# 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
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## References

1. 1.
L. Bader, G. Lunardon, and S. E. Payne, On q-clan geometry, q = 2e, Bull. Belgian Math. Soc, Simon Stevin, Vol. 1 (1994) pp. 301–328.
2. 2.
L. Bader, G. Lunardon, and J. A. Thas, Derivation of flocks of quadratic cones, Forum Math., Vol. 2 (1990) pp. 163–194.
3. 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.
4. 4.
H. Gevaert and N. L. Johnson, Flocks of quadratic cones, generalized quadrangles and translation planes, Geom. Dedicata, Vol. 27 (1988) pp. 301–317.
5. 5.
J. W. P. Hirschfeld, Projective Geometries over Finite Fields, Oxford University Press, Oxford (1979).
6. 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. 7.
W. M. Kantor, Generalized quadrangles associated with G 2(q) .J. Combin. Theory (A), Vol. 29 (1980) pp. 212–219.
8. 8.
W. M. Kantor, Some generalized quadrangles with parameters (q 2, q), Math. Zeit., Vol. 192 (1986) pp. 45–50.
9. 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. 10.
S. E. Payne, A new infinite family of generalized quadrangles, Congressus Numerantium, Vol. 49 (1985) pp. 115–128.
11. 11.
S. E. Payne, An essay on skew translation generalized quadrangles, Geom. Dedicata, Vol. 32 (1989) pp. 93–118.
12. 12.
S. E. Payne, Collineations of the generalized quadrangles associated with q-clans, Annals of Discrete Math., Vol. 52 (1992) pp. 449–461.
13. 13.
S. E. Payne, Collineations of the Subiaco generalized quadrangles, Bull. Belgian Math. Soc., Simon Stevin Vol. 1(1994) pp. 427–438.
14. 14.
S. E. Payne and L. A. Rogers, Local group actions on generalized quadrangles, Simon Stevin, Vol. 64 (1990) pp. 249–284.
15. 15.
S. E. Payne and J. A. Thas, Conical flocks, partial flocks, derivation and generalized quadrangles, Geom. Dedicata, Vol. 38 (1991) pp. 229–243.
16. 16.
S. E. Payne and J. A. Thas, Finite generalized quadrangles, Pitman, (1984).
17. 17.
S. E. Payne and J. A. Thas, Generalized quadrangles, BLT-sets, and Fisher flocks, Congressus Numerantium, Vol. 84 (1991) pp. 161–192.
18. 18.
J. A. Thas, Generalized quadrangles and flocks of cones, European J. Combin., Vol. 8 (1987) pp. 441–452.