On Quasi-Regular Torsion-Free Modules

A torsion-free module is called quasi-regular if each cyclic submodule is a quasi-summand. This article characterizes torsion-free Abelian groups that are quasi-regular as modules over a subring of their endomorphism ring. In particular, if G is a torsion-free Abelian group such that its ring Q E of quasi-endomorphisms is Artinian, then the left E-module G is quasi-regular if and only if the left C-module G is quasi-regular, where C is the center of its endomorphism ring E.     

A-Solvability and Mixed Abelian Groups

An abelian group A is an S-group (S ⁺-group) if every subgroup B ≤ A of finite index is A-generated (A-solvable). This article discusses some of the differences between torsion-free S-groups and mixed S-groups, and studies (mixed) S- and S ⁺-groups, which are self-small and have finite torsion-free rank.     

On pureness in Abelian groups

Torsion-free Abelian groups G and H are called quasi-equal (G ≈ H) if λG ⊂ H ⊂ G for a certain natural number ≈. It is known (see [3]) that the quasi-equality of torsion-free Abelian groups can be represented as the equality in an appropriate factor category. Thus while dealing with certain group properties it is usual to prove that the property under consideration is preserved under the transition to a quasi-equal group. This trick is especially frequently used when the author investigates module properties of Abelian groups; here a group is considered as a left module over its endomorphism ring. On the other hand, a topical problem in the Abelian group theory is the problem of investigation of pureness in the category of Abelian groups (see [4]). We consider the pureness introduced by P. Cohn [2] for Abelian groups as modules over their endomorphism rings. Particularity of the investigation of the properties of pureness for the Abelian group G as the module _E (G)G lies in the fact that this is a more general situation than the investigation of pureness for a unitary module over an arbitrary ring R with the identity element. Indeed, if _R M is an arbitrary unitary left module and M ⁺ is its Abelian group, then each element from R can be identified with an appropriate endomorphism from the ring E(M ⁺) under the canonical ring homomorphism R → E(M ⁺). Then it holds that if _E(M+) N is a pure submodule in _E(M+) M ⁺, then _R N is a pure submodule in _R M. In the present paper the interrelations between pureness, servantness, and quasi-decompositions for Abelian torsion-free groups of finite rank will be investigated. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 2, pp. 225–238, 2004.     

On Chow Groups of G-Graded Rings

Abstract

Let A be a Noetherian ring graded by a finitely generated Abelian group G. It is shown that a Chow group A _?(A) of A is determined by cycles and a rational equivalence with respect to certain G-graded ideals of A. In particular, A _?(A) is isomorphic to the equivariant Chow group of A if G is torsion free.     

Purity and Self-Small Groups

Let A be an abelian group. A group B is A-solvable if the natural map Hom(A, B) ?  _E(A) A → B is an isomorphism. We study pure subgroups of A-solvable groups for a self-small group A of finite torsion-free rank. Particular attention is given to the case that A is in , the class of self-small mixed groups G with G/tG? ? ⁿ for some n < ω. We obtain a new characterization of the elements of , and demonstrate that  differs in various ways from the class  ? of torsion-free abelian groups of finite rank despite the fact that the quasi-category ?  is dual to a full subcategory of ?  ?.     

Torsion-Free Ext

This article investigates homological properties of the class *B of Abelian groups A such that Ext(A, B) is torsion-free. In particular, subgroups and extensions of groups in *B are discussed.     

Quasi-Purifiable Subgroups and Minimal Direct Summands

Let G be an arbitrary Abelian group. A subgroup A of G is said to be quasi-purifiable in G if there exists a pure subgroup H of G containing A such that A is almost-dense in H and H/A is torsion. Such a subgroup H is called a “quasi-pure hull” of A in G. We prove that if G is an Abelian group whose maximal torsion subgroup is torsion-complete, then all subgroups A are quasi-purifiable in G and all maximal quasi-pure hulls of A are isomorphic. Every subgroup A of a torsion-complete p-primary group G is contained in a minimal direct summand of G that is a minimal pure torsion-complete subgroup containing A. An Abelian group G is said to be an “ADE decomposable group” if there exist an ADE subgroup K of G and a subgroup T′ of T(G) such that G = K⊕ T′. An Abelian group whose maximal torsion subgroup is torsion-complete is ADE decomposable. Hence direct products of cyclic groups are ADE decomposable groups.     

Quasi-hopf algebra actions and smash products

A classical result of Magnus asserts that a free group F has a faithful representation in the group of units of a ring of non-commuting formal power series with integral coefficients. We generalize this result to the category of A-groups, where A is an associative ring or an Abelian group. We say that a free A-group F^A has a faithful Magnus representation if there is a ring B containing A as an additive subgroup (or a subring) such that F^A is faithfully represented (exactly as in Magnus' classical result in the case A = Z)in the group of units of the ring of formal power series in non-communting indeterminater over B.The three principal results are: (I) If A is a torsion free Abelian group and F^A is a free A-groupp of Lyndon' type, then F^A has a faithful Magnus representation; (II) If A is an ω‐residually Z ring, then F^A has a faithful Magnus representation;(III) for every nontrivial torsion-free Abelian group A, F^A has a faithful Magnus representation in B[[Y]] for a suitable ring B in and only if F^Q has a faithful Magnus representation in Q[[Y]].     

Generalized Group Algebras of Locally Compact Groups

This article studies the homological properties of generalized group algebra L ¹(G, A) of a locally compact group G over a Banach algebra A with an identity of norm 1. It is shown that if L ¹(G, A) is right continuous then G is finite and A is right continuous. It is also shown that L ¹(G, A) is right self-injective if and only if G is finite and A is right self-injective.     

On Finite Groups with Few Automorphism Orbits

Denote by ω(G) the number of orbits of the action of Aut(G) on the finite group G. We prove that if G is a finite nonsolvable group in which ω(G) ≤5, then G is isomorphic to one of the groups A₅, A₆, PSL(2, 7), or PSL(2, 8). We also consider the case when ω(G) = 6 and show that, if G is a nonsolvable finite group with ω(G) = 6, then either G ≈ PSL(3, 4) or there exists a characteristic elementary abelian 2-subgroup N of G such that G/N ≈ A₅.     

On hereditarily and super R-Hopfian and L-co-Hopfian abelian groups

An Abelian group is said to be R-Hopfian [L-co-Hopfian] if every surjective [injective] endomorphism has a right [left] inverse. An Abelian group G is said to be hereditarily R-Hopfian [hereditarily L-co-Hopfian] if each subgroup of G is R-Hopfian [L-co-Hopfian]; similarly G is super R-Hopfian [super L-co-Hopfian] if each homomorphic image of G is R-Hopfian [L-co-Hopfian]. The various classes of hereditarily and super R-Hopfian and L-co-Hopfian groups are studied and necessary conditions for groups to have these properties are derived; in several, but not all, cases, su?cient conditions are also obtained.     

On Solvable R* Groups of Finite Rank

Abstract

For an element a of a group G,let S(a) denote the semigroup generated by all conjugates of a in G. We prove that if G is solvable of finite rank and 1 ? S(a) for all 1 ≠ a ∈ G,then ?a ^G ?/?b ^G ? is a periodic group for every b ∈ S(a). Conversely if every two generator subgroup of a finitely generated torsion-free solvable group G has this property then G has finite rank,and if every finitely generated subgroup has this property then every partial order on G can be extended to a total order.     

A Note on a Class of Factorized <Emphasis Type="Italic">p</Emphasis>-Groups

In this note we study finite p-groups G = AB admitting a factorization by an Abelian subgroup A and a subgroup B. As a consequence of our results we prove that if B contains an Abelian subgroup of index p ⁿ⁻¹ then G has derived length at most 2n.     

Torsion Abelian RAI-Groups

A subgroup A of an Abelian group G is called its absolute ideal if A is an ideal of any ring on G. An Abelian group is called an RAI-group if there exists a ring on it in which every ideal is absolute. The problem of describing RAI-groups was formulated by L. Fuchs (Problem 93). In this paper, absolute ideals of torsion Abelian groups and torsion Abelian RAI-groups are described.     

On Hypercentral Groups G With |G:G n |<∞

In 1960, Baumslag, following up on work of Cernikov for the 1940s, proved that a hypercentral p-group G with G = G ^p is a divisible Abelian group. In this article, we provide an interesting generalization of this 45 year old result: If a hypercentral p-group G satisfies |G:G ^p |<∞ (of course, it contains G = G ^p ), there exists a normal divisible Abelian subgroup D such that |G:D|<∞.     

The Multiplicity Free Permutation Representations of the Ree Groups 2 G 2(q), the Suzuki Groups 2 B 2(q), and Their Automorphism Groups

Abstract

Let G a simple group of type ² B ₂(q) or ² G ₂(q), where q is an odd power of 2 or 3, respectively. The main goal of this paper is to determine the multiplicity free permutation representations of G and A ≤ Aut(G) where A is a subgroup containing a copy of G. Let B be a Borel subgroup of G. If G = ² B ₂(q) we show that there is only one non-trivial multiplicity free permutation representation, namely the representation of G associated to the action on G/B. If G = ² G ₂(q) we show that there are exactly two such non-trivial representations, namely the representations of G associated to the action on G/B and the action on G/M, where M = UC with U the maximal unipotent subgroup of B and C the unique subgroup of index 2 in the maximal split torus of B. The multiplicity free permutation representations of A correspond to the actions on A/H where H is isomorphic to a subgroup containing B if G = ² B ₂(q), and containing M if G = ² G ₂(q). The problem of determining the multiplicity free representations of the finite simple groups is important, for example, in the classification of distance-transitive graphs.     

Hölder Conditions for Endomorphisms of Hyperbolic Groups

It is proved that an endomorphism φ of a hyperbolic group G satisfies a Hölder condition with respect to a visual metric if and only if φ is virtually injective and Gφ is a quasiconvex subgroup of G. If G is virtually free or torsion-free co-hopfian, then φ is uniformly continuous if and only if it satisfies a Hölder condition if and only if it is virtually injective.     

Induced Representations of Abelian Groups of Finite Rank

We prove that any irreducible faithful representation of an almost torsion-free Abelian group G of finite rank over a finitely generated field of characteristic zero is induced from an irreducible representation of a finitely generated subgroup of the group G.     

Torsion Abelian afi-Groups

This paper is devoted to the study of Abelian afi-groups. A subgroup A of an Abelian group G is called its absolute ideal if A is an ideal of any ring on G. We will call an Abelian group an afi-group if all of its absolute ideals are fully invariant subgroups. In this paper, we will describe afi-groups in the class of fully transitive torsion groups (in particular, separable torsion groups) and divisible torsion groups.     

AlgebraicK-theory of von Neumann algebras

To every von Neumann algebra, one can associate a (multiplicative) determinant defined on the invertible elements of the algebra with range a subgroup of the Abelian group of the invertible elements of the center of the von Neumann algebra. This determinant is a normalization of the usual determinant for finite von Neumann algebras of type I, for the type II₁-case it is the Fuglede-Kadison determinant, and for properly infinite von Neumann algebras the determinant is constant equal to 1. It is proved that every invertible element of determinant 1 is a product of a finite number of commutators. This extends a result of T. Fack and P. de la Harpe for II₁-factors. As a corollary, it follows that the determinant induces an injection from the algebraicK ₁-group of the von Neumann algebra into the Abelian group of the invertible elements of the center. Its image is described. Another group,K ₁ ^w (A), which is generated by elements in matrix algebras overA that induce injective right multiplication maps, is also computed. We use the Fuglede-Kadison determinant to detect elements in the Whitehead group Wh(G).Partially supported by NSF Grant DMS-9103327.