalenz der Dreiecksnomogramme und der Dreigewebe von Loops mit der Isotopie-Isomorphie-Eigenschaft, Aequationes Math. 3 (1969), 86–92.Google Scholar
3. Aczél, J. and McKiernan, M. A., On the characterization of plane projective and complex Moebius transformations, Math. Nachr. 38 (1967), 315–337.Google Scholar
4. André, J., Über nicht-Des argues s che Ebenen mit transitiver Translationsgruppe, Math. Z. 60 (1954), 156–186.Google Scholar
5. Havel, V., On collineations on three and four lines in a projective plane, Aequationes Math. 4 (1970), 51–55.Google Scholar
6. Orbán, B., Extension of collineations defined on certain sets of a Desarguesian projective plane, Aequationes Math. 4 (1970), 65–71.Google Scholar
7. Pickert, G., Projektive Ebenen, Die Grundlehrender mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Beriicksichtigung der Anwendungsgebiete, Bd. 80 (Springer-Verlag, Berlin-Gôttingen-Heidelberg, 1955).Google Scholar
8. Radó, F., Non-injective collineations on some sets in Desarguesian projective planes and extension of non-commutative valuations, Aequationes Math. 4 (1970), 307–321.Google Scholar
9. Rigby, J. F., Collineations on quadrilaterals in projective planes, Mathematica 10 (83) (1968), 369–383.Google Scholar