Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-12T21:39:24.704Z Has data issue: false hasContentIssue false
  • English
  • Français

A translation plane of order 81 and its full collineation group

Published online by Cambridge University Press:  17 April 2009

Vito Abatangelo
Vito Abatangelo
Affiliation:
Dipartimento di Matematica, Via Re David 200, 70125 Bari, Italy.
Rights & Permissions [Opens in a new window]

Abstract

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.

In this paper a new translation plane of order 81 is constructed. Its collineation group is solvable and acts on the line at infinity as a permutation group K which is the product of a group of order 5 belonging to the center of K with a group of order 48. A 2-Sylow subgroup of K is the direct product of a dihedral group of order 8 with a group of order 2. K admits six orbits. They have lengths 4, 6, 12, 12, 24, 24.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 29 , Issue 1 , February 1984 , pp. 19 - 34
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Abatangelo, L.M. and Abatangelo, Vito, “On Bruen's plane of order 25 “, submitted.Google Scholar
[2]Barlotti, Adriano, “Representation and construction of protective planes and other geometric structures from protective spaces”, Jber. Deutsch. Math.-Verein. 77 (1975), 28–38.Google Scholar
[3]Bruck, R.H., “Construction problems in finite projective spaces”, Construction problems in finite projective spaces (C.I.M.E., II Ciclo, Bressanone, 1972. Edizioni Cremonese, Rome, 1973).Google Scholar
[4]Bruen, Aiden A., “Inversive geometry and some new translation planes, I”, Geom. Dedicata 7 (1978), 81–98.CrossRefGoogle Scholar
[5]Capursi, M., “A translation plane of order 112“, J. Combin. Theory Ser. A 35 (1983), 289–300.CrossRefGoogle Scholar
[6]Dembowski, P., Finite geometries (Ergebnisse der Mathematik und ihrer Grenzgebiete, 44. Springer-Verlag, Berlin, Heidelberg, New York, 1968.CrossRefGoogle Scholar
[7]Hughes, Daniel R. and Piper, Fred C., Projective planes (Graduate Texts in Mathematics, 6. Springer-Verlag, New York, Heidelberg, Berlin, 1973).Google Scholar
[8]Korchmáros, Gábor, “The full collineation group of Bruen's plane of order 49”, submitted.Google Scholar
[9]Korchmáros, Gábor, “A translation plane of order 49 with nonsolvable collineation group”, submitted.Google Scholar
[10]Larato, B. and Raguso, G., “Il gruppo delle collineazioni di un piano di ordine 132“, Atti del Convegno di Geometria Combinatoria e di Incidenza, 7 (Passo della Mendola, Trento, Italia, 1982. Rendiconti del Seminario Matematico, Brescia, to appear).Google Scholar
[11]Ostrom, T.G., Finite translation planes (Lecture Notes in Mathematics, 158. Springer-Verlag, Berlin, Heidelberg, New York, 1970).CrossRefGoogle Scholar
[12]Pellegrino, Giuseppe and Korchmaros, Gabor, “Translation planes of order 112”, Combinatorial and geometric structures and their applications, Trento, 1980, 249–264 (North-Holland Mathematical Studies, 63. North-Holland, Amsterdam, 1982).CrossRefGoogle Scholar
[13]Raguso, G., “Un piano di traslazione di ordine 132”, submitted.Google Scholar
[14]Wielandt, Helmut, Finite permutation groups (translated by Bercov, R.. Academic Press, New York, London, 1964).Google Scholar