Showing papers in "Journal of Geometry in 1971"
35 citations
17 citations
TL;DR: In this article, the authors considered those hexagons in which at least two vertices coincide and showed that if every hexagon in this class of 5-point hexagons is Pascalian, then the oval has the PASCALIAN property.
Abstract: An oval C In a projective plane is a set of triply noncollinear points such that each point of C is on exactly one line which contains no other point of C. An oval has the Pascalian property if each of its hexagons has collinear diagonal points. The author considers those hexagons in which at least two vertices coincide and shows that if every hexagon in this class of 5-point hexagons is Pascalian then the oval has the Pascalian property. Therefore, the 5-point Pascalian property is equivalent to the full Pascalian property. The proof makes use of the coordinatization found in Artzy [1, 2].
12 citations
TL;DR: In this paper, it is shown that for desarguesian affine spaces the group is a vectorspace and the partition is the set of all onedimensional subspaces.
Abstract: Translationstructures are generalized affine spaces. They can be described algebraically by partitions of groups. For desarguesian affine spaces the group is a vectorspace and the partition is the set of all onedimensional subspaces. In this case each collineation fixing 0 is a regular semilinear mapping, i.e. an automorphism of the vectorspace. In the general case it is a mapping called equivalence. Each equivalence of a partition is an automorphism iff the set of translations of the group is a normal subgroup of the collineationgroup. The translations form a normal subgroup, if the group is finite or abelian. We prove some theorems for the infinite non abelian case.
11 citations
TL;DR: In this paper, the affine line geometry of free modules over rings is characterized geometrically, and the way to solve this problem leads to the concept of the so called "affine line geometry".
Abstract: This paper is concerned with the problem how to characterize geometrically the affine structure of free modules over rings (associative, with unit-element). The way to solve this problem leads to the concept of the so called “affine line-geometry”.
9 citations
TL;DR: In this article, a characterisation of reflection groups by group spaces is given, by replacing one of the axioms by its negation, and an algebraic description of these geometries is given.
Abstract: Recently H. Karzel gave a characterisation of reflection groups by group spaces. We obtain new geometries by replacing one of the axioms by its negation. An algebraic description of these geometries is given.
4 citations
TL;DR: In this article, the relation between the geometrical ordering of a Miquelian Mobius-plane and the ordering of the underlying field K is investigated, and it is shown that every semi-ordering of K induces an order function of M(K, Q).
Abstract: In this paper we investigate the relations between the geometrical ordering of a Miquelian Mobius-planeM(K, Q) and the ordering of the underlying field K. Every semi-ordering of K induces an order function ofM(K, Q), and conversely, if ¦k¦ >2, every semi-ordering of K is induced by exactly two (∞)-normal order functions ofM(K, Q). Moreover the normal order functions correspond to. those semi-orderings > of K for which is Q(x, y) >0 for all (x, y) e (K × K){(0, 0)}. Full orderings ofM(K, Q) are defined similar to those in projective geometry.
4 citations
TL;DR: In this article, it was proved that any collineation defined on a quasi-anchor can be extended to a collining of the T-closure of a setA in a projective translation plane is defined as the image of A under the group generated by translationsτ such that there exist proper points A, B withτ(A) = B.
Abstract: The T-closure of a setA in a projective translation plane is defined as the image ofA under the group generated by translationsτ such that there exist proper points A, B withτ(A) = B. The sets, called quasi-anchors, make up the concepts of anchors introduced in [11]. It is proved that any collineation defined on a quasi-anchor can be extended to a collineation of the T-closure ofA. An application to the problem of uniqueness of the nomography is provided in a special case.
3 citations
3 citations
TL;DR: In this paper, a finite group G is generated by a subsetS of involutions satisfying the theorem of the three reflections: ifa,b,x,y,z ∈ S, ab ≠ 1 and ifabx,aby,abz are involutions, thenxyz ∆ S. ThenS is a class of conjugate elements of G if and only ifG/Z(G) is a nonabelian simple group.
Abstract: LetG be a finite group which is generated by a subsetS of involutions satisfying the theorem of the three reflections: Ifa,b,x,y,z ∈ S, ab ≠ 1 and ifabx,aby,abz are involutions, thenxyz ∈ S. Assume thatS contains three elements which generate a four-group. ThenS is a class of conjugate elements ofG if and only ifG/Z(G) is a non-abelian simple group. Moreover,G/Z(G) is a nonabelian simple group ifG is not isomorphic to any PGL2(n).
2 citations
TL;DR: In this paper, the results of applications of solutions of functional equations or of methods used in the theory of functional equation to the following subjects are discussed: 1. Extensions of homomorphisms from sub-semigroups to groups generated by them. 2. Determination of all Cremona transformations which reduce linear transformations with triangular matrices to translations.
Abstract: The results of applications of solutions of functional equations or of methods used in the theory of functional equations to the following subjects are discussed in this paper. Determination of all Cremona transformations which reduce linear transformations with triangular matrices to translations. One-parameter subsemigroups of affine transformations and their homomorphisms. Extensions of homomorphisms from sub-semigroups to groups generated by them. Determination of all collineations on subsets of general projective planes and their extensions to the entire plane.
TL;DR: In this article, a 3-net is mapped onto itself in a way which also induces a map of a coordinatizing loop onto another, "isostrophic" loop (Q*, O).
Abstract: When in a 3-net the three line pencils are permuted, then with some additional requirements the net is mapped onto itself in a way which also induces a map of a coordinatizing loop (Q*), onto another, “isostrophic” loop (Q*, O). Every identity ab = aOb induces a net collineation and, simultaneously, a loop law. This procedure is used for finding collineations of a translation plane such that their existence is a necessary and sufficient condition for the validity of some laws in the multiplicative loop of a coordinatizing quasifield. Proofs can be found in [1, 2, 3].
TL;DR: In this article, it was shown 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.
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 (Spharischer 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*.