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Page 8908 of Mathematical Reviews Vol. , Issue 2002M
[page]

2002
*
Mathematical Reviews
*

There are many other results

*in*this paper—*one*of the few papers*on*the classification of mixed*groups*of*torsion*-*free*rank |*in*re- cent times. ... A*subset*is periodic if it is the direct*product*of a nontrivial subgroup and a*subset*of G.” 2002m:20089 20K15 Arnold, David M. (1-WACO; Waco, TX) Direct sums of local*torsion*-*free*abelian*groups*. ...##
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Rips–Segev torsion-free groups without the unique product property

2015
*
Journal of Algebra
*

We explain and generalize Rips-Segev's construction of

doi:10.1016/j.jalgebra.2015.05.004
fatcat:jl4k4eo7hjey5dxgj55ceclwbq
*torsion*-*free**groups*without the unique*product*property by viewing these*groups*as given by graphical*small*cancellation presentations over the*free*... We construct uncountably many non-isomorphic*torsion*-*free**groups*without the unique*product*property. ... problem, revisions of first versions of the manuscript, many valuable remarks and discussions; and to his colleagues Christopher Cashen and Dominik Gruber for many valuable remarks and many discussions*on*...##
###
Page 39 of Mathematical Reviews Vol. , Issue 2000j
[page]

2000
*
Mathematical Reviews
*

*In*the final section of the paper the authors consider the com- pletely decomposable

*groups*

*in*ZY, i.e. those

*groups*of the form €B A; where each A; is a reduced mixed

*group*

*in*& of

*torsion*-

*free*rank

*one*... a bounded p-

*group*and A a reduced

*torsion*-

*free*

*group*that is not p-divisible. ...

##
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Semi-rigid classes of cotorsion-free abelian groups

1985
*
Journal of Algebra
*

*On*the other hand, it is not hard to show that there are rigid classes of strongly-] G]-

*free*

*groups*G

*in*ZFC + V = L; cf. also Section 5. ... Hence

*one*cannot expect to prove the existence of a rigid system which is a proper class. ... CONSTRUCTION OF HOMOGENEOUS COTORSION-

*FREE*

*GROUPS*WHICH HAVE ONLY

*SMALL*SUBGROUPS

*WITH*COTORSION-

*FREE*QUOTIENT 3.2. Let 1 E i. ...

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Page 4805 of Mathematical Reviews Vol. , Issue 93i
[page]

1993
*
Mathematical Reviews
*

Summary: “A finite abelian

*group*is said to be factored by its*subsets*if it is written as a direct*product*of these*subsets*. ... From the introduction: “This paper began*with*the question, “What submonoids of a*free*monoid*on**one*generator (Z*, or @) are equalizers?’ ...##
###
On products of primary abelian groups

1992
*
Journal of Algebra
*

The t-

doi:10.1016/0021-8693(92)90091-y
fatcat:ijhnnjxi7zbfbadpu67kdbrnvm
*product*is,*in*the homological sense, the direct*product**in*the category of abelian p-*groups*. ... As is frequently the case*with*homomorphisms defined*on**products*, the index sets will be assumed to be non-measurable. ... Then there is a cocountable*subset*J, of J such that the composition of 4*with*the projection of C onto aiEJ, C, is*small*. Proof Note that G is the t-*product*of bounded*groups*. ...##
###
Rips construction without unique product
[article]

2014
*
arXiv
*
pre-print

Given a finitely presented

arXiv:1407.2441v1
fatcat:amfg2n64ujd77hpxi6bgbdgmhq
*group*Q, we produce a short exact sequence 1→ N G Q → 1 such that G is a*torsion*-*free*Gromov hyperbolic*group*without the unique*product*property and N is without the unique ... As an application, we obtain Tarski monster*groups*without the unique*product*property. ... Dahmani, who, after they learned from our results, pointed out that Ol'shanskii's*small*cancellation theory helps to produce more Property (T)*groups*without the unique*product*property. ...##
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Finite index subgroups without unique product in graphical small cancellation groups: Figure 1

2015
*
Bulletin of the London Mathematical Society
*

We construct

doi:10.1112/blms/bdv040
fatcat:6qj7l36yhbh6bhcw6g7d2sqloi
*torsion*-*free*hyperbolic*groups*without unique*product*whose subgroups up to some given finite index are themselves non-unique*product**groups*. ... This is achieved by generalising a construction of Comerford to graphical*small*cancellation presentations, showing that for every subgroup H of a graphical*small*cancellation*group*there exists a*free*... Such*groups*can be realised as finitely presented graphical*small*cancellation*groups*over a*free**product*of*torsion*-*free*hyperbolic*groups*[Ste13] , thus providing the first examples of*torsion*-*free*...##
###
Page 647 of Mathematical Reviews Vol. 21, Issue 6
[page]

1960
*
Mathematical Reviews
*

The paper concludes

*with*a result which shows that there is an “immense number of*torsion*-*free**groups*..., even for very*small*ra: F. Haimo (St. Louis, Mo.) 3482: Fuchs, L. ... If 8 isa*torsion**group*andif X isa*torsion*-*free**group*, &*one*-to-*one*correspondence is set up from the set of mixed*groups*M*with**torsion*subgroup S and*torsion*- quotient*group*X to Hom(Q @ X, Ext(Q/Z, ...##
###
Page 7031 of Mathematical Reviews Vol. , Issue 2001J
[page]

2001
*
Mathematical Reviews
*

Let A be a finite

*subset*of the*free**group*F of rank n. ... We say that G has the*small*square property*on*k- sets if |X?| < k? for any k-element*subset*X of G. ...##
###
On the cardinality of sumsets in torsion-free groups

2012
*
Bulletin of the London Mathematical Society
*

Let A, B be finite

doi:10.1112/blms/bds032
fatcat:ip5ipkk3wfdhtpkogg4rd227cy
*subsets*of a*torsion*-*free**group*G. ... We obtain c(k)<c_0k^6 for arbitrary*torsion*-*free**groups*, and c(k)<c_0k^3 for*groups**with*the unique*product*property, where c_0 is an absolute constant. ... Hamidoune for fruitful discussions*on*the problem addressed*in*this paper, and Peter A. Linnell for providing*in*depth information*on*unique*product**groups*. ...##
###
Cardinality of product sets in torsion-free groups and applications in group algebras
[article]

2019
*
arXiv
*
pre-print

Also, we prove that if C is a finite

arXiv:1808.08708v3
fatcat:62vc6twh5jff7g2nrs7nmfwheq
*subset*containing the identity element of a*torsion*-*free**group*G such that |C| = 3 and 〈 C〉 is not abelian, then for all*subsets*B of G*with*|B|≥ 7, |BC|≥ |B| + 5. ... Let G be a unique*product**group*, i.e., for any two finite*subsets*A and B of G there exists x∈ G which can be uniquely expressed as a*product*of an element of A and an element of B. ... first shown*in*[29] that not all*torsion*-*free**groups*have the unique*product*property by constructing a*group**with*the use of a generalization of*small*cancellation theory so-called graphical*small*...##
###
Page 1798 of Mathematical Reviews Vol. , Issue 80E
[page]

1980
*
Mathematical Reviews
*

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. ... The author gives a characterization of*torsion*-*free*factor-splitting*groups*. ...##
###
Page 3467 of Mathematical Reviews Vol. , Issue 88g
[page]

1988
*
Mathematical Reviews
*

*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. ... Then

*one*uses standard results characterizing

*torsion*

*in*

*small*cancellation

*groups*. Stephen J. Pride (Glasgow) 88g:20072 20F10 20F36 Makanina, T. A.

*On*a system of equations

*in*a braid

*group*. ...

##
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Homomorphisms and duality for torsion-free groups

1968
*
Mathematische Zeitschrift
*

Let A be a self-

doi:10.1007/bf01110257
fatcat:q6ddt7tx5faavk3jm5ozz7iieu
*small**torsion*-*free*abelian*group*which is faithfully flat as an E(A)-module. ... By Lemma 3T, A satisfies ML*with*respect to V/ A . • Lemma 3.3. Let A be a self-*small**torsion*-*free*abelian*group*which is faithfully flat as an E(A)-module. ...
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