Disjoint circles: a classification

Author:
Gary L. Ebert

Journal:
Trans. Amer. Math. Soc. **232** (1977), 83-109

MSC:
Primary 50D45; Secondary 05B25

DOI:
https://doi.org/10.1090/S0002-9947-1977-0442819-9

MathSciNet review:
0442819

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Abstract | References | Similar Articles | Additional Information

Abstract: For *q* a prime-power, let denote the miquelian inversive plane of order *q*. The classification of certain translation planes of order , called subregular, has been reduced to the classification of sets of disjoint circles in . While R. H. Bruck has extensively studied triples of disjoint circles, this paper is concerned with sets of four or more circles in . In a previous paper, the author has shown (for odd *q*) that the number of quadruples of disjoint circles in is asymptotic to . Hence a judicious approach to the classification problem is to study ``interesting'' quadruples. In general, let be a nonlinear set of *n* disjoint circles in . Let *H* be the subgroup of the collineation group of composed of collineations that permute the 's among themselves, and let *K* be that subgroup composed of collineations fixing each of the . An interesting set of n disjoint circles would be one for which . It is shown that if and only if

() |

When and under mild restrictions on *q*, an algorithm is developed that finds all nonlinear quadruples of disjoint circles satisfying the orthogonality conditions and having nontrivial group *H*. Given such a quadruple, the algorithm determines exactly what group *H* is acting. It is also shown that most quadruples in , for large *q*, do indeed satisfy the conditions . In addition, the cases when or 7 are explored to a lesser degree.

**[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]**R. H. Bruck,*Construction problems in finite projective spaces*, Finite geometric structures and their applications (Centro Internaz. Mat. Estivo (C.I.M.E.), II Ciclo, Bressanone, 1972) Edizioni Cremonese, Rome, 1973, pp. 105–188. MR**0342413****[3]**Peter Dembowski,*Möbiusebenen gerader Ordnung*, Math. Ann.**157**(1964), 179–205 (German). MR**0177344**, https://doi.org/10.1007/BF01362432**[4]**P. Dembowski and D. R. Hughes,*On finite inversive planes*, J. London Math. Soc.**40**(1965), 171–182. MR**0172156**, https://doi.org/10.1112/jlms/s1-40.1.171**[5]**G. L. Ebert,*Translation planes of order*:*asymptotic estimates*, Trans. Amer. Math. Soc. (submitted).**[6]**Heinz Lüneburg,*Die Suzukigruppen und ihre Geometrien*, Springer-Verlag, Berlin-New York, 1965 (German). MR**0207820****[7]**W. F. Orr,*The miquelian inversive plane**and the associated projective planes*, Dissertation, Univ. of Wisconsin, Madison, Wis., 1973.**[8]**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/S0002-9947-1977-0442819-9

Keywords:
Finite miquelian inversive plane,
disjoint circles,
linear sets of circles,
collineation group,
orthogonality,
inversion,
conjugate pairs of points,
linear fractional transformations,
triple transitivity,
cycle structure in symmetric groups,
matrix representation of circles,
nonzero squares in finite fields,
Sylow theorems

Article copyright:
© Copyright 1977
American Mathematical Society