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The Fundamental Theorem of q-Clan Geometry

  • S. E. Payne

Abstract

Let q be any prime power, F = GF(q). A q-clan is a set C = {A t : t ∈ F}of q, 2 × 2 matrices over F such that their pairwise differences are all anisotropic, i.e., for distinct \( s,t \in F,\left( {a,b} \right)\left( {{A_s} - {A_t}} \right)\left( {\begin{array}{*{20}{c}} a \\ b \\ \end{array} } \right) = 0 \) has only the trivial solution a = b = 0. Starting with a q-clan C, there are at least the following geometries associated with C in a canonical way (cf. [18]): a generalized quadrangle GQ(C) with parameters (q 2 , q); a flock F(C) of a quadratic cone in PG(3, q); a line spread S(C) of PG(3, q); a translation plane T(C) of dimension at most 2 over its kernel. Starting with a natural definition of equivalence for q-clans, the Fundamental Theorem of q-clan geometry (F.T.) interprets the equivalence of q-clans C 1 and C 2 as an isomorphism between G(C 1 ) and G(C 2 ), where G(C i ) is any of the geometries (mentioned above) associated with C i . The F.T. was first recognized in its present form in [1], but it was stated there in detail only for q = 2 e , and the proof was claimed to be only a slightly revised version of the proof given in [14] of an important special case of the F.T. However, the proof in [14] starts off by assuming a fairly technical result from [11] where it is embedded in a more general theory.

Keywords

Fundamental Theorem Prime Power Generalize Quadrangle Translation Plane Collineation Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Kluwer Academic Publishers, Boston 1996

Authors and Affiliations

  • S. E. Payne
    • 1
  1. 1.Deptartment of MathematicsCU-DenverDenverUSA

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