Unitals in Finite Desarguesian Planes
A. Cossidente
, G.L. Ebert2
and G. Korchmáros3
2Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA G. KORCHM ÁROS
DOI: 10.1023/A:1011981711246
Abstract
We show that a suitable 2-dimensional linear system of Hermitian curves of PG(2, q 2) defines a model for the Desarguesian plane PG(2, q). Using this model we give the following group-theoretic characterization of the classical unitals. A unital in PG(2, q 2) is classical if and only if it is fixed by a linear collineation group of order 6( q + 1) 2 that fixes no point or line in PG(2, q 2).
Pages: 119–125
Keywords: unitals; Hermitian curves; Desarguesian planes; unitary groups
Full Text: PDF
References
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2. V. Abatangelo, “On Buekenhout-Metz unitals in PG(2, q2), q even,” Arch. Math. 59 (1992), 197-203.
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4. E.F. Assmus, Jr. and J.D. Key, Designs and their Codes, Cambridge Tracts in Mathematics, Vol. 103, Cambridge University Press, Cambridge, 1992.
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9. J. Cannon and C. Playoust, An Introduction to MAGMA, University of Sidney, Sidney, Australia, 1993.
10. A. Cossidente, G.L. Ebert, and G. Korchmáros, “A group-theoretic characterization of classical unitals,” Arch. Math. 74 (2000), 1-5.
11. P. Dembowski, Finite Geometries, Springer-Verlag, 2nd edition, 1997.
12. G.L. Ebert, Hermitian arcs, Special Issue entitled “Recent Progress in Geometry,” E. Ballico and G. Korchmáros (Eds.), Rend. Circolo Mat. Palermo III 51 (1998), 87-105.
13. G.L. Ebert and K. Wantz, “A group theoretic characterization of Buekenhout-Metz unital,” J. Combin. Designs 4 (1996), 143-152.
14. W. Feit and J.G. Thompson, “Solvability of groups of odd order,” Pac. Jour. Math. 13 (1963), 775-1029.
15. R.W. Hartley, “Determination of the ternary collineation groups whose coefficients lie in GF(2n),” Ann. of Math. 27 (1926), 140-158.
16. C. Hering, “On the structure of finite collineation groups of projective planes,” Abh. Math. Sem. Univ. Hamburg 49 (1979), 155-182.
17. A.R. Hoffer, “On unitary collineation groups,” J. Algebra 22 (1972), 211-218.
18. J.W.P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford University Press, Oxford, 1985.
19. H.H. Mitchell, “Determination of the ordinary and modular ternary linear groups,” Trans. Amer. Math. Soc. 12 (1911), 207-242.
20. A.D. Thomas and G.V. Wood, Group Tables, Shiva Publishing Co., Kent, 1980.
2. V. Abatangelo, “On Buekenhout-Metz unitals in PG(2, q2), q even,” Arch. Math. 59 (1992), 197-203.
3. V. Abatangelo and B. Larato, “A group theoretic characterization of parabolic Buekenhout-Metz unitals,” Boll. U.M.I. 5-A(7) (1991), 195-206.
4. E.F. Assmus, Jr. and J.D. Key, Designs and their Codes, Cambridge Tracts in Mathematics, Vol. 103, Cambridge University Press, Cambridge, 1992.
5. M. Biliotti and G. Korchmáros, “Collineation groups preserving a unital of a projective plane of odd order,” J. Algebra 122 (1989), 130-149.
6. M. Biliotti and G. Korchmáros, “Collineation groups preserving a unital of a projective plane of even order,” Geom. Dedicata 31 (1989), 333-344.
7. A. Blokhuis, A. Brouwer, and H. Wilbrink, “Hermitian unitals are code words,” Discrete Math. 97 (1991), 63-68.
8. D.M. Bloom, “The subgroups of PSL(3, q) for q odd,” Trans. Amer. Math. Soc. 127 (1967), 150-178.
9. J. Cannon and C. Playoust, An Introduction to MAGMA, University of Sidney, Sidney, Australia, 1993.
10. A. Cossidente, G.L. Ebert, and G. Korchmáros, “A group-theoretic characterization of classical unitals,” Arch. Math. 74 (2000), 1-5.
11. P. Dembowski, Finite Geometries, Springer-Verlag, 2nd edition, 1997.
12. G.L. Ebert, Hermitian arcs, Special Issue entitled “Recent Progress in Geometry,” E. Ballico and G. Korchmáros (Eds.), Rend. Circolo Mat. Palermo III 51 (1998), 87-105.
13. G.L. Ebert and K. Wantz, “A group theoretic characterization of Buekenhout-Metz unital,” J. Combin. Designs 4 (1996), 143-152.
14. W. Feit and J.G. Thompson, “Solvability of groups of odd order,” Pac. Jour. Math. 13 (1963), 775-1029.
15. R.W. Hartley, “Determination of the ternary collineation groups whose coefficients lie in GF(2n),” Ann. of Math. 27 (1926), 140-158.
16. C. Hering, “On the structure of finite collineation groups of projective planes,” Abh. Math. Sem. Univ. Hamburg 49 (1979), 155-182.
17. A.R. Hoffer, “On unitary collineation groups,” J. Algebra 22 (1972), 211-218.
18. J.W.P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford University Press, Oxford, 1985.
19. H.H. Mitchell, “Determination of the ordinary and modular ternary linear groups,” Trans. Amer. Math. Soc. 12 (1911), 207-242.
20. A.D. Thomas and G.V. Wood, Group Tables, Shiva Publishing Co., Kent, 1980.