INTRODUCTION

Near 2n-gons, as defined by Shult and Yanushka [9], form a class of geometries including the familiar generalised 2n-gons. It also includes the dual polar spaces, for which a system of axioms was given by Cameron [3] and refined by Shult [8].

The concept of a flat embedding of a geometry in a projective space is abstracted from the classification of antiflag transitive collineation groups of projective spaces by Cameron and Kantor [4]. Geometries possessing flat embeddings include symplectic geometries, generalised hexagons of type G2(q), and dual orthogonal geometries of type 0(2n+l, q). The embeddings in the last case can be constructed using spinors. Note that the second and third types are near 2n-gons.

This paper is directed towards determining the flat embeddings of near 2n-gons. Theorems 4.1 and 4.2 give this determination under additional hypotheses. As preliminaries, a summary of the theory of near 2n-gons and the axioms for dual polar spaces is given in Section 2, and the embedding theorem of Cameron and Kantor is stated in Section 3. The final section gives an account of the spinor embedding of dual 0(2n+l, q).

I am grateful to A.L. Wells Jr. for discussions about the topic of this paper.