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Degrees of orders on torsion-free Abelian groups

2013
*
Annals of Pure and Applied Logic
*

We show that if H is an effectively completely decomposable computable

doi:10.1016/j.apal.2013.01.005
fatcat:egwxvckiqbg3fn4gvw5v6a6u3a
*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. ...##
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COMPUTABLE ABELIAN GROUPS

2014
*
Bulletin of Symbolic Logic
*

We provide an introduction to methods and recent results

doi:10.1017/bsl.2014.32
fatcat:utyr6wtenjh5bg7vm4kkinsocu
*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. ...##
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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. ...##
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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. ...##
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Π10 classes and orderable groups

2002
*
Annals of Pure and Applied Logic
*

It is known that the spaces

doi:10.1016/s0168-0072(01)00097-5
fatcat:axbvt77cyjddjgm5al55v4qnmi
*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*. ...##
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On linear noetherian groups

1969
*
Journal of Number Theory
*

A constructive proof is given and it is shown that conversely every noetherian

doi:10.1016/0022-314x(69)90026-2
fatcat:eytd46sferg2vdhpdd3yi53wfa
*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*. ...##
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FREE CENTRE-BY-NILPOTENT-BY-ABELIAN LIE RINGS OF RANK 2

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

doi:10.1017/s1446788715000051
fatcat:db2vsnxfxfasvlptap4aa5f2zi
*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. ...##
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Finitely generated residually torsion-free nilpotent groups. I

1999
*
Journal of the Australian Mathematical Society
*

The object

doi:10.1017/s1446788700002032
fatcat:cun3u2fvvbgolg2ptbmg6bfgd4
*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*. ...##
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Torsion in free center-by-nilpotent-by-abelian Lie rings of rank 2

2017
*
Communications in Algebra
*

For c ≥ 2, the

doi:10.1080/00927872.2017.1350699
fatcat:e6d7mzsu7raq3g3dpyrvf2ks3m
*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 ...##
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Torsion in free centre-by-nilpotent-by-abelian Lie rings of rank 2
[article]

2017
*
arXiv
*
pre-print

For c≥ 2, the

arXiv:1701.02594v1
fatcat:yed5gok7wnccnbwm57wvq5hjam
*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 ...##
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Galois-theoretic features for 1-smooth pro-p groups

2021
*
Canadian mathematical bulletin
*

We prove that every 1-smooth pro-p

doi:10.4153/s0008439521000461
fatcat:dpbssymsxrcp3n4xkf42t62vbe
*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*. ...##
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Galois-theoretic features for 1-smooth pro-p groups
[article]

2021
*
arXiv
*
pre-print

We prove that every 1-smooth pro-p

arXiv:2004.12605v5
fatcat:hoacka7v5je6ngek4zgc7tjk7u
*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*. ...##
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A representation of the wreath product of two torsion-free abelian groups in a power series ring

1966
*
Proceedings of the American Mathematical Society
*

Mal'cev [9] ) the wreath product E wr F

doi:10.1090/s0002-9939-1966-0207809-8
fatcat:mo6tyll5fjhyrffzvyp7p6fn7i
*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*...##
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A Representation of the Wreath Product of Two Torsion-Free Abelian Groups in a Power Series Ring

1966
*
Proceedings of the American Mathematical Society
*

Mal'cev [9] ) the wreath product E wr F

doi:10.2307/2036111
fatcat:37wamqv5obh3jozvctfjbw5zy4
*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*...##
###
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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