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Transactions of the American Mathematical Society

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Translation planes of order $ q\sp{2}$: asymptotic estimates

Author: Gary L. Ebert
Journal: Trans. Amer. Math. Soc. 238 (1978), 301-308
MSC: Primary 05B25; Secondary 50D30
MathSciNet review: 0480096
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Abstract: R. H. Bruck has pointed out the one-to-one correspondence between the isomorphism classes of certain translation planes, called subregular, and the equivalence classes of disjoint circles in a finite miquelian inversive plane $ IP(q)$. The problem of determining the number of isomorphism classes of translation planes is old and difficult. Let q be an odd prime-power. In this paper, a study of sets of disjoint circles in $ IP(q)$ enables the author to find an asymptotic estimate of the number of isomorphism classes of translation planes of order $ {q^2}$ which are subregular of index 3 or 4. It is conjectured (and proved for $ n \leqslant 3$) that, given a set of n disjoint circles in $ IP(q)$, the numbers of circles disjoint from each of the given n circles is asymptotic to $ {q^3}/{2^n}$. This conjecture, if true, would allow one to estimate the number of subregular translation planes of order $ {q^2}$ with any positive index.

References [Enhancements On Off] (What's this?)

  • [1] R. H. Bruck, Construction problems of finite projective planes, Combinatorial Mathematics and its Applications, Univ. of North Carolina Press, Chapel Hill, 1969, pp. 426-514. MR 0250182 (40:3422)
  • [2] -, Construction problems in finite projective spaces, Finite Geometric Structures and Their Applications, Centro Internazionale Matematico Estivo, Bressanone, 1972, pp 107-188.
  • [3] R. H. Bruck and R. C. Bose, The construction of translation planes from projective spaces, J. Algebra 1 (1964), 85-102. MR 0161206 (28:4414)
  • [4] P. Dembowski, Möbiusebenen gerader Ordnung, Math. Ann. 157 (1964), 179-205. MR 0177344 (31:1607)
  • [5] H. Luneburg, Die Suzukigruppen und ihre Geometrien, Lecture Notes in Math., no. 10, Springer-Verlag, Berlin and New York, 1965. MR 0207820 (34:7634)
  • [6] W. F. Orr, The miquelian inversive plane $ IP(q)$ and the associated projective planes, Dissertation, Univ. of Wisconsin, Madison, Wisconsin, 1973.
  • [7] B. L. Van der Waerden and L. J. Smid, Eine Axiomatik der Kreisgeometrie und der Laguerre-geometrie, Math. Ann. 110 (1935), 753-776. MR 1512968

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Keywords: Subregular translation planes, finite miquelian inversive planes, disjoint circles, bundle, pencil, inversion, conjugate pairs of points, linear sets of circles, projective linear group, asymptotic estimates
Article copyright: © Copyright 1978 American Mathematical Society

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