scholarly journals On the Reidemeister spectrum of an Abelian group

2019 ◽  
Vol 31 (1) ◽  
pp. 199-214
Author(s):  
Brendan Goldsmith ◽  
Fatemeh Karimi ◽  
Noel White
Keyword(s):  
Abelian Group ◽  
Quotient Group ◽  
Abelian Groups ◽  
Full Spectrum ◽  
Direct Sums ◽  
Reidemeister Number ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

Abstract The Reidemeister number of an automorphism ϕ of an Abelian group G is calculated by determining the cardinality of the quotient group {G/(\phi-1_{G})(G)} , and the Reidemeister spectrum of G is precisely the set of Reidemeister numbers of the automorphisms of G. In this work we determine the full spectrum of several types of group, paying particular attention to groups of torsion-free rank 1 and to direct sums and products. We show how to make use of strong realization results for Abelian groups to exhibit many groups where the Reidemeister number is infinite for all automorphisms; such groups then possess the so-called {R_{\infty}} -property. We also answer a query of Dekimpe and Gonçalves by exhibiting an Abelian 2-group which has the {R_{\infty}} -property.

On the Square Subgroup of a Mixed SI-Group

2018 ◽  
Vol 61 (1) ◽  
pp. 295-304 ◽  
Author(s):  
R. R. Andruszkiewicz ◽  
M. Woronowicz
Keyword(s):  
Abelian Group ◽  
Quotient Group ◽  
Abelian Groups ◽  
Additive Group ◽  
Negative Solution ◽  
New Class ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free ◽  
Group Modulo

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.

QUASI-ISOMORPHISMS AND GROUPS OF QUASI-HOMOMORPHISMS

2009 ◽  
Vol 08 (05) ◽  
pp. 617-627
Author(s):  
ULRICH ALBRECHT ◽  
SIMION BREAZ
Keyword(s):  
Abelian Group ◽  
Dual Problem ◽  
Abelian Groups ◽  
Mixed Abelian Group ◽  
Free Rank ◽  
Mixed Groups ◽  
Torsion Free Rank ◽  
Torsion Free

This paper investigates to which extent a self-small mixed Abelian group G of finite torsion-free rank is determined by the groups Hom (G,C) where C is chosen from a suitable class [Formula: see text] of Abelian groups. We show that G is determined up to quasi-isomorphism if [Formula: see text] is the class of all self-small mixed groups C with r0(C) ≤ r0(G). Several related results are given, and the dual problem of orthogonal classes is investigated.

SELF-SMALL ABELIAN GROUPS

2009 ◽  
Vol 80 (2) ◽  
pp. 205-216 ◽  
Author(s):  
ULRICH ALBRECHT ◽  
SIMION BREAZ ◽  
WILLIAM WICKLESS
Keyword(s):  
Small Groups ◽  
Abelian Groups ◽  
Direct Sums ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

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.

scholarly journals Spectral synthesis on discrete Abelian groups

2007 ◽  
Vol 143 (1) ◽  
pp. 103-120 ◽  
Author(s):  
M. LACZKOVICH ◽  
L. SZÉKELYHIDI
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Spectral Synthesis ◽  
Free Rank ◽  
Discrete Abelian Group ◽  
Torsion Free Rank ◽  
Torsion Free

AbstractWe prove that spectral synthesis holds on a discrete Abelian group G if and only if the torsion free rank of G is finite.

GENERALIZED ENDOPRIMAL ABELIAN GROUPS

2006 ◽  
Vol 05 (01) ◽  
pp. 1-17 ◽  
Author(s):  
U. ALBRECHT ◽  
S. BREAZ ◽  
W. WICKLESS
Keyword(s):  
Abelian Group ◽  
Simple Form ◽  
Abelian Groups ◽  
Free Rank ◽  
Torsion Groups ◽  
Mixed Groups ◽  
Torsion Free Rank ◽  
Torsion Free ◽  
Nil Radical ◽  
Natural Way

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.

scholarly journals Localization in non-noetherian group rings

1980 ◽  
Vol 21 (1) ◽  
pp. 151-163 ◽  
Author(s):  
P. F. Smith
Keyword(s):  
Abelian Group ◽  
Prime Ideal ◽  
Noetherian Ring ◽  
General Setting ◽  
Group Rings ◽  
Regular Local Ring ◽  
Characteristic P ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

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>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.

scholarly journals Countable ℵ0-indecomposable mixed abelian groups of finite torsion-free rank

1986 ◽  
Vol 100 (2) ◽  
pp. 421-429 ◽  
Author(s):  
Saharon Shelah ◽  
Alexander Soifer
Keyword(s):  
Abelian Groups ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free
Sign in / Sign up

Export Citation Format

Share Document