On the structure of finite collineation groups containing symmetries of generalized quadrangles

Abstract

In this paper we begin to investigate the structure of a generalized quadrangle (~, ~2) which admits a finite collineation group G containing non-identity symmetries. Of course, when starting such an investigation, it is natural to concentrate attention on those cases where G is assumed to satisfy certain irreducibility criteria. The nature of these possible restrictions on the action of G is suggested by examining the substructures of (~, ~). This is done in Section 2. In Section 4 we consider what can be said about the composition series for G. Under relatively weak irreducibility conditions reasonably detailed information is obtained. This is contained in Theorem (4.1). These results are then used in the final section of this article to tackle the problem when both non-trivial axial and central symmetries are present in G. This places surprisingly severe restrictions on the geometry and, consequently, our strongest result is obtained here as (5.1). As a corollary to this theorem we obtain the following characterization: