## 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## Preview

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## References

- 1.L. Bader, G. Lunardon, and S. E. Payne, On q-clan geometry,
*q*= 2^{e},*Bull. Belgian Math. Soc, Simon Stevin*, Vol. 1 (1994) pp. 301–328.MathSciNetMATHGoogle Scholar - 2.L. Bader, G. Lunardon, and J. A. Thas, Derivation of flocks of quadratic cones,
*Forum Math*., Vol. 2 (1990) pp. 163–194.MathSciNetMATHCrossRefGoogle Scholar - 3.F. De Clerck, H. Gevaert, and J. A. Thas, Flocks of a quadratic cone in
*PG(3, q), q ≤ 8, Geom. Dedicata*, Vol. 26 (1988) pp. 215–230.MATHGoogle Scholar - 4.H. Gevaert and N. L. Johnson, Flocks of quadratic cones, generalized quadrangles and translation planes,
*Geom. Dedicata*, Vol. 27 (1988) pp. 301–317.MathSciNetMATHCrossRefGoogle Scholar - 5.J. W. P. Hirschfeld,
*Projective Geometries over Finite Fields*, Oxford University Press, Oxford (1979).MATHGoogle Scholar - 6.N. L. Johnson, Derivation of partial flocks of quadratic cones,
*Rend. Mat. Appl*. (7) Vol. 12, No. 4 (1992) (1993) pp. 817–848.Google Scholar - 7.W. M. Kantor, Generalized quadrangles associated with
*G*_{2}(*q*) .*J. Combin. Theory*(A), Vol. 29 (1980) pp. 212–219.MathSciNetMATHCrossRefGoogle Scholar - 8.W. M. Kantor, Some generalized quadrangles with parameters (
*q*^{2},*q*),*Math. Zeit*., Vol. 192 (1986) pp. 45–50.MathSciNetMATHCrossRefGoogle Scholar - 9.G. Lunardon, A remark on the derivation of flocks,
*Advances in Finite Geometries and Designs*(eds. J. W. P. Hirschfeld, et al.), Oxford University Press (1991) pp. 299–309.Google Scholar - 10.S. E. Payne, A new infinite family of generalized quadrangles,
*Congressus Numerantium*, Vol. 49 (1985) pp. 115–128.MathSciNetGoogle Scholar - 11.S. E. Payne, An essay on skew translation generalized quadrangles,
*Geom. Dedicata*, Vol. 32 (1989) pp. 93–118.MathSciNetMATHCrossRefGoogle Scholar - 12.S. E. Payne, Collineations of the generalized quadrangles associated with
*q*-clans,*Annals of Discrete Math*., Vol. 52 (1992) pp. 449–461.CrossRefGoogle Scholar - 13.S. E. Payne, Collineations of the Subiaco generalized quadrangles,
*Bull. Belgian Math. Soc., Simon Stevin*Vol. 1(1994) pp. 427–438.MATHGoogle Scholar - 14.S. E. Payne and L. A. Rogers, Local group actions on generalized quadrangles,
*Simon Stevin*, Vol. 64 (1990) pp. 249–284.MathSciNetMATHGoogle Scholar - 15.S. E. Payne and J. A. Thas, Conical flocks, partial flocks, derivation and generalized quadrangles,
*Geom. Dedicata*, Vol. 38 (1991) pp. 229–243.MathSciNetMATHCrossRefGoogle Scholar - 16.S. E. Payne and J. A. Thas,
*Finite generalized quadrangles*, Pitman, (1984).MATHGoogle Scholar - 17.S. E. Payne and J. A. Thas, Generalized quadrangles, BLT-sets, and Fisher flocks,
*Congressus Numerantium*, Vol. 84 (1991) pp. 161–192.MathSciNetGoogle Scholar - 18.J. A. Thas, Generalized quadrangles and flocks of cones
*, European J. Combin*., Vol. 8 (1987) pp. 441–452.MathSciNetMATHGoogle Scholar