Abstract
The symmetric group on k symbols is made to operate on a certain set of matrices in such a way that its orbits are in one-to-one correspondence with the orbits of the k-arcs of an N-dimensional projective space under the group of projectivities. This leads to a formula for the number of such orbits.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Glynn, D. G., ‘Rings of geometries II’, J. Combin. Theory Ser. A 49 (1988), 26–66.
Gordon, C. E. and Killgrove, R., ‘Representative arcs in field planes of prime order’, Congr. Numer. 71 (1990), 73–86.
Gordon, C. E., ‘Collineation classes of arcs in field planes of prime order’, unpublished, 1989.
Hirschfeld, J. W. P., Projective Geometries over Finite Fields, Clarendon Press, Oxford, 1979.
Kallaher, M., Affine Planes with Transitive Collineation Groups, North-Holland, New York, 1982.
Ledermann, W., Finite Groups, Oliver and Boyd, Edinburgh and London, 1961, p. 73.
van Leijenhorst, D. C., ‘Orbits on the projective line’, J. Combin. Theory Ser. A 31 (1981), 146–154.
Moore, E. H., ‘Concerning the abstract groups of order k! and 1/2k! holohedrically isomorphic with the symmetric and the alternating substitution-groups on k letters’, Proc. London Math. Soc. 28 (1896), 357–366.
Niven, I. and Zuckerman, H. S., An Introduction to the Theory of Numbers (2nd edn), Wiley, New York, London, Sydney, 1966.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gordon, C.E. Orbits of arcs in PG(N, K) under projectivities. Geom Dedicata 42, 187–203 (1992). https://doi.org/10.1007/BF00147548
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00147548