94 research outputs found
Monothetic algebraic groups
No full textWe call an algebraic group monothetic if it possesses a dense cyclic subgroup. For an arbitrary field k
we describe the structure of all, not necessarily affine, monothetic k-groups G and determine in which
cases G has a k-rational generator
On free constructions
No full textFree constructions are used frequently in geometry in order to construct geometric models with sometimes opposite properties. The authors develop a unified treatment of the subject, including all first-order classes of incidence geometries, e.g., projective or affine planes, generalized n-gons, Benz planes, etc
Algebraic (2,2)-transformation groups
No full textIn this paper we determine all algebraic trans-formation groups G, defined over an algebraical-ly closed field k, which operate transitively, but not primitively, on a variety M, provided the following conditions are fulfilled. We ask that the (non-effective) action of G on the va-riety of blocks is sharply 2-transitive, as well as the action on a block X of the normalizer Gx. Also we require sharp transitivity on pairs (X,Y)of independent points of M, i.e. points con-tained in different blocks
Gruppenuniversalität und Homogenisierbarkeit
No full textA class C of structures is said to be group universal if every group is the full automorphism group of some structure in C. In the present paper it is shown that each of the following classes is group universal: affine planes, projective planes, (m,n)-planes, (k,n)-Steiner systems, LP-spaces, covering geometries, commutative loops, quasigroups, commutative division algebras over fields and other related classes. In the proof the authors first select for a given group G a graph D with G=Aut(D). Then to each node and each edge of D a rigid structure is attached. Using free constructions, all these structures are amalgamated to a structure in C whose automorphism group is G. The proof uses geometric techniques as well as model-theoretic machinery
Free constructions
No full textFree extensions are often used in geometry to show the existence of models for a given theory and to construct examples whose properties are contrary to those of common models. Ever since Hall (1943) introduced free extensions for projective planes, the increasing interest for other classes of incidence geometries has led to repeated variations of Hall's ideas, where the constructions themselves only had to be adapted according to the different situations given by the language, and the axioms of the geometry under consideration.
Besides a survey on significant results obtained thus far, the main purpose of this article is the development of a unifying treatment including all classes of incidence geometries which can be characterized by a set Σ of axioms formulated in a first-order language L (e.g., projective planes, affine planes, generalized n-gons, Benz planes, etc.). By using rather simple model-theoretic tools, we can define the notions of (hyper-)free, open, confined, closed, (hyper-)free extensions, and degenerate geometries without knowing Σ and L explicitly. To a vast extent, we succeed in reformulating and proving the main results concerning free extensions within this general frame. The application of these results reduces to an easy verification of some model-theoretic conditions on the axioms. Thus it becomes clear that the real nature of free extensions in fact lies beyond geometry. We hope that this insight will contribute to stop splitting research on that subject. On the other hand, our treatment yields new results, in particular, on the generalized n-gons.
Surprisingly, groups of projectivities also prove themselves to be sensitive to our unifying point of view, as far as their algebraic structure as a free group or connection with the group of automorphisms is concerned (cf. Theorems 10, 11, and Theorem 15). In addition, repercussions arise about traditional definitions for the group of projectivities. So, for affine planes, central perspectivities with parallel carriers reveal themselves as natural as the usual parallel perspectivities (cf. Theorem 13).
In the last section, in order to illustrate applicability of hyperfree extensions, we combine them with certain amalgamation techniques and broach three rather archetypical questions: Is every group G the full automorphism group of some model in a given class of geometries? May every model be embedded into some homogeneous model enjoying nice transitivity properties? For which classes of geometries do there exist highly transitive groups of projectivities
NEAR-RINGS AND GROUPS OF AFFINE MAPPINGS
No full textWe classify semi-topological locally compact and semi-algebraic
near-rings R where the set of non-invertible elements of R forms an ideal I
of R such that the multiplicative group of R/I acts sharply transitively on
I\{0}. To achieve our results we use as a main tool the classi cation of
locally compact and algebraic (2; 2)-transformation groups given in two previuos papers
Locally compact (2, 2)-transformation groups
No full textWe determine all locally compact imprimitive transformation groups acting sharply 2-transitively on a non-totally disconnected quotient space of blocks inducing on any block a sharply 2-transitive group and satisfying the following condition: if Δ1, Δ2 are two distinct blocks and Pi, Qi ∈ Δi (i = 1, 2), then there is just one element in the inertia subgroup which maps Pi onto Qi. These groups are natural generalizations of the group of affine mappings of the line over the algebra of dual numbers over the field of real or complex numbers or over the skew-field of quaternions. For imprimitive locally compact groups, our results correspond to the classical results of Kalscheuer for primitive locally compact groups (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Multiplicative Loops of Quasifields Having Complex Numbers as Kernel
No full textWe determine the multiplicative loops of locally compact connected
4-dimensional quasifields Q having the field of complex numbers
as their kernel. In particular, we turn our attention to multiplicative loops
which have either a normal subloop of dimension one or which contain a
subgroup isomorphic to Spin3(R). Although the 4-dimensional semifields
Q are known, their multiplicative loops have interesting Lie groups generated
by left or right translations. We determine explicitly the quasifields
Q which coordinatize locally compact translation planes of dimension 8
admitting an at least 16-dimensional Lie group as automorphism group
Multiplicative loops of 2-dimensional topological quasifields
No full textWe determine the algebraic structure of the multiplicative loops for locally compact 2-dimensional topological connected quasifields. In particular, our attention turns to multiplicative loops which have either a normal subloop of positive dimension or which contain a 1-dimensional compact subgroup. In the last section, we determine explicitly the quasifields which coordinatize locally compact translation planes of dimension 4 admitting an at least 7-dimensional Lie group as collineation group
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