On the structure of Ext(A,Z) in ZFC+

1985 ◽  
Vol 50 (2) ◽  
pp. 302-315 ◽  
Author(s):  
G. Sageev ◽  
S. Shelah
Keyword(s):  
Abelian Groups ◽  
Fundamental Problem ◽  
Free Groups ◽  
Torsion Group ◽  
Torsion Subgroup ◽  
Cardinal Numbers ◽  
Structure Problem ◽  
Image Position ◽  
Elementary Fact ◽  
Torsion Free

A fundamental problem in the theory of abelian groups is to determine the structure of Ext(A, Z) for arbitrary abelian groups A. This problem was raised by L. Fuchs in 1958, and since then has been the center of considerable activity and progress.We briefly summarize the present state of this problem. It is a well-known fact thatwhere tA denotes the torsion subgroup of A. Thus the structure problem for Ext(A, Z) breakdown to the two distinct cases, torsion and torsion free groups. For a torsion group T,which is compact and reduced, and its structure is known explicitly [12].For torsion free A, Ext(A, Z) is divisible; hence it has a unique representationThus Ext(A, Z) is characterized by countably many cardinal numbers, which we denote as follows: ν0(A) is the rank of the torsion free part of Ext(A, Z), and νp(A) are the ranks of the p-primary parts of Ext(A, Z), Extp(A, Z).If A is free it is an elementary fact that Ext(A, Z) = 0. The second named author has shown [16] that in the presence of V = L the converse is also true. For countable torsion free, nonfree A, C. Jensen [13] has shown that νp(A) is either finite or and νp(A) ≤ ν0(A). Therefore, the case for uncountable, nonfree, torsion free groups A remains to be studied.

scholarly journals Locally soluble groups with min-n.

1974 ◽  
Vol 17 (3) ◽  
pp. 305-318 ◽  
Author(s):  
H. Heineken ◽  
J. S. Wilson
Keyword(s):  
Soluble Group ◽  
Free Groups ◽  
Torsion Group ◽  
Minimal Condition ◽  
Normal Subgroups ◽  
Soluble Groups ◽  
Isomorphism Classes ◽  
Torsion Groups ◽  
Torsion Free ◽  
General Method

It was shown by Baer in [1] that every soluble group satisfying Min-n, the minimal condition for normal subgroups, is a torsion group. Examples of non-soluble locally soluble groups satisfying Min-n have been known for some time (see McLain [2]), and these examples too are periodic. This raises the question whether all locally soluble groups with Min-n are torsion groups. We prove here that this is not the case, by establishing the existence of non-trivial locally soluble torsion-free groups satisfying Min-n. Rather than exhibiting one such group G, we give a general method for constructing examples; the reader will then be able to see that a variety of additional conditions may be imposed on G. It will follow, for instance, that G may be a Hopf group whose normal subgroups are linearly ordered by inclusion and are all complemented in G; further, that the countable groups G with these properties fall into exactly isomorphism classes. Again, there are exactly isomorphism classes of countable groups G which have hypercentral nonnilpotent Hirsch-Plotkin radical, and which at the same time are isomorphic to all their non-trivial homomorphic images.

On Torsion-Free Groups of Finite Rank

1984 ◽  
Vol 36 (6) ◽  
pp. 1067-1080 ◽  
Author(s):  
David Meier ◽  
Akbar Rhemtulla
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Finite Rank ◽  
Maximal Element ◽  
Free Groups ◽  
Solvable Groups ◽  
Finitely Generated ◽  
Isolated Subgroup ◽  
Image Position ◽  
Torsion Free

This paper deals with two conditions which, when stated, appear similar, but when applied to finitely generated solvable groups have very different effect. We first establish the notation before stating these conditions and their implications. If H is a subgroup of a group G, let denote the setWe say G has the isolator property if is a subgroup for all H ≦ G. Groups possessing the isolator property were discussed in [2]. If we define the relation ∼ on the set of subgroups of a given group G by the rule H ∼ K if and only if , then ∼ is an equivalence relation and every equivalence class has a maximal element which may not be unique. If , we call H an isolated subgroup of G.

scholarly journals Characterization of the automorphisms having the lifting property in the category of abelianp-groups

2003 ◽  
Vol 2003 (71) ◽  
pp. 4511-4516
Author(s):  
S. Abdelalim ◽  
H. Essannouni
Keyword(s):  
Homomorphic Image ◽  
Abelian Groups ◽  
Torsion Group ◽  
Divisible Group ◽  
Torsion Free

Letpbe a prime. It is shown that an automorphismαof an abelianp-groupAlifts to any abelianp-group of whichAis a homomorphic image if and only ifα=π idA, withπan invertiblep-adic integer. It is also shown that ifAis torsion group or torsion-freep-divisible group, thenidAand−idAare the only automorphisms ofAwhich possess the lifting property in the category of abelian groups.

scholarly journals THE COMPLEXITY OF THE EMBEDDABILITY RELATION BETWEEN TORSION-FREE ABELIAN GROUPS OF UNCOUNTABLE SIZE

2018 ◽  
Vol 83 (2) ◽  
pp. 703-716
Author(s):  
FILIPPO CALDERONI
Keyword(s):  
Abelian Groups ◽  
Free Ring ◽  
Uncountable Cardinal ◽  
Free Abelian Groups ◽  
Quasi Order ◽  
Image Position ◽  
Torsion Free ◽  
The Continuum

AbstractWe prove that for every uncountable cardinal κ such that κ<κ = κ, the quasi-order of embeddability on the κ-space of κ-sized graphs Borel reduces to the embeddability on the κ-space of κ-sized torsion-free abelian groups. Then we use the same techniques to prove that the former Borel reduces to the embeddability relation on the κ-space of κ-sized R-modules, for every $\mathbb{S}$-cotorsion-free ring R of cardinality less than the continuum. As a consequence we get that all the previous are complete $\Sigma _1^1$ quasi-orders.

On cardinal numbers associated with locally compact Abelian groups

1965 ◽  
Vol 61 (1) ◽  
pp. 75-79 ◽  
Author(s):  
J. B. Reade
Keyword(s):  
Topological Group ◽  
Abelian Groups ◽  
Cardinal Number ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Cardinal Numbers ◽  
Compact Abelian Groups ◽  
Image Position

We are concerned in this work with the following question:Suppose that i is a continuous algebraic isomorphism from the topological group H onto a subgroup of the topological group G and suppose that the image i(H) is not closed in G; then what can we say about the cardinal numberWe observe two easy results.

Tensor Products and the Splitting of Abelian Groups

1971 ◽  
Vol 23 (5) ◽  
pp. 764-770 ◽  
Author(s):  
D. A. Lawver ◽  
E. H. Toubassi
Keyword(s):  
Positive Integer ◽  
Basic Assumption ◽  
Abelian Groups ◽  
Primary Component ◽  
Torsion Subgroup ◽  
Rank One ◽  
Splitting Problem ◽  
Torsion Subgroups ◽  
Free Element ◽  
Torsion Free

In [2], Irwin, Khabbaz, and Rayna discuss the splitting problem for abelian groups through the use of the tensor product. Throughout the paper they make a basic assumption, namely, that the torsion subgroup contains but one primary component. Under this restriction they introduce the concept of “splitting length”, which is a positive integer indicator of how far a group is from splitting. The results obtained along these lines may be extended to groups whose torsion subgroups contain any finite number of primary components by applying the work of Oppelt [4].Irwin, Khabbaz, and Rayna [2] define the notion of a p-sequence and show that for groups A where T(A) is p-primary and A/T(A) has rank one, the existence of a torsion-free element with a p-sequence is sufficient for the group to split.

On autocommutators and the autocommutator subgroup in infinite abelian groups

2018 ◽  
Vol 30 (4) ◽  
pp. 877-885
Author(s):  
Luise-Charlotte Kappe ◽  
Patrizia Longobardi ◽  
Mercede Maj
Keyword(s):  
Automorphism Group ◽  
Abelian Groups ◽  
Torsion Group ◽  
Nilpotency Class ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Minimal Order ◽  
Autocommutator Subgroup ◽  
Free Abelian Groups ◽  
Torsion Free

Abstract It is well known that the set of commutators in a group usually does not form a subgroup. A similar phenomenon occurs for the set of autocommutators. There exists a group of order 64 and nilpotency class 2, where the set of autocommutators does not form a subgroup, and this group is of minimal order with this property. However, for finite abelian groups, the set of autocommutators is always a subgroup. We will show in this paper that this is no longer true for infinite abelian groups. We characterize finitely generated infinite abelian groups in which the set of autocommutators does not form a subgroup and show that in an infinite abelian torsion group the set of commutators is a subgroup. Lastly, we investigate torsion-free abelian groups with finite automorphism group and we study whether the set of autocommutators forms a subgroup in those groups.

scholarly journals A Note on Additive Groups of Some Specific Associative Rings

2016 ◽  
Vol 30 (1) ◽  
pp. 219-229
Author(s):  
Mateusz Woronowicz
Keyword(s):  
Associative Ring ◽  
Abelian Groups ◽  
Free Groups ◽  
Additive Group ◽  
Direct Sums ◽  
Rank One ◽  
Elementary Methods ◽  
Associative Rings ◽  
Torsion Free

AbstractAlmost complete description of abelian groups (A, +, 0) such that every associative ring R with the additive group A satisfies the condition: every subgroup of A is an ideal of R, is given. Some new results for SR-groups in the case of associative rings are also achieved. The characterization of abelian torsion-free groups of rank one and their direct sums which are not nil-groups is complemented using only elementary methods.

A Problem on Relative Projectivity for Abelian Groups

1986 ◽  
Vol 29 (1) ◽  
pp. 114-122 ◽  
Author(s):  
S. Feigelstock ◽  
R. Raphael
Keyword(s):  
Abelian Groups ◽  
Free Groups ◽  
Mixed Case ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Free Case ◽  
Torsion Groups ◽  
Torsion Free ◽  
Whitehead Groups

AbstractThe article studies the class of abelian groups G such that in every direct sum decomposition G = A ⊕ B, A is 5-projective. Such groups are called pds groups and they properly include the quasi-projective groups.The pds torsion groups are fully determined.The torsion-free case depends on a lemma that establishes freedom in the non-indecomposable case for several classes of groups. There is evidence suggesting freedom in the general reduced torsion-free case but this is not established and prompts a logical discussion. It is shown, for example, that pds torsion-free groups must be Whitehead if they are not indecomposable, but that there exists Whitehead groups that are not pds if there exist non-free Whitehead groups.The mixed case is characterized and examples are given.

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