Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-13T04:07:18.868Z Has data issue: false hasContentIssue false
  • English
  • Français

Collineation groups preserving an oval in a projective place of odd order

Part of: Permutation groups Finite geometry and special incidence structures Geometry Linear incidence geometry

Published online by Cambridge University Press:  09 April 2009

Mauro Biliotti  and
Gabor Korchmaros
Mauro Biliotti
Affiliation:
Dipartimento di Matematica, Università di LecceVia Arnesano 73100 Lecce, Italia
Gabor Korchmaros
Affiliation:
Istituto di Matematica Università della BasilicataVia N. Sauro 34 85100 Potenza, Italia
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 we investigate the structure of a collineation group G of a finite projective plane Π of odd order, assuming that G leaves invariant an oval Ω of Π. We show that if G is nonabelian simple, then G ≅ PSL(2, q) for q odd. Several results about the structre and the action of G are also obtained under the assumptions that n ≡ 1 (4) and G is transitive on the points of Ω.

MSC classification

Secondary: 51E15: Affine and projective planes 51A10: Homomorphism, automorphism and dualities 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 48 , Issue 1 , February 1990 , pp. 156 - 170
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Biliotti, M. and Korchmaros, G., ‘Collineqeation groups strongly irreducible on an oval’, Ann. Discrete Math. 30 (1986), 85–97.Google Scholar
[2]Biliotti, M. and Korchmaros, G., ‘Collineation groups which are primitive on an oval of a projective plane of odd order’, J. London Math. Soc. (2) 33 (1986), 525–534.CrossRefGoogle Scholar
[3]Biliotti, M. and Korchmaros, G., ‘Hyperovals with a transitive collineation group’, Geom. Dedicata 24 (1987), 269–281.CrossRefGoogle Scholar
[4]Buekenhout, F., ‘The geometry of finite simple groups’, Buildings and the geometry of diagrams, (Lecture Notes in Mathematics, vol. 1181, Springer-Verlag, Berlin, Heidelberg, New York and Tokyo, 1986).CrossRefGoogle Scholar
[5]Dembowski, P., Finite geometries, (Springer-Verlage, Berlin, Heidelberg and New York, 1968).CrossRefGoogle Scholar
[6]Fischer, B., ‘A characterization of the symmetric groups on 4 and 5 letters’, J. Algebra 3 (1966), 88–98.CrossRefGoogle Scholar
[7]Glauberman, G., ‘Central elements in core-free groups’, J. Algebra 4 (1966), 403–420.CrossRefGoogle Scholar
[8]Gorenstein, D., Finite groups, (Harper & Row, New York, 1968).Google Scholar
[9]Gorenstein, D., The classification of finite simple groups, (Plenum Press, New York and London, 1983).CrossRefGoogle Scholar
[10]Hering, C., ‘On the structure of finite collineation groups of projective planes’, Abh. Math. Sem. Univ. Hamburg 49 (1979), 155–182.CrossRefGoogle Scholar
[11]Huppert, B., Endliche Gruppen I, (Springer-Verlag, Berlin, Heidelberg and New York, 1979).Google Scholar
[12]Huppert, B. and Blackburn, N., Finite groups III, (Springer-Verlag, Berlin, Heidelberg and New York, 1982).Google Scholar
[13]Kantor, W. M., ‘On the structure of collineation groups of finite projective planes’, Proc. London Math. Soc. (3) 32 (1976), 385–402.CrossRefGoogle Scholar
[14]Mitchell, H. H., ‘Determination of the ordinary and modular liner groups’, Trans. Amer. Math. Soc. 12 (1911), 207–242.CrossRefGoogle Scholar
[15]Reifart, A. and Stroth, G., ‘On finite simple groups containing perspectivities’, Geom. Dedicata 13 (1982), 7–46.CrossRefGoogle Scholar