Summary
LetJ be a finite inversive plane of odd orderq. If for at least one pointp ofJ the internal affine planeJ p is Desarguesian, thenJ is Miquelian. Other formulation: the finite Desarguesian affine plane of odd orderq has a unique one point extension; this extension is the Miquelian inversive plane of orderq. It follows that there is a unique inversive plane of orderq, withq∈{3, 5, 7}.
This is a preview of subscription content, log in via an institution to check access.
References
L. Bader: Some new examples of flocks ofQ + (3,q). Geom. Dedicata27 (1988) 213–218
L. Bader, G. Lunardon. On the flocks ofQ +(3,q). Geom. Dedicata29 (1989) 177–183
R.D. Baker, G.L. Ebert: A nonlinear flock in the Minkowski plane of order 11. Congr. Numer.58 (1987) 75–81
A. Bonisoli, G. Korchmáros: Flocks of hyperbolic quadrics and linear groups containing homologies. Geom. Dedicata42 (1992) 295–309
Y. Chen. The Steiner systemS(3, 6, 26). J. Geom.2 (1972) 7–28
P. Dembowski. Finite geometries. Berlin Heidelberg New York: Springer 1968
R.H.F. Denniston. Uniqueness of the inversive plane of order 5. Manus. Math.8 (1973) 11–19
R.H.F. Denniston: Uniqueness of the inversive plane of order 7. Manus. Math.8 (1973) 21–23
N. Durante: Piani inversivi e flock di quadriche, Tesi di Laurea in Matematica 1991–92, Università degli Studi di Napoli e Universiteit Gent
J.C. Fisher, J.A. Thas. Flocks inPG(3,q). Math. Z.169 (1979) 1–11
H.-R. Halder, W. Heise, Einführung in die Kombinatorik. München: Carl Hanser 1976
J.W.P. Hirschfeld: Projective geometries over finite fields. Oxford: Oxford University Press 1975
J.W.P. Hirschfeld: Finite projective spaces of three dimensions, Oxford: Oxford University Press 1985
N.L. Johnson: Flocks of hyperbolic quadrics and translation planes admitting affine homologies. J. Geom.34 (1989) 50–73
J.A. Thas: Flocks of nonsingular ruled quadrics inPG(3,q). Atti Accad. Naz. Lincei59, (1975) 83–85
J.A. Thas: Some results on quadrics and a new class of partial geometries. Simon Stevin55 (1981) 129–139
J.A. Thas: Flocks, maximal exterior sets and inversive planes. In: Finite geometries and combinatorial designs. American Mathematical Society 1990, Providence, RI, pp. 187–218
J.A. Thas: Solution of a classical problem on finite inversive planes. In: Finite buildings, related geometries and applications. Oxford: Oxford University Press 1990, pp. 145–159
E. Witt: Uber Steinersche Systeme. Abh. Math. Sem. Univ. Hamburg12 (1938) 265–275
Author information
Authors and Affiliations
Additional information
Oblatum 23-X-1992 & 24-I-1994
Rights and permissions
About this article
Cite this article
Thas, J.A. The affine planeAG(2,q),q odd, has a unique one point extension. Invent Math 118, 133–139 (1994). https://doi.org/10.1007/BF01231529
Issue Date:
DOI: https://doi.org/10.1007/BF01231529