Translation planes of order : asymptotic estimates

Author:
Gary L. Ebert

Journal:
Trans. Amer. Math. Soc. **238** (1978), 301-308

MSC:
Primary 05B25; Secondary 50D30

DOI:
https://doi.org/10.1090/S0002-9947-1978-0480096-4

MathSciNet review:
0480096

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Abstract | References | Similar Articles | Additional Information

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 . 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 enables the author to find an asymptotic estimate of the number of isomorphism classes of translation planes of order which are subregular of index 3 or 4. It is conjectured (and proved for ) that, given a set of *n* disjoint circles in , the numbers of circles disjoint from each of the given *n* circles is asymptotic to . This conjecture, if true, would allow one to estimate the number of subregular translation planes of order with any positive index.

**[1]**R. H. Bruck,*Construction problems of finite projective planes*, Combinatorial Mathematics and its Applications (Proc. Conf., Univ. North Carolina, Chapel Hill, N.C., 1967) Univ. North Carolina Press, Chapel Hill, N.C., 1969, pp. 426–514. MR**0250182****[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**161206**, https://doi.org/10.1016/0021-8693(64)90010-9**[4]**Peter Dembowski,*Möbiusebenen gerader Ordnung*, Math. Ann.**157**(1964), 179–205 (German). MR**177344**, https://doi.org/10.1007/BF01362432**[5]**Heinz Lüneburg,*Die Suzukigruppen und ihre Geometrien*, Springer-Verlag, Berlin-New York, 1965 (German). MR**0207820****[6]**W. F. Orr,*The miquelian inversive plane**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 Laguerregeometrie*, Math. Ann.**110**(1935), no. 1, 753–776 (German). MR**1512968**, https://doi.org/10.1007/BF01448057

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1978-0480096-4

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