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Degrees of orders on torsion-free Abelian groups

Asher M. Kach, Karen Lange, Reed Solomon
2013 Annals of Pure and Applied Logic  
We show that if H is an effectively completely decomposable computable torsion-free abelian group, then there is a computable copy G of H such that G has computable orders but not orders of every (Turing  ...  ) degree.  ...  Describe the possible degree spectra of orders X(G) on a computable 150 presentation G of a computable torsion-free abelian group. 151 Our notation is mostly standard.  ... 
doi:10.1016/j.apal.2013.01.005 fatcat:egwxvckiqbg3fn4gvw5v6a6u3a

COMPUTABLE ABELIAN GROUPS

ALEXANDER G. MELNIKOV
2014 Bulletin of Symbolic Logic  
We provide an introduction to methods and recent results on infinitely generated abelian groups with decidable word problem.  ...  There exists a non-empty Π 0 1 -class such that no computable torsion-free abelian group realizes this class as degrees of linear orders on the group.  ...  Every torsion-free abelian group of finite rank has a jump degree. Proof.  ... 
doi:10.1017/bsl.2014.32 fatcat:utyr6wtenjh5bg7vm4kkinsocu

Page 1828 of Mathematical Reviews Vol. 50, Issue 6 [page]

1975 Mathematical Reviews  
If B is of bounded order, or if all torsion-free submodules of A are free, then A-elongations of A by B are unique.  ...  J. 13315 Rings on completely decomposable torsion-free abelian groups. Comment. Math. Univ. Carolinae 15 (1974), 381-392.  ... 

Page 2983 of Mathematical Reviews Vol. , Issue 80H [page]

1980 Mathematical Reviews  
For a torsion-free abelian group A, let A+ denote the class of torsion-free abelian groups B for which Hom(A, B)=0.  ...  For a torsion-free abelian group A, let D be the set of (isomorphism classes of) rank-one factor groups of A, G the set of rank-one pure subgroups of A.  ... 

Π10 classes and orderable groups

Reed Solomon
2002 Annals of Pure and Applied Logic  
It is known that the spaces of orders on orderable computable fields can represent all Π 0 1 classes up to Turing degree.  ...  We show that the spaces of orders on orderable computable abelian and nilpotent groups cannot represent Π 0 1 classes in even a weak manner.  ...  Every torsion free abelian group is orderable. 3. Every torsion free nilpotent group is orderable.  ... 
doi:10.1016/s0168-0072(01)00097-5 fatcat:axbvt77cyjddjgm5al55v4qnmi

On linear noetherian groups

Hans Zassenhaus
1969 Journal of Number Theory  
A constructive proof is given and it is shown that conversely every noetherian group of matrices of finite degree over any field contains a polycyclic subgroup of finite index.  ...  Swan any group containing a polycyclic subgroup of finite index can be faithfully represented by integral matrices.  ...  For example every nilpotent (or every locally nilpotent) group has a torsion subgroup. A group without elements of finite order > 1 is said to be torsion free.  ... 
doi:10.1016/0022-314x(69)90026-2 fatcat:eytd46sferg2vdhpdd3yi53wfa

FREE CENTRE-BY-NILPOTENT-BY-ABELIAN LIE RINGS OF RANK 2

MARIA ALEXANDROU, RALPH STÖHR
2015 Journal of the Australian Mathematical Society  
We show that the quotient${\it\gamma}_{c}(L^{\prime })+L^{\prime \prime \prime }/[{\it\gamma}_{c}(L^{\prime }),L]+L^{\prime \prime \prime }$is a direct sum of a free abelian group and a torsion group of  ...  We study the free Lie ring of rank$2$in the variety of all centre-by-nilpotent-by-abelian Lie rings of derived length$3$.  ...  Lemma 3. 2 . 2 Let A be a free U -module, c 2. Then the tensor product M c (A)⊗ U Z is a direct sum of a free abelian group and a torsion group of exponent dividing c.  ... 
doi:10.1017/s1446788715000051 fatcat:db2vsnxfxfasvlptap4aa5f2zi

Finitely generated residually torsion-free nilpotent groups. I

Gilbert Baumslag
1999 Journal of the Australian Mathematical Society  
The object of this paper is to study the sequence of torsion-free ranks of the quotients by the terms of the lower central series of a finitely generated group.  ...  This gives rise to the introduction into the study of finitely generated, residually torsion-free nilpotent groups of notions relating to the Gelfand-Kirillov dimension.  ...  Let F be a subgroup of the group of automorphisms of the torsion-free Abelian group A.IfT acts nilpotently on A, then F is torsion-free.  ... 
doi:10.1017/s1446788700002032 fatcat:cun3u2fvvbgolg2ptbmg6bfgd4

Torsion in free center-by-nilpotent-by-abelian Lie rings of rank 2

Ralph Stöhr
2017 Communications in Algebra  
For c ≥ 2, the free centre-by-(nilpotent-of-class-c-1)-by abelian Lie ring on a set X is the quotient L/[(L ′ ) c , L] where L is the free Lie ring on X, and (L ′ ) c denotes the cth term of the lower  ...  In this paper we give a complete description of the torsion subgroup of its additive group in the case where |X| = 2 and c is a prime number. 2010 Mathematics Subject Classification.  ...  by c on M c (A): sums of a free abelian group and a torsion group of exponent dividing c, (ii) for k ≥ 1 the homology groups H k (L c (A)) and H k (M c (A)) are torsion groups of exponent dividing c  ... 
doi:10.1080/00927872.2017.1350699 fatcat:e6d7mzsu7raq3g3dpyrvf2ks3m

Torsion in free centre-by-nilpotent-by-abelian Lie rings of rank 2 [article]

Ralph Stöhr
2017 arXiv   pre-print
For c≥ 2, the free centre-by-(nilpotent-of-class-c-1)-by abelian Lie ring on a set X is the quotient L/[(L')^c,L] where L is the free Lie ring on X, and (L')^c denotes the cth term of the lower central  ...  In this paper we give a complete description of the torsion subgroup of its additive group in the case where |X|=2 and c is a prime number.  ...  by c on M c (A): sums of a free abelian group and a torsion group of exponent dividing c, (ii) for k ≥ 1 the homology groups H k (L c (A)) and H k (M c (A)) are torsion groups of exponent dividing c  ... 
arXiv:1701.02594v1 fatcat:yed5gok7wnccnbwm57wvq5hjam

Galois-theoretic features for 1-smooth pro-p groups

Claudio Quadrelli
2021 Canadian mathematical bulletin  
We prove that every 1-smooth pro-p group contains a unique maximal closed abelian normal subgroup, in analogy with a result by Engler and Koenigsmann on maximal pro-p Galois groups of fields, and that  ...  if a 1-smooth pro-p group is solvable, then it is locally uniformly powerful, in analogy with a result by Ware on maximal pro-p Galois groups of fields.  ...  Weigel for working together on maximal prop Galois groups and their cohomology; and P. Guillot and I. Snopce for the interesting discussions on 1-smooth pro-p groups.  ... 
doi:10.4153/s0008439521000461 fatcat:dpbssymsxrcp3n4xkf42t62vbe

Galois-theoretic features for 1-smooth pro-p groups [article]

Claudio Quadrelli
2021 arXiv   pre-print
We prove that every 1-smooth pro-p group contains a unique maximal closed abelian normal subgroup, in analogy with a result by Engler and Koenigsmann on maximal pro-p Galois groups of fields, and that  ...  if a 1-smooth pro-p group is solvable, then it is locally uniformly powerful, in analogy with a result by Ware on maximal pro-p Galois groups of fields.  ...  Weigel for working together on maximal pro-p Galois groups and their cohomology; and P. Guillot and I. Snopce for the discussions on 1-smooth pro-p groups.  ... 
arXiv:2004.12605v5 fatcat:hoacka7v5je6ngek4zgc7tjk7u

A representation of the wreath product of two torsion-free abelian groups in a power series ring

Gilbert Baumslag
1966 Proceedings of the American Mathematical Society  
Mal'cev [9] ) the wreath product E wr F of an arbitrary pair of torsion-free abelian groups E and F is residually torsion-free nilpotent.  ...  Notice that a torsion-free divisible group is a direct sum of copies of the additive group of rational numbers; the number of copies of the rationals involved is called the rank of the torsion-free abelian  ... 
doi:10.1090/s0002-9939-1966-0207809-8 fatcat:mo6tyll5fjhyrffzvyp7p6fn7i

A Representation of the Wreath Product of Two Torsion-Free Abelian Groups in a Power Series Ring

Gilbert Baumslag
1966 Proceedings of the American Mathematical Society  
Mal'cev [9] ) the wreath product E wr F of an arbitrary pair of torsion-free abelian groups E and F is residually torsion-free nilpotent.  ...  Notice that a torsion-free divisible group is a direct sum of copies of the additive group of rational numbers; the number of copies of the rationals involved is called the rank of the torsion-free abelian  ... 
doi:10.2307/2036111 fatcat:37wamqv5obh3jozvctfjbw5zy4

Page 44 of Mathematical Reviews Vol. 26, Issue 1 [page]

1963 Mathematical Reviews  
P. 217 Separability of complete direct sums of torsion-free Abelian groups of rank one. (Russian) Dokl. Akad. Nauk SSSR 143 (1962), 275-276.  ...  The free group on d generators has & for these groups (I) and (II) and no other Abelian groups.  ... 
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