A finite generalized quadrangle is a point-line geometry characterized by the following property: for a point p and a line L, p not incident with L, there exists exactly one line M incident with p and one point t incident with L, such that M meets L in t. Classical examples of finite generalized quadrangles are the geometries of totally isotropic or totally singular vector spaces with relation to a sesquilinear or quadratic form of Witt index 2. Nonclassical finite generalized quadrangles are examples of Ahrens and Szekeres [1]. It is noteworthy that there exists a synthetic geometrical connection between these examples and some classical examples [2].

The paper first reconsiders a method of Kantor [3] to construct generalized quadrangles as group coset geometries. The nonclassical examples of Ahrens and Szekeres are called AS-configurations. Some algebraic characterizations of an AS-configuration are proved. Finally, the above-mentioned geometrical connection is explored in a purely algebraic way. The final theorem states that a symplectic AS-configuration is always classical.

Readers who are familiar with the synthetic study of finite generalized quadrangles will find interesting algebraic results connected with these geometries. Conversely, readers who are familiar with (finite) group theory will find interesting geometrical properties of group coset geometries in this paper. As such, the results are an interesting contribution in the intersection of incidence geometry, group theory, and algebra.