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

Abstract

Suppose $\mathcal{S}$ is a generalized quadrangle (GQ) of order $\left( {s,t} \right),s,t \ne 1$, 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 $\left( {s,t} \right),s,t \ne 1$, 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.