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A Family of Unitals in the Hughes Plane

Published online by Cambridge University Press:  20 November 2018

Barbu C. Kestenband
Barbu C. Kestenband*
Affiliation:
New York Institute of Technology, Wheatley Rd., Old Westbury N.Y. 11568, U.S.A.
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We construct a family of unitals in the Hughes plane. We prove that they are not isomorphic with the classical unitals, and in so doing we exhibit a configuration that exists in the latter, but not in the former. This new configurational property of the classical unitals might serve in the future again as an isomorphism test.

A particular instance of our construction has appeared in [11]. But it only concerns itself with the case where the matrix involved is the identity, whereas the present article treats the general case of symmetric matrices over a suitable field. Furthermore, [11] does not answer the isomorphism question. It states that (the English translation is ours) “It remains to be seen whether the unitary designs constructed in this note are isomorphic or not with known designs”.

Keywords

51E2005B30
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 42 , Issue 6 , 01 December 1990 , pp. 1067 - 1083
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Brouwer, A.E., Some unitals on 28 points and their embeddings in projective planes of order 9, Lecture Notes in Mathematics 893, p. 183–188.Google Scholar
2. Buekenhout, F., Existence of unitals in finite translation planes of order q2 with a kernel of order q, Geometriae Dedicate 5 (1976), p. 189–194.Google Scholar
3. Dembowski, P., Finite Geometries, Springer-Verlag N.Y., Inc., 1968.10.1007/978-3-642-62012-6CrossRefGoogle Scholar
4. Ganley, M.J., A class of unitary block designs, Math. Zeitschrift 128, p. 34-42 (1972).Google Scholar
5. Hughes, D.R., Piper, F.C., Projective Planes, Springer-Verlag, 1973.Google Scholar
6. Metz, R., On a class of unitals, Geometriae Dedicata 8 (1979), p. 125-126.Google Scholar
7. O'Nan, M.E., Automorphisms of unitary block designs, Journal of Algebra 20 (1972), p. 495- 511.Google Scholar
8. Piper, F.C., Polarities of finite projective planes, Atti del Convegno di Geometria Combinatoria e sue Applicazioni (univ. degli studi di Perugia, Perugia) 1970, p. 373-376,1st. Mat. Univ. Perugia, Perugia 1971.Google Scholar
9. Room, T.G., Polarities and ovals in the Hughes plane, J. Austr. Math. Soc. 13 (1972), p. 196–204.Google Scholar
10. Rosati, L.A., I gruppi di collineazioni dei piani di Hughes, Boll. Un. Mat. Ital. 13 (1958), p. 505–513.Google Scholar
11. Rosati, L.A., Disegni unitari nei piani di Hughes, Geometriae Dedicata 27 (1988), p. 295–299.Google Scholar