Symplectic geometries, transvection groups, and modules (Q1317454)

Let \((P,L)\) be a connected partial linear space in which any pair of intersecting lines is contained in a subspace isomorphic to a symplectic plane. Assume that \((P,L)\) contains at least two such subspaces and a line with more than three points. The author extends to infinite spaces a result of J. I. Hall. He proves that \((P,L)\) is isomorphic to the geometry of hyperbolic lines of a symplectic geometry. Then he uses this to characterize the subgroups of the symplectic group that are generated by transvection subgroups. He also gives a characterization of the natural modules for this groups.