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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Disjoint circles: a classification

Author: Gary L. Ebert
Journal: Trans. Amer. Math. Soc. 232 (1977), 83-109
MSC: Primary 50D45; Secondary 05B25
MathSciNet review: 0442819
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Abstract: For q a prime-power, let $ {\text{IP}}(q)$ denote the miquelian inversive plane of order q. The classification of certain translation planes of order $ {q^2}$, called subregular, has been reduced to the classification of sets of disjoint circles in $ {\text{IP}}(q)$. While R. H. Bruck has extensively studied triples of disjoint circles, this paper is concerned with sets of four or more circles in $ {\text{IP}}(q)$. In a previous paper, the author has shown (for odd q) that the number of quadruples of disjoint circles in $ {\text{IP}}(q)$ is asymptotic to $ {q^{12}}/1536$. Hence a judicious approach to the classification problem is to study ``interesting'' quadruples. In general, let $ {C_1}, \ldots ,{C_n}$ be a nonlinear set of n disjoint circles in $ {\text{IP}}(q)$. Let H be the subgroup of the collineation group of $ {\text{IP}}(q)$ composed of collineations that permute the $ {C_i}$'s among themselves, and let K be that subgroup composed of collineations fixing each of the $ {C_i}'s$. An interesting set of n disjoint circles would be one for which $ K = 1$. It is shown that $ K = 1$ if and only if

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{\text{(i)}}\;{\text{there}}\;{... ...}}\;{\text{other}}\;n - 1\;{\text{circles}}{\text{.}}} \\ \end{array} } \right.$ ($ *$)

When $ n = 4$ and under mild restrictions on q, an algorithm is developed that finds all nonlinear quadruples of disjoint circles satisfying the orthogonality conditions $ ( \ast )$ 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 $ {\text{IP}}(q)$, for large q, do indeed satisfy the conditions $ ( \ast )$. In addition, the cases when $ n = 5,6,$ or 7 are explored to a lesser degree.

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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

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