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Decidability and Computability of Certain Torsion-Free Abelian Groups

Rodney G. Downey, Sergei S. Goncharov, Asher M. Kach, Julia F. Knight, Oleg V. Kudinov, Alexander G. Melnikov, Daniel Turetsky
<span title="">2010</span> <i title="Duke University Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/un57kpou5jawnm7yjqoeumbno4" style="color: black;">Notre Dame Journal of Formal Logic</a> </i> &nbsp;
We study completely decomposable torsion-free abelian groups of the form G S := ⊕ n∈S Qp n for sets S ⊆ ω.  ...  We show that G S has a decidable copy if and only if S is Σ 0 2 and has a computable copy if and only if S is Σ 0 3 .  ...  Thus, special classes of torsion-free abelian groups classes are central objects of study in the area.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1215/00294527-2010-006">doi:10.1215/00294527-2010-006</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nivejewytzanxak4ki4awl4pma">fatcat:nivejewytzanxak4ki4awl4pma</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170922212313/http://homepages.mcs.vuw.ac.nz/~downey/publications/cdTFAG.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/f0/1a/f01a0e80e7811e2c4df4f37de13eb68c5108b4a2.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1215/00294527-2010-006"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Page 4022 of Mathematical Reviews Vol. , Issue 84j [page]

<span title="">1984</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
rings and A-projective torsion-free abelian groups (pp. 209-227); Otto Mutzbauer, Type graph (pp. 228-252); Claudia Metelli, On type-related properties of torsion-free abelian groups (pp. 253-267); K.  ...  The present paper deals with the problems of computing generators for the normalizer of a subgroup and of deciding whether two given subgroups are conjugate, two problems which may be NP-hard [C. M.  ... 
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Page 15 of Mathematical Reviews Vol. , Issue 82a [page]

<span title="">1982</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The heart of the result given here is the proof that the construction of divisible hulls of abelian groups is not concrete (although it is concrete when restricted to torsion-free abelian groups).  ...  They prove that the theories of the following classes of structures are rich: groups, torsion-free groups, semi- groups, torsion-free semigroups, cancellation semigroups, inverse semigroups, associative  ... 
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Page 1798 of Mathematical Reviews Vol. , Issue 80E [page]

<span title="">1980</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
He concludes with an example of a nonsplitting abelian group of infinite torsion-free rank such that every pure subgroup of finite torsion-free rank which contains the torsion subgroup splits. E.  ...  Bican, Ladislav Factor-splitting abelian groups of arbitrary rank. Comment. Math. Univ. Carolin. 19 (1978), no. 4, 653-672. Let G be a torsion-free abelian group.  ... 
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Page 2086 of Mathematical Reviews Vol. , Issue 90D [page]

<span title="">1990</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
It was noted [Cutler, “The existence of certain pure dense subgroups of abelian p-groups is not decidable in ZFC”, submitted] that every p”*'-projective group A of cardinality x, a regular car- dinal,  ...  Throughout let G denote a torsion-free abelian group of finite rank.  ... 
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Torsion homology and cellular approximation

Ramón Flores, Fernando Muro
<span title="2019-02-06">2019</span> <i title="Mathematical Sciences Publishers"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/zcokwzr5jnc2lol6jqxbob7l4e" style="color: black;">Algebraic and Geometric Topology</a> </i> &nbsp;
Our examples include interesting families of non-commutative infinite groups, including Burnside groups, certain solvable examples and the first Grigorchuk group.  ...  In this note we describe the role of the Schur multiplier in the structure of the p-torsion of discrete groups.  ...  of a finite rank non-minimal abelian group, and Jérôme Scherer for his careful reading of the draft.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2140/agt.2019.19.457">doi:10.2140/agt.2019.19.457</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/r2xx35vy2jeppbvr2jk3rlvxf4">fatcat:r2xx35vy2jeppbvr2jk3rlvxf4</a> </span>
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On the Ext-computability of Serre quotient categories

Mohamed Barakat, Markus Lange-Hegermann
<span title="">2014</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/h7qx4qsc2zf7hiupv27dke5fk4" style="color: black;">Journal of Algebra</a> </i> &nbsp;
limit of Ext-groups in the ambient Abelian category A.  ...  To develop a constructive description of Ext in categories of coherent sheaves over certain schemes, we establish a binatural isomorphism between the Ext-groups in Serre quotient categories A/C and a direct  ...  ACKNOWLEDGMENTS We thank Markus Perling and Greg Smith for discussions on the range of applicability of Theorem 1.1 to coherent sheaves.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jalgebra.2014.08.004">doi:10.1016/j.jalgebra.2014.08.004</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hpplzrsaibgn5jfvinty32pb44">fatcat:hpplzrsaibgn5jfvinty32pb44</a> </span>
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A Practical Algorithm for Finding Matrix Representations for Polycyclic Groups

Eddie H. Lo, Gretchen Ostheimer
<span title="">1999</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ezljl2d3lzga5efenbxdvvfcpa" style="color: black;">Journal of symbolic computation</a> </i> &nbsp;
The examples which we studied included a collection of free nilpotent groups, and our results here led us to a theoretical result concerning such groups.  ...  We describe a new algorithm for nding matrix representations for polycyclic groups given by nite presentations.  ...  { 7 { 7 Proof Since H is free abelian, it is torsion-free and is nilpotent of class 1.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/jsco.1999.0286">doi:10.1006/jsco.1999.0286</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rchbywucozd3berz52ghw2nm7e">fatcat:rchbywucozd3berz52ghw2nm7e</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190304040824/http://pdfs.semanticscholar.org/c080/02e461928c92e89f2416bfb4fe71dd76b54a.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c0/80/c08002e461928c92e89f2416bfb4fe71dd76b54a.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/jsco.1999.0286"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Page 7743 of Mathematical Reviews Vol. , Issue 2004j [page]

<span title="">2004</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The same holds for the class TFA (TFA,) of all torsion-free abelian groups of finite rank (of rank n for n € N).  ...  This article is about the classification problem of countable torsion- free abelian groups and, in fact, corresponds to a talk delivered at the abelian group conference in Honolulu.  ... 
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Page 1630 of Mathematical Reviews Vol. , Issue 99c [page]

<span title="">1991</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
99¢:20075 composable and Murley. Then a torsion-free reduced group M of finite rank is a Butler R-module if and only if M = R®z B for some Butler group B. C.  ...  The main theorem of the paper, which states that for a regular uncountable cardinal / the existence of a A-free not 4*-free abelian group implies the existence of a A-free, A-separable not /4*-free abelian  ... 
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Page 3000 of Mathematical Reviews Vol. , Issue 2001E [page]

<span title="">2001</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The category TF(Z,) is the category of p-local torsion-free abelian groups of finite rank with the usual homomorphisms.  ...  A torsion-free (abelian) group of cardinality / is said to be almost free if every subgroup of For the web version of Mathematical Reviews, see http: //www.ams.org/mathscinet  ... 
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Page 168 of Mathematical Reviews Vol. , Issue 2002A [page]

<span title="">2002</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Summary: “For torsion-free abelian groups G we discuss the class J@(G) of all torsion groups 7 satisfying Ext(G, 7) = 0, that is, the subclass of all torsion elements of the cotorsion class cogenerated  ...  Comput. 9 (2000), no. 6, 513-518 Let G be an abelian group and suppose that A is a finite subset of G. We might as well assume that A is a generating set for G and that 0 € A.  ... 
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Page 6716 of Mathematical Reviews Vol. , Issue 99j [page]

<span title="">1999</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The second paper deals with the case in which A = Q, but G is an arbitrary torsion-free hyperbolic group. Theorem: Let G be a torsion-free hyperbolic group.  ...  Then there exists an algorithm that decides whether a given finite system of equations over a free Q,-group has a solution, and if it does, finds a solution.  ... 
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Groups with decidable word problem that do not embed in groups with decidable conjugacy problem [article]

Arman Darbinyan
<span title="2019-11-28">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show the existence of finitely presented torsion-free groups with decidable word problem that cannot be embedded in any finitely generated group with decidable conjugacy problem.  ...  This answers a well-known question of Collins from the early 1970's.  ...  The groupĠ is a semidirect product of two torsion-free abelian groups, hence it is torsion-free and metabelian.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1708.09047v2">arXiv:1708.09047v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/z4dc5ciq5rbf5gh4vqw5do674m">fatcat:z4dc5ciq5rbf5gh4vqw5do674m</a> </span>
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Page 3467 of Mathematical Reviews Vol. , Issue 88g [page]

<span title="">1988</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
products of a free group of rank 4 and a free group of rank 3 amalgamating a subgroup of index 2 in the first factor with a subgroup of index 3 in the second factor.  ...  This article deals with the factor groups of the lower central series of groups which are free products of a finite number of finitely generated abelian groups.  ... 
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