Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T07:42:41.479Z Has data issue: false hasContentIssue false
  • English
  • Français

Collineation groups which are sharply transitive on an oval

Published online by Cambridge University Press:  17 April 2009

P.B. Kirkpatrick
P.B. Kirkpatrick
Affiliation:
Department of Pure Mathematics, University of Sydney, Sydney, New South Wales.
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.

Let G be a group of collineations in a protective plane Π of order n. Suppose that one of the point orbits of G is an oval of Π, and that G acts regularly on this orbit. We prove that G fixes a non-incident point-line pair if either n is even, or n is odd and G is abelian, or n ≠ 11, 23, 59 is odd and is a pseudo-conic. It is then easy to deduce information about the lengths of the other orbits of G, and about the structure of G as an abstract group.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 11 , Issue 2 , October 1974 , pp. 197 - 211
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Baer, Reinhold, “Polarities in finite projective planes”, Bull. Amer. Math. Soc. 52 (1916), 77–93.CrossRefGoogle Scholar
[2]Baer, Reinhold, “Projectivities with fixed points on every line of the plane”, Bull. Amer. Math. Soc. 52 (1916), 273–286.CrossRefGoogle Scholar
[3]Brauer, Richard and Suzuki, Michio, “On finite groups of even order whose 2-Sylow group is a quaternion group”, Proa. Nat. Acad. Sci. U.S.A. 45 (1959), 1757–1759.CrossRefGoogle Scholar
[4]Burnside, W., Theory of groups of finite order, 2nd ed. (Cambridge University Press, Cambridge, 1911; reprinted Dover, New York, 1955).Google Scholar
[5]Dembowski, P., Finite geometries (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44. Springer-Verlag, Berlin, Heidelberg, New York, 1968).CrossRefGoogle Scholar
[6]Dembowski, Peter, “Verallgemeinerungen von Transitivitätsklassen endlicher projektiver Ebenen”, Math. Z. 69 (1958), 59–89.CrossRefGoogle Scholar
[7]Feit, Walter and Thompson, John G., “Solvability of groups of odd order”, Pacific J. Math. 13 (1963), 773–1029.Google Scholar
[8]Foulser, David A. and Sandler, Reuben, “Certain properties of orbits under collineation groups”, J. Combinatorial Theory 2 (1967), 546–570.CrossRefGoogle Scholar
[9]Gorenstein, Daniel, Finite groups (Harper and Row, New York, Evanston, London, 1968).Google Scholar
[10]Hall, Marshall Jr, The theory of groups (The Macmillan Company, New York, 1959).Google Scholar
[11]Hering, Christoph, “Eine Bemerkung über Automorphismengruppen von endlichen projektiven Ebenen und Möbiusebenen”, Arch. Math. 18 (1967), 107–110.CrossRefGoogle Scholar
[12]Hughes, D.R., “Collineations and generalized incidence matrices”, Trans. Amer. Math. Soc. 86 (1957), 284–296.CrossRefGoogle Scholar
[13]Hughes, D[aniel] R., Piper, F[red] C., Protective planes (Graduate Texts in Mathematics, 6. Springer-Verlag, New York, Heidelberg, Berlin, 1973).Google Scholar
[14]Ostrom, T.G., “Ovals, dualities, and Desargues's theorem”, Canad. J. Math. 7 (1955), 417–431.CrossRefGoogle Scholar
[15]Parker, E.T., “On collineations of symmetric designs”, Proc. Amer. Math. Soc. 8 (1957), 350–351.CrossRefGoogle Scholar
[16]Piper, Fred, “The orbit structure of collineation groups of finite projective planes”, Math. Z. 103 (1968), 318–332.CrossRefGoogle Scholar
[17]Qvist, B., “Some remarks concerning curves of the second degree in a finite plane”, Ann. Acad. Sci. Fennicae. Ser. A.I. Math.-Phys. 134 (1952), 1–27.Google Scholar
[18]Singer, James, “A theorem in finite projective geometry and some applications to number theory”, Trans. Amer. Math. Soc. 43 (1938), 377–385.CrossRefGoogle Scholar
[19]Wielandt, Helmut, Finite permutation groups (Academic Press, New York and London, 1961).Google Scholar