PERFECTLY BOUNDED CLASSES OF ABELIAN GROUPS

2013 ◽  
Vol 12 (05) ◽  
pp. 1250208 ◽  
Author(s):  
PATRICK W. KEEF
Keyword(s):  
Abelian Groups ◽  
Direct Sums ◽  
Proper Subclass ◽  
Infinite Height ◽  
Reduced Group

Let [Formula: see text] be the class of abelian p-groups. A non-empty proper subclass [Formula: see text] is bounded if it is closed under subgroups, additively bounded if it is also closed under direct sums and perfectly bounded if it is additively bounded and closed under filtrations. If [Formula: see text], we call the partition of [Formula: see text] given by [Formula: see text] a B/U-pair. We state most of our results not in terms of bounded classes, but rather the corresponding B/U-pairs. Any additively bounded class contains a unique maximal perfectly bounded subclass. The idea of the length of a reduced group is generalized to the notion of the length of an additively bounded class. If λ is an ordinal or the symbol ∞, then there is a natural largest and smallest additively bounded class of length λ, as well as a largest and smallest perfectly bounded class of length λ. If λ ≤ ω, then there is a unique perfectly bounded class of length λ, namely the pλ-bounded groups that are direct sums of cyclics; however, this fails when λ > ω. This parallels results of Dugas, Fay and Shelah (1987) and Keef (1995) on the behavior of classes of abelian p-groups with elements of infinite height. It also simplifies, clarifies and generalizes a result of Cutler, Mader and Megibben (1989) which states that the pω + 1-projectives cannot be characterized using filtrations.

scholarly journals Algebraic entropies, Hopficity and co-Hopficity of direct sums of Abelian Groups

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Brendan Goldsmith ◽  
Ketao Gong
Keyword(s):  
Abelian Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Direct Sums ◽  
Direct Sum ◽  
Zero Entropy ◽  
Necessary And Sufficient

AbstractNecessary and sufficient conditions to ensure that the direct sum of two Abelian groups with zero entropy is again of zero entropy are still unknown; interestingly the same problem is also unresolved for direct sums of Hopfian and co-Hopfian groups.We obtain sufficient conditions in some situations by placing restrictions on the homomorphisms between the groups. There are clear similarities between the various cases but there is not a simple duality involved.

Criteria for Total Projectivity

1981 ◽  
Vol 33 (4) ◽  
pp. 817-825 ◽  
Author(s):  
Paul Hill
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Abelian Groups ◽  
General Criterion ◽  
Direct Sums ◽  
Primary Abelian Group ◽  
Cyclic Groups ◽  
Direct Sum ◽  
Totally Projective Groups ◽  
Torsion Groups

All groups herein are assumed to be abelian. It was not until the 1940's that it was known that a subgroup of an infinite direct sum of finite cyclic groups is again a direct sum of cyclics. This result rests on a general criterion due to Kulikov [7] for a primary abelian group to be a direct sum of cyclic groups. If G is p-primary, Kulikov's criterion presupposes that G has no elements (other than zero) having infinite p-height. For such a group G, the criterion is simply that G be the union of an ascending sequence of subgroups Hn where the heights of the elements of Hn computed in G are bounded by some positive integer λ(n). The theory of abelian groups has now developed to the point that totally projective groups currently play much the same role, at least in the theory of torsion groups, that direct sums of cyclic groups and countable groups played in combination prior to the discovery of totally projective groups and their structure beginning with a paper by R. Nunke [11] in 1967.

Sur les Sous-Groupes Purifiables des Groupes Abéliens Primaires

1990 ◽  
Vol 33 (1) ◽  
pp. 11-17 ◽  
Author(s):  
K. Benabdallah ◽  
C. Piché
Keyword(s):  
Abelian Groups ◽  
Divisible Group ◽  
Direct Sums ◽  
Cyclic Subgroups ◽  
Cyclic Groups ◽  
Direct Sum

AbstractThe class of primary abelian groups whose subsocles are purifiable is not yet completely characterized and it contains the class of direct sums of cyclic groups and torsion complete groups. In sharp constrast with this, the class of groups whose p2-bounded subgroups are purifiable consist only of those groups which are the direct sum of a bounded and a divisible group. Various tools are developed and a short application to the pure envelopes of cyclic subgroups is given in the last section.

scholarly journals Primary Abelian Groups and Direct Sums of Cyclics

1993 ◽  
Vol 159 (2) ◽  
pp. 387-399 ◽  
Author(s):  
J. Irwin ◽  
P. Keef
Keyword(s):  
Abelian Groups ◽  
Direct Sums

COMMUTING PROPERTIES OF EXT

2013 ◽  
Vol 94 (2) ◽  
pp. 276-288 ◽  
Author(s):  
PHILL SCHULTZ
Keyword(s):  
Abelian Groups ◽  
Direct Products ◽  
Direct Sums

AbstractWe characterize the abelian groups $G$ for which the functors $\mathrm{Ext} (G, - )$ or $\mathrm{Ext} (- , G)$ commute with or invert certain direct sums or direct products.

scholarly journals Automorphisms of nilpotent groups of class two with small rank

1985 ◽  
Vol 39 (1) ◽  
pp. 121-131 ◽  
Author(s):  
Thomas A. Fournelle
Keyword(s):  
Automorphism Groups ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Factor Group ◽  
Direct Sums ◽  
Matrix Computations ◽  
Central Factor ◽  
Rank One ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractRational abelian groups, that is, torsion-free abelian groups of rank one, are characterized by their types. This paper characterizes rational nilpotent groups of class two, that is, nilpotent groups of class two in which the center and central factor group are direct sums of rational abelian groups. This characterization is done according to the types of the summands of the center and the central factor group. Using these types and some cohomological techniques it is possible to determine the automorphism group of the nilpotent group in question by performing essentially matrix computations.In particular, the automorphism groups of rational nilpotent groups of class two and rank three are completely described. Specific examples are given of semicomplete and pseudocomplete nilpotent groups.

Direct summands of unrestricted direct sums of Abelian groups

1959 ◽  
Vol 10 (1) ◽  
pp. 403-408 ◽  
Author(s):  
Gilbert Baumslag ◽  
Norman Blackburn
Keyword(s):  
Abelian Groups ◽  
Direct Sums
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