Abstract
We investigate collineations of finite 2-affine planes and show that with the exception of some 2-affine planes of order at most 4 each collineation is induced by a collineation of its projective completion. We further deal in more detail with 2-affine planes of type II and their collineations that fix each long line.
Similar content being viewed by others
References
Baer R.: Projectivities with fixed points on every line of the plane. Bull Am. Math. Soc. 52, 273–286 (1946).
Baer R.: Projectivities of finite projective planes. Am. J. Math. 69, 653–684 (1947).
Batten L., Totten J.: On a class of linear spaces with two consecutive line degrees. Ars Combin. 10, 107–114 (1980).
Beutelspacher A.: Embedding linear spaces with two line degrees in finite projective planes. J. Geom. 26, 43–61 (1986).
Beutelspacher A., Metsch K.: Embedding finite linear spaces in projective planes. Combinatorics ’84 (Bari, 1984), In: North-Holland Mathematics Studies, vol. 123, pp. 39–56. North-Holland, Amsterdam (1986).
Beutelspacher A., Metsch K.: Embedding finite linear spaces in projective planes. II. Discret Math. 66, 219–230 (1987).
Gorenstein D.: Finite Groups. Harper & Row, New York (1968).
Hughes D.R., Piper F.C.: Projective Planes. Springer, New York (1973).
Jungnickel D., Vedder K.: Generalized homologies. Mitt. Math. Sem. Giess. 166, 103–125 (1984).
Metsch K.: An improved bound for the embedding of linear spaces into projective planes. Geom. Dedic. 26, 333–340 (1988).
Oehler M.: Endliche bi-affine inzidenzebenen. Geom. Dedic. 4, 419–436 (1975).
Pickert G.: Projektive Ebenen, 2nd edn. Springer, Berlin (1975).
Stanton R.G.: A combinatorial approach to small projective geometries. Bull. Inst. Combin. Appl. 17, 63–70 (1996).
Steinke G.F.: Finite Laguerre near-planes of odd order admitting Desarguesian derivations. Eur. J. Combin. 21, 543–554 (2000).
Totten J.: Embedding the complement of two lines in a finite projective plane. J. Aust. Math. Soc. 22(Series A), 27–34 (1976).
Vedder K.: Generalised elations. Bull. Lond. Math. Soc. 18, 573–579 (1986).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Mitchell.
Rights and permissions
About this article
Cite this article
Steinke, G.F. Collineations of finite 2-affine planes. Des. Codes Cryptogr. 85, 107–120 (2017). https://doi.org/10.1007/s10623-016-0292-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-016-0292-6