On the foundations of polar geometry

Geometry of Polar Grassmann Spaces

“…It holds true except for one special case (3) and, roughly speaking, it states that automorphisms of <Sfc, automorphisms of any of adjacencies, automorphisms of perpendicularity, and automorphisms of 2Jt acting on subspaces of SUt all coincide 1 . Besides, this exceptional case (3), which corresponds to the geometry of singular lines in a 7-dimensional metric projective space with maximal singular subspaces being 3-subspaces seems to be interesting in itself.…”

A characterization of the symplectic groups PSp(2m, q) as rank 3 permutation groups

“…This fact implies that G a is a rank 3 permutation group on {R(ab): b G a λ -α}, and the set of "totally singular lines" carry q + 1 points. We then show that X together with its totally singular lines forms a nondegenerate Shult space [1] of rank ^ 3. Next we use a theorem of Buekenhout and Shult [1] to conclude that X is isomorphic to the set of points of a classical geometry of symplectic type.…”

“…By Proposition 3.7 and Theorem 4 of Buekenhout and Shult [1], X is a polar space of rank ^ 3 in which lines carry q + 1^3 points. Since \X\ = v r is finite, by Theorem 1 of Buekenhout and Shult [1], X is isomorphic to the set of singular points of a classical symplectic, unitary or orthogonal geometry. Because a line of X carries q + 1 points and corresponds to a totally singular line of a classical geometry, it follows that q is a prime power.…”

On Hyperplanes of the Geometry D4,2 and their Related Codes

“…This study presented a description for two classes of hyperplanes of point-line geometry of type D 4,2 which was characterized completely in [7] . For the following definitions [6] . A given set I, geometry Γ over I is an ordered triple Γ= (X, D), where X is a set, D is a partition {X i } of X indexed by I, X i are called components,  is a symmetric and reflexive relation on X called incidence relation such that: A point-line geometry (P, L) is simply a geometry for which I = 2, one of the two types is called points, in this notation the points are the members of Pand the other type is called lines.…”