Projektive G-Faserräume

Abstract

Let σ > 0 be an integer. A projective σ-fibre space is formed by a covering of a projective geometry with σ-1 isomorphic geometries. The double elliptic space (Sphärischer Raum) is an example of a 2-fibre space. This note deals with projective σ-fibre spaces which are structured by multy valued orderfunctions (This notion was introduced by W. JUNKERS [2] for projective geometries) the range of which is a group G. If such an ordered σ-fibre space has the property ¦G¦=σ, it is called projective G-fibre space. It is proved that the desarguesian projective G-fibre spaces V are exactly those, which are induced by a vector space S over a field K (commutative or not) having a normal subgroup P ⊲ K^*(·) with K^*′ ⊂P such that G≅K^*/P and S≅V^*/P. This theorem is a generalization of the well-known case P=K^*.