On subgroups of totally projective primary abelian groups and direct sums of cyclic groups

2012 ◽  
pp. 205-215 ◽  
Author(s):  
Patrick Keef
Keyword(s):  
Abelian Groups ◽  
Direct Sums ◽  
Cyclic Groups

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.

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.

scholarly journals Subgroups of finite direct sums of valuated cyclic groups

1988 ◽  
Vol 114 (1) ◽  
pp. 1-15 ◽  
Author(s):  
David M Arnold ◽  
Fred Richman
Keyword(s):  
Direct Sums ◽  
Cyclic Groups

The intersection graph of a group

2015 ◽  
Vol 14 (05) ◽  
pp. 1550065 ◽  
Author(s):  
S. Akbari ◽  
F. Heydari ◽  
M. Maghasedi
Keyword(s):  
Disjoint Union ◽  
Abelian Groups ◽  
Solvable Groups ◽  
Intersection Graph ◽  
Free Graph ◽  
Intersection Graphs ◽  
Cyclic Groups ◽  
Vertex Set

Let G be a group. The intersection graph of G, denoted by Γ(G), is the graph whose vertex set is the set of all nontrivial proper subgroups of G and two distinct vertices H and K are adjacent if and only if H ∩ K ≠ 1. In this paper, we show that the girth of Γ(G) is contained in the set {3, ∞}. We characterize all solvable groups whose intersection graphs are triangle-free. Moreover, we show that if G is finite and Γ(G) is triangle-free, then G is solvable. Also, we prove that if Γ(G) is a triangle-free graph, then it is a disjoint union of some stars. Among other results, we classify all abelian groups whose intersection graphs are complete. Finally, we study the intersection graphs of cyclic groups.

scholarly journals On Keller's conjecture for certain cyclic groups

1979 ◽  
Vol 22 (1) ◽  
pp. 17-21 ◽  
Author(s):  
A. D. Sands
Keyword(s):  
Prime Power ◽  
Abelian Groups ◽  
Prime Power Order ◽  
Finite Abelian Groups ◽  
Cyclic Groups

Keller (6) considered a generalisation of a problem of Minkowski (7) concerning the filling of Rn by congruent cubes. Hajós (4) reduced Minkowski's conjecture to a problem concerning the factorization of finite abelian groups and then solved this problem. In a similar manner Hajós (5) reduced Keller's conjecture to a problem in the factorization of finite abelian groups, but this problem remains unsolved, in general. It occurs also as Problem 80 in Fuchs (3). Seitz (10) has obtained a solution for cyclic groups of prime power order. In this paper we present a solution for cyclic groups whose order is the product of two prime powers.

scholarly journals The Number of Subgroup Chains of Finite Nilpotent Groups

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1537 ◽  
Author(s):  
Lingling Han ◽  
Xiuyun Guo
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Recursive Formula ◽  
Nilpotent Groups ◽  
Classification Problem ◽  
Finite Abelian Groups ◽  
Counting Problem ◽  
Cyclic Groups ◽  
Fuzzy Subgroups

In this paper, we mainly count the number of subgroup chains of a finite nilpotent group. We derive a recursive formula that reduces the counting problem to that of finite p-groups. As applications of our main result, the classification problem of distinct fuzzy subgroups of finite abelian groups is reduced to that of finite abelian p-groups. In particular, an explicit recursive formula for the number of distinct fuzzy subgroups of a finite abelian group whose Sylow subgroups are cyclic groups or elementary abelian groups is given.

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