On AS-configurations, skew-translation generalised quadrangles of even order and a conjecture of Payne

Abstract

Generalised quadrangles, as a special case of generalised polygons, were introduced by Jacques Tits [14] in 1959. About twenty years later Kantor [7] showed that an elation generalised quadrangle is completely characterized by its so-called 4-gonal family or Kantor family.Motivated by the Ahrens and Szekeres Quadrangle [1], Ghinelli [4] established a variation of the procedure of Kantor [7], by introducing the notion of an AS-configuration of order n. Such a configuration gives rise to a skew-translation generalised quadrangle of order (n,n), and conversely, outlined in J. Bamberg, S.P. Glasby, E. Swartz [2]. The only known groups of even order admitting an AS-configuration are elementary abelian 2-groups. Moreover, Payne [8] conjectured that there are no other examples. Indeed, he found a proof for n=4. Moreover it is also true for n=8, compare J. Bamberg, S.P. Glasby, E. Swartz [2].The purpose of this paper is to proveTheoremA group of even order admitting an AS-configuration of order n is elementary abelian, if n is not a square. In other words, the only skew-translation generalised quadrangles of order (n,n), n≡0(mod2), are translation generalised quadrangles, if n is not a square.K. Thas claims in his preprint [12] that he found among other things a proof of Payne's conjecture. Unfortunately, in the main part of his proof there is a gap not easy to fill, [13].