By I.G. Macdonald
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Additional resources for Algebraic geometry : introduction to schemes
With minimal k). Then the elements Xi are mutually distinct (except that Xo = X(2k+I)d)' We have 8(xo, Xkd) = kd = 8(xo, X(k+l)d)' by minimality of k. Indeed, if 8(xo, Xkd) = ed, e < k, then e == k mod 2 gives us a circuit oflength no more than (f+k+l)d < (2k+l)d. If f == (k+l) mod 2, then we obtain a circuit of length no more than (k + l)d < (2k + 1)d. Consider an apartment 1: which contains Xo and Xkd+l. Since Xkd+l is not thick (remembering d ~ 2), we infer that Xkd and X(k+l)d also belong to ~.
Then s 2': t, and if r is finite, then p is projective if and -0 only if s = t. 40 Chapter 1. Basic Concepts and Results As a preliminary study, we now take a closer look at the cases n = 4,5,6. 6 Regular points in generalized quadrangles For generalized quadrangles it is obvious that the notions of "distance-2-regular point" and "regular point" coincide. So let p be a regular point in a generalized quadrangle. We define the following geometry r~. The points of r~ are the perps xl- of points x collinear with p.
7. Finite and semi-finite generalized polygons 27 1. 7 Remarks on orders of projective planes Apart from the above results, not too much is known about the orders of finite generalized polygons. All finite projective planes presently known have prime power order. Of course, there is the celebrated result of BRUCK & RYSER  which states that, if the order s of a projective plane is congruent to 1 or 2 modulo 4, then s can be written as the sum of two perfect squares, or, equivalently, the square-free part of n contains no prime divisors equal to 3 modulo 4.