Disjoint circles: a classification
Gary L. Ebert
Trans. Amer. Math. Soc. 232 (1977), 83-109
Primary 50D45; Secondary 05B25
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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.
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Finite miquelian inversive plane,
linear sets of circles,
conjugate pairs of points,
linear fractional transformations,
cycle structure in symmetric groups,
matrix representation of circles,
nonzero squares in finite fields,
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