Groups Of Finite Torsion-free Rank Research Articles

Semi-generalized co-Bassian groups

As a common nontrivial generalization of the notion of a generalized co-Bassian group, recently defined by the third author, we introduce the notion of a semi-generalized co-Bassian group and initiate its comprehensive study. Specifically, we give a complete characterization of these groups in the cases of [Formula: see text]-torsion groups and groups of finite torsion-free rank by showing that these groups can be completely determined in terms of generalized finite [Formula: see text]-ranks and also depends on their quotients modulo the maximal torsion subgroup. Surprisingly, for [Formula: see text]-primary groups, the concept of a semi-generalized co-Bassian group is closely related to that of a generalized co-Bassian group.

Finitely generated metabelian groups arising from integer polynomials

Abstract It is shown that there is a finitely generated metabelian group of finite torsion-free rank associated with each non-constant integer polynomial. It is shown how many structural properties of the group can be detected by inspecting the polynomial.

On the structure of groups admitting faithful modules with certain conditions of primitivity

In the paper we study structure of soluble-by-finite groups of finite torsion-free rank which admit faithful modules with conditions of primitivity. In particular, we prove that under some additional conditions if an infinite finitely generated linear group $G$ of finite rank admits a fully primitive fully faithful module then $G$ has infinite $FC$-centre.

On the Square Subgroup of a MixedSI-Group

AbstractThe relation between the structure of a ring and the structure of its additive group is studied in the context of some recent results in additive groups of mixed rings. Namely, the notion of the square subgroup of an abelian group, which is a generalization of the concept of nil-group, is considered mainly for mixed non-splitting abelian groups which are the additive groups only of rings whose all subrings are ideals. A non-trivial construction of such a group of finite torsion-free rank no less than two, for which the quotient group modulo the square subgroup is not a nil-group, is given. In particular, a new class of abelian group for which an old problem posed by Stratton and Webb has a negative solution, is indicated. A new, far from obvious, application of rings in which the relation of being an ideal is transitive, is obtained.

Dualities for self–small groups

We construct a family of dualities on some subcategories of the quasi-category S \mathcal {S} of self-small groups of finite torsion-free rank which cover the class S \mathcal {S} . These dualities extend several of those in the literature. As an application, we show that a group A ∈ S A\in \mathcal {S} is determined up to quasi–isomorphism by the Q \mathbb {Q} –algebras { Q Hom ⁡ ( C , A ) : C ∈ S } \{\mathbb {Q}\operatorname {Hom}(C,A):\,C\in \mathcal {S}\} and { Q Hom ⁡ ( A , C ) : C ∈ S } \{\mathbb {Q}\operatorname {Hom}(A,C):\,C\in \mathcal {S}\} . We also generalize Butler’s Theorem to self-small mixed groups of finite torsion-free rank.

Direct decompositions of mixed abelian groups

We consider a sufficiently large subcategory of the category of mixed abelian groups of finite torsion-free rank and its quotient catego ry obtained by annihilating those homomorphisms which factor through the torsion. We prove that the second category provides a good approximation to the first category, but is much simpler: the groups of morphisms in the second category are finite rank torsion-free groups. This renders it possible to exam ine direct decompositions of mixed groups by the same methods as in the case of finite rank torsion-free groups.

SELF-SMALL ABELIAN GROUPS

AbstractThis paper investigates self-small abelian groups of finite torsion-free rank. We obtain a new characterization of infinite self-small groups. In addition, self-small groups of torsion-free rank 1 and their finite direct sums are discussed.

Cancellation properties for quotient divisible groups

We investigate cancellation, the unit lifting property, and 1 in the stable range for mixed self-small abelian groups of finite torsion-free rank. Some of our results generalize results previously obtained for torsion-free finite rank groups by Fuchs, Kravchenko, Loonstra, and Stelzer. In particular, we prove Stelzer's main cancellation result for the class of mixed quotient divisible abelian groups.

GENERALIZED ENDOPRIMAL ABELIAN GROUPS

An n-ary endofunction on an abelian group G is a function f : Gn → G such that f(θg1,…,θgn) = θ f(g1,…,gn) for all endomorphisms θ of G. A group G is endoprimal if, for each natural number n, each n-ary endofunction has the following simple form: [Formula: see text] for some collection of integers {li : 1 ≤ i ≤ n}. The notion of endoprimality arises from universal algebra in a natural way and has been applied to the study of abelian groups in papers Davey and Pitkethly (97), Kaarli and Marki (99) and Göbel, Kaarli, Marki, and Wallutis (to appear). These papers make the case that the notion of endoprimality can give rise to interesting and tractable classes of abelian groups. We continue working along these lines, adapting our definition to make it more suitable for working with general classes of abelian groups. We study generalized endoprimal (ge) abelian groups. Here every n-ary endofunction is required to be of the form [Formula: see text] for some collection of central endomorphisms {λi : 1 ≤ i ≤ n} of G. (Note that such a sum is always an endofunction.) We characterize generalized endoprimal abelian groups in a number of cases, in particular for torsion groups, torsion-free finite rank groups G such that E(G) has zero nil radical, and self-small mixed groups of finite torsion-free rank.

The finite quasi-Baer property

This paper discusses extensions of the Baer splitting property to the class of self-small abelian groups of finite torsion-free rank. This class is extensive, containing the classes of torsion-free abelian groups of finite rank and mixed quotient divisible groups. The latter in particular includes the class G . Several applications of the main results are given.

Splitting Mixed Groups of Finite Torsion-Free Rank#

First, we give a necessary and sufficient condition for torsion-free finite rank subgroups of arbitrary abelian groups to be purifiable. An abelian group G is said to be a strongly ADE decomposable group if there exists a purifiable T(G)-high subgroup of G. We use a previous result to characterize ADE decomposable groups of finite torsion-free rank. Finally, in an extreme case of strongly ADE decomposable groups, we give a necessary and sufficient condition for abelian groups of finite torsion-free rank to be splitting.

Radical modules over hyperabelian groups of finite torsion-free rank

Radical modules over hyperabelian groups of finite torsion-free rank

Representative functions on discrete solvable groups

The topic here is the representation of discrete groups as automorphisms of finite-dimensional vector spaces over a field. The results here are mainly generalizations to the class of solvable groups of finite torsion-free rank of results of Mostow concerning polycyclic groups. In discussing the finite-dimensional representations of a group f over a field k, one is led to consider the “continuous dual” k[r]O of the group algebra, where “continuous” refers to the topology on k[r] in which a fundamental system of neighborhoods of zero is the family of kernels of finitedimensional representations of r over k. k[r]” has the structure of a coalgebra--indeed, of a Hopf algebra-and the locally finite-dimensional k[f]-modules correspond to the k[f]’ comodules. The advantage of looking at k[f]” is that any finite-dimensional k[F]“-comodule is a subcomodule of the direct sum of finitely many copies of k[r]“. A further advantage is that, when k is algebraically closed, k[fJ” is the polynomial algebra of a pro-affme algebraic group Gk(IJ whose rational representations correspond exactly to the locally finite-dimensional representations of f over k. This brings to bear the structural results on pro-afftne algebraic groups, which are strongest when the base field k is of characteristic zero. In Section 1, we examine the unipotent radical U,(T) of the pro-affine group Gk(IJ associated with a discrete group r, and we prove that, on the category of solvable groups of finite torsion-free rank (i.e., the groups of type A, in Mal’cev’s classification [lo] ), the assignment of Uk(ZJ to f is an exact functor to the category of unipotent affrne algebraic groups. In Section 2, we consider a certain homomorphic image Bk(I) of G,(T) which we call the basic k-group associated with I-; this group is the “lowest” homomorphic image of Gk(r) to “preserve” the unipotent radical U,(f). The assignment of Bk(f) is functorial on a subcategory of the category of 272 OO21-8693/87 $3.00

Countable ℵ 0-indecomposable mixed abelian groups of finite torsion-free rank

A group is called Et,-indecomposable if it does not have a decomposition into a direct sum of N, non-zero summands. This work is devoted to the study of K,-indecomposable countable mixed abelian groups of finite torsion-free rank in the most general setting. Theorem 1’ proven here has main results of [6, Theorem 23 and [4, Theorem 33 as corollaries: let T be a countable reduced torsion abelian group, and R a torsion-free abelian group of finite rank n. There exists an Et,-indecomposable abelian group G with the torsion part T and G/T% R if and only if a basic subgroup B of T can be decomposed into a direct sum B = @ ;= O Bi such that 1 B, 1 < K,,

Localization in non-Noetherian group rings

Let k be a field and G an Abelian group of finite torsion-free rank. Brewer, Costa and Lady [1, Theorem A] showed that if k has characteristic 0 then each localization of the group algebra kG at a prime ideal is a regular local ring. They also showed (in the same theorem) that if k has characteristic p &gt; 0, then kG is locally Noetherian (i.e. each localization of kG at a prime ideal is a Noetherian ring) if and only if G is an extension of a finitely generated group by a torsion p′-group. The purpose of this note is to examine this theorem in a more general setting.

Localization in non-noetherian group rings

Let k be a field and G an Abelian group of finite torsion-free rank. Brewer, Costa and Lady [1, Theorem A] showed that if k has characteristic 0 then each Localization of the group algebra kG at a prime ideal is a regular local ring. They also showed (in the same theorem) that if k has characteristic p&gt;0, then kG is locally Joetherian (i.e. each localization of kG at a prime ideal is a Noetherian ring) if and only if G is an extension of a finitely generated group by a torsion p′-group. The purpose of this note is to examine this theorem in a more general setting.