Abelian quotients and orbit sizes of linear groups

Let G be a finite group,and let V be a completely reducible faithful finite G-module(i.e.,G ≤GL(V),where V is a finite vector space which is a direct sum of irreducible G-submodules).It has been known for a long time that if G is abelian,then G has a regular orbit on V.In this paper we show that G has an orbit of size at least |G/G′| on V.This generalizes earlier work of the authors,where the same bound was proved under the additional hypothesis that G is solvable.For completely reducible modules it also strengthens the 1989 result |G/G′| |V| by Aschbacher and Guralnick.     

On the enumeration of finite Abelian and solvable groups

Let a(n) be the number of nonisomorphic abelian groups of order n. We obtain a short interval result for the local density of a(n). More generally, we get short interval version of results of Ivi? on the local density of prime independent multiplicative functions. Also we prove a short interval version of the theorem of Erdös and Szekeres on the summatory function of a(n) and the theorem of Greenberg and Newman on the enumeration of a certain type of finite solvable groups.     

A1-homotopy groups,excision, and solvable quotients

We study some properties of ${\mathbb{A}}^1$-homotopy groups: geometric interpretations of connectivity, excision results, and a re-interpretation of quotients by free actions of connected solvable groups in terms of covering spaces in the sense of ${\mathbb{A}}^1$-homotopy theory. These concepts and results are well suited to the study of certain quotients via geometric invariant theory. As a case study in the geometry of solvable group quotients, we investigate ${\mathbb{A}}^1$-homotopy groups of smooth toric varieties. We give simple combinatorial conditions (in terms of fans) guaranteeing vanishing of low degree ${\mathbb{A}}^1$-homotopy groups of smooth (proper) toric varieties. Finally, in certain cases, we can actually compute the “next” non-vanishing ${\mathbb{A}}^1$-homotopy group (beyond ${\pi }_1^{{\mathbb{A}}^1}$) of a smooth toric variety. From this point of view, ${\mathbb{A}}^1$-homotopy theory, even with its exquisite sensitivity to algebro-geometric structure, is almost “as tractable” (in low degrees) as ordinary homotopy for large classes of interesting varieties.     

Finitep-periodic quotients of general linear groups

Mathematische Annalen -     

Large orbits of subgroups of solvable linear groups

Suppose that G is a finite solvable group and V is a finite, faithful and completely reducible G-module. Let H be a nilpotent subgroup of G; then H has at least three regular orbits on V ⊕ V. Let H be a subgroup of G and 3 ? |H|; then H has at least three regular orbits on V ⊕ V. Let H be a subgroup of G and assume the Sylow 2-subgroups of the semidirect product HV are abelian; then H has at least two regular orbits on V ⊕ V.     

Harmonic functions of linear growth on solvable groups

We answer a question posed by Bonilla and Grosse-Erdmann by showing that the operators P(D) on the space of entire functions H(C), where D is the differentiation operator and P is a polynomial, do not possess a frequently hypercyclic subspace.     

Rationality of quotients by linear actions of affine groups

LetG = SLn(C)Cn be the (special) affine group. In this paper we study the representation theory of G and in particular the question of rationality for V/G, where V is a generically free G-representation. We show that the answer to this question is positive (Theorem 6.1) if the dimension of V is sufficiently large and V is indecomposable. We explicitly characterize two-step extensions 0 → S → V → Q → 0, with completely reducible S and Q, whose rationality cannot be obtained by the methods presented here (The...     

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.     

Abelian quotients of monoids of homology cylinders

A homology cylinder over a surface consists of a homology cobordism between two copies of the surface and markings of its boundary. The set of isomorphism classes of homology cylinders over a fixed surface has a natural monoid structure and it is known that this monoid can be seen as an enlargement of the mapping class group of the surface. We now focus on abelian quotients of this monoid. We show that both the monoid of all homology cylinders and that of irreducible homology cylinders are not finitely generated and moreover they have big abelian quotients. These properties contrast with the fact that the mapping class group is perfect in general. The proof is given by applying sutured Floer homology theory to homologically fibered knots studied in a previous paper.     

Finite quotients of rings and applications to subgroup separability of linear groups

In this paper we apply results from algebraic number theory to subgroup separability of linear groups. We then state applications to subgroup separability of free products with amalgamation of hyperbolic -manifold groups.

     

Characterization of classical groups by orbit sizes on the natural module

We show that if is a finite vector space, and is a subgroup of having the same orbit sizes on 1-spaces as an orthogonal or unitary group on , then, with a few exceptions, is itself an orthogonal or unitary group on .

     

Criterion of completeness of topological linear spaces and topological Abelian groups

A criterion of completeness, established earlier by the author for locally convex spaces, is generalized to arbitrary topological linear spaces and topological Abelian groups.     

Abelian groups

The present, fourth survey of review articles on Abelian groups includes works reviewed in the years 1979–1984. Also, as in the preceding surveys, no attention has been given to questions involving finite Abelian groups, topological groups, ordered groups, group algebras, modules, or the structure of subgroups, or to questions connected with logic. The word group throughout is understood to mean Abelian group (except in the case of the group of automorphisms of an Abelian group). Concepts and notation not defined in this survey can be found in the books [117, 118]. The letterZ throughout denotes the group (or ring) of integers,Q the group (or field) of rational numbers,Q _p the group (ring) of all rational numbers whose denominator is not divisible by the primep, I_p the group of allp-adic integers. The torsion part of an Abelian groupG is throughout denoted bytG. Translated from Itogi Nauki i Tekhniki. Seriya Algebra, Topologiya, Geometriya, Vol. 23, pp. 51–118.