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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.  ... 

Rips–Segev torsion-free groups without the unique product property

Markus Steenbock
2015 Journal of Algebra  
We explain and generalize Rips-Segev's construction of 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  ... 
doi:10.1016/j.jalgebra.2015.05.004 fatcat:jl4k4eo7hjey5dxgj55ceclwbq

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.  ... 

Semi-rigid classes of cotorsion-free abelian groups

Rüdiger Göbel, Saharon Shelah
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.  ... 
doi:10.1016/0021-8693(85)90178-4 fatcat:lanhcuumxza33pu762cmln7bsy

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

Patrick Keef
1992 Journal of Algebra  
The t-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.  ... 
doi:10.1016/0021-8693(92)90091-y fatcat:ijhnnjxi7zbfbadpu67kdbrnvm

Rips construction without unique product [article]

Goulnara Arzhantseva, Markus Steenbock
2014 arXiv   pre-print
Given a finitely presented 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.  ... 
arXiv:1407.2441v1 fatcat:amfg2n64ujd77hpxi6bgbdgmhq

Finite index subgroups without unique product in graphical small cancellation groups: Figure 1

D. Gruber, A. Martin, M. Steenbock
2015 Bulletin of the London Mathematical Society  
We construct 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  ... 
doi:10.1112/blms/bdv040 fatcat:6qj7l36yhbh6bhcw6g7d2sqloi

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

Károly J. Böröczky, Péter P. Pálfy, Oriol Serra
2012 Bulletin of the London Mathematical Society  
Let A, B be finite 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.  ... 
doi:10.1112/blms/bds032 fatcat:ip5ipkk3wfdhtpkogg4rd227cy

Cardinality of product sets in torsion-free groups and applications in group algebras [article]

Alireza Abdollahi, Fatemeh Jafari
2019 arXiv   pre-print
Also, we prove that if C is a finite 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  ... 
arXiv:1808.08708v3 fatcat:62vc6twh5jff7g2nrs7nmfwheq

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.  ... 

Homomorphisms and duality for torsion-free groups

R. B. Warfield
1968 Mathematische Zeitschrift  
Let A be a self-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.  ... 
doi:10.1007/bf01110257 fatcat:q6ddt7tx5faavk3jm5ozz7iieu
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