Some generalized quadrangles with parameters \(q^ 2,q\) (Q1075590)

The paper is dedicated to Professor Helmut Wielandt to commemorate his seventy-fifth birthday. The following main result concerning generalized quadrangles is proved: If \(q=p^ e\) is a power of an odd prime p, then (i) there are [(e-1)/2] pairwise nonisomorphic generalized quadrangles with parameters \(q^ 2\), q not isomorphic to any previously known generalized quadrangle in the special case, when \(e>1\), and (ii) there is a generalized quadrangle with parameters \(q^ 2\), q not isomorphic to any quadrangle in (i) nor to any previously known generalized quadrangle in the special case, when \(q>3\) and \(q\equiv \pm 2\) (mod 5). According to this result, the paper follows the previously published paper [J. Comb. Theory, Ser. A 29, 212-219 (1980; Zbl 0465.51007)] due to the author and the paper of \textit{S. E. Payne} [Congr. Numerantium 29, 717-734 (1980; Zbl 0453.05015)].