Group theoretic characterizations of Buekenhout-Metz unitals in PG(2, q 2) \mathop{\mathrm{PG}}(2,q^{2})
Giorgio Donati
and Nicola Durante
DOI: 10.1007/s10801-010-0250-8
Abstract
Let G be the group of projectivities stabilizing a unital U \mathcal{U} in PG(2, q 2) \mathop{\mathrm{PG}}(2,q^{2}) and let A, B be two distinct points of U \mathcal{U}. In this paper we prove that, if G has an elation group of order q with center A and a group of projectivities stabilizing both A and B of order a divisor of q - 1 greater than 2( Ö q -1) 2(\sqrt{q}-1), then U \mathcal{U} is an ovoidal Buekenhout-Metz unital. From this result two group theoretic characterizations of orthogonal Buekenhout-Metz unitals are given.
Pages: 401–407
Keywords: keywords unitals; projectivities; elations
Full Text: PDF
References
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2. Abatangelo, V., Larato, B.: A group-theoretical characterization of parabolic Buekenhout-Metz unitals. Boll. Unione Mat. Ital. 5A(7), 195-206 (1991)
3. André, J.: Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe. Math. Z. 60, 156-186 (1954)
4. Baker, R.D., Ebert, G.L.: Intersections of unitals in the Desarguesian plane. Congr. Numer. 70, 87-94 (1990)
5. Baker, R.D., Ebert, G.L.: On Buekenhout-Metz unitals of odd order. J. Comb. Theory Ser. A 60, 67-84 (1992)
6. Barlotti, A.: Un'estensione del teorema di Segre-Kustaaheimo. Boll. Unione Mat. Ital. 10, 498-506 (1955)
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8. Barwick, S.G., Ebert, G.L.: Unitals in Projective Planes. Springer Monographs in Mathematics. Springer, New York (2008)
9. Bruck, R.H., Bose, R.C.: The construction of translation planes from projective spaces. J. Algebra 1, 85-102 (1964)
10. Bruck, R.H., Bose, R.C.: Linear representation of projective planes in projective spaces. J. Algebra 4, 117-172 (1966)
11. Buekenhout, F.: Existence of unitals in finite translation planes of order q2 with a kernel of order q. Geom. Dedic. 5, 189-194 (1976)
12. Casse, L.R.A., O'Keefe, C.M., Penttila, T.: Characterizations of Buekenhout-Metz unitals. Geom. Dedic. 59, 29-42 (1996)
13. Drudge, K.: On the orbits of Singer groups and their subgroups. Electron. J. Comb. 9(1) (2002), Research Paper 15, 10 pp. (electronic)
14. Ebert, G.L.: On Buekenhout-Metz unitals of even order. Eur. J. Comb. 13, 109-117 (1992)
15. Ebert, G.L., Wantz, K.: A group theoretic characterization of Buekenhout-Metz unitals. J. Comb. Des. 4, 143-152 (1996)
16. Hirschfeld, J.W.P.: Finite Projective Spaces of Three Dimensions. Oxford University Press, London (1985)
17. Hirschfeld, J.W.P.: Projective Geometries over Finite Fields. Oxford University Press, London (1998)
18. Hirschfeld, J.W.P., Sz\Acute\Acute onyi, T.: Sets in a finite plane with few intersection numbers and a distinguished point. Discrete Math. 97, 229-242 (1991)
19. Metz, R.: On a class of unitals. Geom. Dedic. 8, 125-126 (1979)
20. O'Keefe, C.M., Penttila, T.: Ovoids in PG(3, 16) are elliptic quadrics. II. J. Geom. 44, 140-159 (1992)
21. Panella, G.: Caratterizzazione delle quadriche di uno spazio (tri-dimensionale) lineare sopra un corpo finito. Boll Unione Mat. Ital. 10, 507-513 (1955)
22. Quinn, C.T., Casse, L.R.A.: Concerning a characterization of Buekenhout-Metz unitals. J. Geom. 52, 159-167 (1995)
23. Segre, B.: Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane. Ann. Mat. Pura Appl.
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