Hall plane of order 9
The Hall plane of order 9 is one of the three smallest examples of a finite non-Desarguesian projective plane, along with its dual and the Hughes plane of order 9. As its name suggests, it is a Hall plane, and is the smallest member of that family.
Construction
While usually constructed in the same way as other Hall planes, the Hall plane of order 9 was actually found earlier by Oswald Veblen and Joseph Wedderburn in 1907.[1] There are four quasifields of order nine which can be used to construct the Hall plane of order nine. Three of these are Hall systems generated by the irreducible polynomials [math]\displaystyle{ f(x) = x^2 + 1 }[/math], [math]\displaystyle{ g(x) = x^2 - x - 1 }[/math] or [math]\displaystyle{ h(x) = x^2 + x - 1 }[/math].[2] The first of these produces an associative quasifield,[3] that is, a near-field, and it was in this context that the plane was discovered by Veblen and Wedderburn. This plane is often referred to as the nearfield plane of order nine.
Properties
Automorphism Group
The Hall plane of order 9 is the unique projective plane, finite or infinite, which has Lenz-Barlotti class IVa.3.[4] Its automorphism group acts on its (necessarily unique) translation line imprimitively, having 5 pairs of points that the group preserves set-wise; the automorphism group acts as [math]\displaystyle{ S_5 }[/math] on these 5 pairs.[5]
Unitals
The Hall plane of order 9 admits four inequivalent embedded unitals.[6] Two of these unitals arise from Buekenhout's[7] constructions: one is parabolic, meeting the translation line in a single point, while the other is hyperbolic, meeting the translation line in 4 points. The latter of these two unitals was shown by Grüning[8] to also be embeddable in the dual Hall plane. Another of the unitals arises from the construction of Barlotti and Lunardon.[9] The fourth has an automorphism group of order 8 isomorphic to the quaternions, and is not part of any known infinite family.
References
- ↑ Veblen, Oscar; Wedderburn, Joseph H.M. (1907), "Non-Desarguesian and non-Pascalian geometries", Transactions of the American Mathematical Society 8: 379–388, doi:10.2307/1988781, http://www.ams.org/tran/1907-008-03/S0002-9947-1907-1500792-1/S0002-9947-1907-1500792-1.pdf
- ↑ Stevenson, Frederick W. (1972), Projective Planes, San Francisco: W.H. Freeman and Company, ISBN 0-7167-0443-9 Pages 333–334.
- ↑ D. Hughes and F. Piper (1973). Projective Planes. Springer-Verlag. ISBN 0-387-90044-6. Page 186.
- ↑ Dembowski, Peter (1968). Finite Geometries : Reprint of the 1968 Edition. Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN 978-3-642-62012-6. OCLC 851794158. https://www.worldcat.org/oclc/851794158. Page 126.
- ↑ André, Johannes (1955-12-01). "Projektive Ebenen über Fastkörpern" (in de). Mathematische Zeitschrift 62 (1): 137–160. doi:10.1007/BF01180628. ISSN 1432-1823. https://doi.org/10.1007/BF01180628.
- ↑ Penttila, Tim; Royle, Gordon F. (1995-11-01). "Sets of type (m, n) in the affine and projective planes of order nine" (in en). Designs, Codes and Cryptography 6 (3): 229–245. doi:10.1007/BF01388477. ISSN 1573-7586. https://doi.org/10.1007/BF01388477.
- ↑ Buekenhout, F. (July 1976). "Existence of unitals in finite translation planes of order [math]\displaystyle{ q^2 }[/math] with a kernel of order [math]\displaystyle{ q }[/math]" (in en). Geometriae Dedicata 5 (2). doi:10.1007/BF00145956. ISSN 0046-5755. http://link.springer.com/10.1007/BF00145956.
- ↑ Grüning, Klaus (1987-06-01). "A class of unitals of order [math]\displaystyle{ q }[/math] which can be embedded in two different planes of order [math]\displaystyle{ q^2 }[/math]" (in en). Journal of Geometry 29 (1): 61–77. doi:10.1007/BF01234988. ISSN 1420-8997. https://doi.org/10.1007/BF01234988.
- ↑ Barlotti, A.; Lunardon, G. (1979). "Una classe di unitals nei [math]\displaystyle{ \Delta }[/math]-piani". Rivisita di Matematica della Università di Parma 4: 781–785.
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