Sharply transitive sets in quasigroup actions

This paper forms part of the general development of the theory of quasigroup permutation representations. Here, the concept of sharp transitivity is extended from group actions to quasigroup actions. Examples of nontrivial sharply transitive sets of quasigroup actions are constructed. A general theorem shows that uniformity of the action is necessary for the existence of a sharply transitive set. The concept of sharp transitivity is related to two pairwise compatibility relations and to maximal cliques within the corresponding compatibility graphs.     

Analytic Dirac approximation for real linear algebraic groups

For a real linear algebraic group G let A(G){\mathcal{A}(G)} be the algebra of analytic vectors for the left regular representation of G on the space of superexponentially decreasing functions. We present an explicit Dirac sequence in A(G){\mathcal{A}(G)}. Since A(G){\mathcal{A}(G)} acts on E for every Fréchet-representation (π, E) of moderate growth, this yields an elementary proof of a result of Nelson that the space of analytic vectors is dense in E.     

Typical equivalence of linear groups and other algebraic systems

The paper is devoted to the notion of typical equivalence introduced by B. I. Plotkin. We give some examples of elementarily equivalent objects that are not typically equivalent and show two ways to construct nonisomorphic typically equivalent algebras. We also prove A. I. Maltsev??s theorem on elementary equivalence of linear groups over fields for the case of typical equivalence.     

A new approach to rank one linear algebraic groups

One can develop the basic structure theory of linear algebraic groups (the root system, Bruhat decomposition, etc.) in a way that bypasses several major steps of the standard development, including the self-normalizing property of Borel subgroups.     

Barely transitive permutation groups

Part of this work was done while the author was the recipient of RGC Grant 1423 awarded by the University of Alabama Research Grants Committee.     

Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups

We give a new proof that compact infra-solvmanifolds with isomorphic fundamental groups are smoothly diffeomorphic. More generally, we prove rigidity results for manifolds which are constructed using affine actions of virtually polycyclic groups on solvable Lie groups. Our results are derived from rigidity properties of subgroups in solvable linear algebraic groups.     

Triviality of differential Galois cohomology of linear differential algebraic groups

For a partial differential field K, we show that the triviality of the first differential Galois cohomology of every linear differential algebraic group over K is equivalent to K being algebraically, Picard–Vessiot, and linearly differentially closed. This cohomological triviality condition is also known to be equivalent to the uniqueness up to an isomorphism of a Picard–Vessiot extension of a linear differential equation with parameters.     

On the modality of parabolic subgroups of linear algebraic groups

For a linear algebraic group G over an algebraically closed field k and a parabolic subgroup P of G the modality of P is defined to be the maximal number of parameters upon which a family of G-orbits on Lie P _u depends and it is denoted by mod P, where P _u is the unipotent radical of P. The principal aim of this note is a generalization of two basic “monotonicity” results from [19] to positive characteristic: (1) If Θ is a semisimple automorphism of G and P is Θ-stable, then mod P ^\Θ≤ mod P. (2) If G is reductive, char k is a good prime for G, and H is a closed reductive subgroup of G normalized by a maximal torus T⊂P of G, then mod (P∩H)≤ mod P. Received: 22 April 1998 / Revised version: 3 July 1998     

Fully transitive torsion-free Abelian groups

Translated from Algebra i Logika, Vol. 29, No. 5, pp. 549–560, September–October, 1990.     

Finite groups with transitive semipermutability

A group G is said to be a T-group (resp. PT-group, PST-group), if normality (resp. permutability, S-permutability) is a transitive relation. In this paper, we get the characterization of finite solvable PST-groups. We also give a new characterization of finite solvable PT-groups.      

Algebras with transitive automorphism groups

Presented by I. Rosenberg.