scholarly journals A correction of “The nilstufe of the direct sum of rank 1 torsion free groups”

1975 ◽  
Vol 26 (1-2) ◽  
pp. 221-221
Author(s):  
S. Feigelstock
Keyword(s):  
Free Groups ◽  
Direct Sum ◽  
Torsion Free

Existentially complete torsion-free nilpotent groups

1978 ◽  
Vol 43 (1) ◽  
pp. 126-134 ◽  
Author(s):  
D. Saracino
Keyword(s):  
Free Groups ◽  
Nilpotent Groups ◽  
Generic Model ◽  
Model Companion ◽  
Generic Models ◽  
Direct Sum ◽  
Carry Over ◽  
Torsion Free

This paper continues the study of existentially complete nilpotent groups initiated in [6]. Following [6], we let Kn denote the theory of groups nilpotent of class ≤ n and let Kn+ denote the theory of torsion-free groups nilpotent of class ≤ n. The principal results of [6] were that for n ≥ 2, neither Kn nor Kn+ has a model companion, and the classes E, F, and G of existentially complete, finitely generic and infinitely generic models of Kn are all distinct. The question of the relationships between these classes in the context of Kn was left open, however, and the proof of their distinctness for Kn+ obviously did not carry over to Kn+, because it made strong use of torsion elements.In this paper we establish the relationships between E, F, and G for K2+. We show that all three classes are distinct. We also show that there is only one countable finitely generic model, and only one countable infinitely generic model, and that all the countable existentially complete models can be arranged in a sequence N1 ⊆ N2 ⊆ N3 ⊆ … ⊆ Nω, where Z(Nn) is the direct sum of n copies of Q. Another result is that the finite and infinite forcing companions of K2+ differ by an ∀∃∀ sentence. Finally, we show that there exist finitely generic models of K2+ in all infinite cardinalities.

A Problem on Relative Projectivity for Abelian Groups

1986 ◽  
Vol 29 (1) ◽  
pp. 114-122 ◽  
Author(s):  
S. Feigelstock ◽  
R. Raphael
Keyword(s):  
Abelian Groups ◽  
Free Groups ◽  
Mixed Case ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Free Case ◽  
Torsion Groups ◽  
Torsion Free ◽  
Whitehead Groups

AbstractThe article studies the class of abelian groups G such that in every direct sum decomposition G = A ⊕ B, A is 5-projective. Such groups are called pds groups and they properly include the quasi-projective groups.The pds torsion groups are fully determined.The torsion-free case depends on a lemma that establishes freedom in the non-indecomposable case for several classes of groups. There is evidence suggesting freedom in the general reduced torsion-free case but this is not established and prompts a logical discussion. It is shown, for example, that pds torsion-free groups must be Whitehead if they are not indecomposable, but that there exists Whitehead groups that are not pds if there exist non-free Whitehead groups.The mixed case is characterized and examples are given.

scholarly journals Self-splitting Abelian groups

2001 ◽  
Vol 64 (1) ◽  
pp. 71-79 ◽  
Author(s):  
P. Schultz
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Free Module ◽  
Research Report ◽  
Original Form ◽  
Original Research ◽  
Direct Sum ◽  
The University ◽  
Torsion Free ◽  
Made In

G is reduced torsion-free A belian group such that for every direct sum ⊕G of copies of G, Ext(⊕G, ⊕G) = 0 if and only if G is a free module over a rank 1 ring. For every direct product ΠG of copies of G, Ext(ΠG,ΠG) = 0 if and only if G is cotorsion.This paper began as a Research Report of the Department of Mathematics of the University of Western Australia in 1988, and circulated among members of the Abelian group community. However, it was never submitted for publication. The results have been cited, widely, and since copies of the original research report are no longer available, the paper is presented here in its original form in Sections 1 to 5. In Section 6, I survey the progress that has been made in the topic since 1988.

Torsion-Free Groups with All Subgroups 4-Subnormal

2005 ◽  
Vol 33 (12) ◽  
pp. 4567-4585
Author(s):  
Howard Smith ◽  
Gunnar Traustason
Keyword(s):  
Free Groups ◽  
Torsion Free

scholarly journals Locally soluble groups with min-n.

1974 ◽  
Vol 17 (3) ◽  
pp. 305-318 ◽  
Author(s):  
H. Heineken ◽  
J. S. Wilson
Keyword(s):  
Soluble Group ◽  
Free Groups ◽  
Torsion Group ◽  
Minimal Condition ◽  
Normal Subgroups ◽  
Soluble Groups ◽  
Isomorphism Classes ◽  
Torsion Groups ◽  
Torsion Free ◽  
General Method

It was shown by Baer in [1] that every soluble group satisfying Min-n, the minimal condition for normal subgroups, is a torsion group. Examples of non-soluble locally soluble groups satisfying Min-n have been known for some time (see McLain [2]), and these examples too are periodic. This raises the question whether all locally soluble groups with Min-n are torsion groups. We prove here that this is not the case, by establishing the existence of non-trivial locally soluble torsion-free groups satisfying Min-n. Rather than exhibiting one such group G, we give a general method for constructing examples; the reader will then be able to see that a variety of additional conditions may be imposed on G. It will follow, for instance, that G may be a Hopf group whose normal subgroups are linearly ordered by inclusion and are all complemented in G; further, that the countable groups G with these properties fall into exactly isomorphism classes. Again, there are exactly isomorphism classes of countable groups G which have hypercentral nonnilpotent Hirsch-Plotkin radical, and which at the same time are isomorphic to all their non-trivial homomorphic images.

scholarly journals On a Nonsingular Equation of Length 9 Over Torsion Free Groups

2019 ◽  
Vol 12 (2) ◽  
pp. 590-604
Author(s):  
M. Fazeel Anwar ◽  
Mairaj Bibi ◽  
Muhammad Saeed Akram
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Torsion Free Group ◽  
Torsion Free

In \cite{levin}, Levin conjectured that every equation is solvable over a torsion free group. In this paper we consider a nonsingular equation $g_{1}tg_{2}t g_{3}t g_{4} t g_{5} t g_{6} t^{-1} g_{7} t g_{8}t \\ g_{9}t^{-1} = 1$ of length $9$ and show that it is solvable over torsion free groups modulo some exceptional cases.

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