Abstract
We show that if U is a Buekenhout-Metz unital (with respect to a point P) in any translation plane of order q 2 with kernel containing GF(q), then U has an associated 2-(q2,q+1,q) design which is the point-residual of an inversive plane, generalizing results of Wilbrink, Baker and Ebert. Further, our proof gives a natural, geometric isomorphism between the resulting inversive plane and the (egglike) inversive plane arising from the ovoid involved in the construction of the Buekenhout-Metz unital. We apply our results to investigate some parallel classes and partitions of the set of blocks of any Buekenhout-Metz unital.
Similar content being viewed by others
References
ANDRÉ J.Über nicht-Desarguessche Ebenen mit transitiver Translationgruppe. Math. Z.,60 (1954) 156–186.
BAKER R. D. and EBERT G. L.On Buekenhout-Metz Unitals of Odd Order. J. Comb. Theory, Series A,60 (1992) 67–84.
BRUCK R. H. and BOSE R. C.The Construction of Translation Planes from Protective Spaces. J. Algebra,1 (1964) 85–102.
BUEKENHOUT F.Existence of Unitals in Finite Translation Planes of Order q2 with a Kernel of Order q. Geom. Dedicata,5 (1976) 189–194.
DEMBOWSKI P.Finite Geometries. Springer-Verlag, 1968.
EBERT G. L.On Buekenhout-Metz Unitals of Even Order. Europ. J. Combin.13 (1992) 109–117.
HIRSCHFELD J. W. P.Finite Projective Spaces of Three Dimensions. Oxford University Press, 1985.
KESTENBAND B. C.New families of balanced incomplete block designs with block size q+1. J. Statistical Planning and Inference,30 (1992) 83–91.
METZ R.On a Class of Unitals. Geom. Dedicata,8 (1979) 125–126.
ORR W.F.The Miquelian inversive plane IP(q) and the associated projective planes. PhD thesis, Madison, Wisconsin, 1973.
THAS J. A.Flocks of finite egglike inversive planes. Finite geometric structures and their applications (ed A Barlotti) Ed Cremonese, Roma (1973) 189–191.
WILBRINK H.A Characterization of the Classical Unital. Finite Geometries, Lecture Notes in Pure and Applied Math.,82 Marcel-Dekker, 1983.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Barwick, S.G., O'Keefe, C.M. Unitals and inversive planes. J Geom 58, 43–52 (1997). https://doi.org/10.1007/BF01222925
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01222925