A Theorem Concerning Nets Arising from Generalized Quadrangles with a Regular Point

Abstract

Suppose \mathcal{S} is a generalized quadrangle (GQ) of order (s, t), s, t \ne 1, with a regular point. Then there is a net which arises from this regular point. We prove that if such a net has a proper subnet with the same degree as the net, then it must be an affine plane of order t. Also, this affine plane induces a proper subquadrangle of order t containing the regular point, and we necessarily have that s = t^2. This result has many applications, of which we give one example. Suppose \mathcal{S} is an elation generalized quadrangle (EGQ) of order (s, t), s, t \ne 1, with elation point p. Then \mathcal{S} is called a skew translation generalized quadrangle (STGQ) with base-point p if there is a full group of symmetries about p of order t which is contained in the elation group. We show that a GQ \mathcal{S} of order s is an STGQ with base-point p if and only if p is an elation point which is regular.