Finite Generalized Quadrangles (Ems Series of Lectures in by Stanley E. Payne and Joseph A. Thas

By Stanley E. Payne and Joseph A. Thas

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Ahrens and G. Szekeres. 3; q/: (i) x D , y D a, z D b; denoted by Œ ; a; b. (ii) x D a, y D , z D b; denoted by Œa; ; b. (iii) x D c 2 b C a, y D 2c C b, z D ; denoted by Œc; b; a. q/. c; b/. q/g. P 0 ; B 0 ; I0 / in the following way. c; b/, and the points of P . The elements of B 0 are the lines of type (ii) and (iii) of B, the elements Œc, and Œ1. q/, u I0 L iff u I L for all u 2 P and all lines L of type (ii) or (iii) of B. It is easily checked that each point of P 0 is incident with q C 1 lines of B 0 , and each line of B 0 is incident with q C 1 points of P 0 .

1 C t / is an eigenvalue of QM . From m0 D 1 it follows that this eigenvalue of QM has multiplicity 1. s C t /. For each divisor d of n, let Ud denote the sum of all primitive d -th roots of unity. Then Ud is an integer [87]. For each divisor d of n, the primitive d -th roots of unity all contribute the same number of times to eigenvalues  of QM with j j D s C t . s C t / as an eigenvalue P of QM , with d jn and d a primitive d -th root of unity. 1 C t /. mod s C t /. Let f and g be as given in the theorem.

Since y I y  , clearly y  D xy D L D x  , implying x D y, a contradiction. So each line of S is incident with at most one absolute point of  . A line L is absolute iff L I L iff L is absolute. , L I L . If L I M I u I L, then L I u I M  I L, hence u D M and M  D u. Consequently u and M are absolute, and we have proved that each line L is incident with at least one absolute point. It follows that the set of absolute points of  is an ovoid. Dually, the set of all absolute lines is a spread.

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