Abstract
In this note, we investigate finite connected flat geometries of type C 3 of uniform odd order in which the lines and the planes form a projective space. We give an algebraic description of such geometries by sets of alternating bilinear forms or, equivalently, by sets of skew symmetric matrices. These sets of skew symmetric matrices will be identified in a natural way with the full point set of a projective plane. By a general result concerning two-valued functions defined on projective planes of order 5, we finally prove the non-existence of finite connected flat geometries of type C 3 of uniform order 5 in which the lines and the planes form a projective space.
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