The direct sum of cyclic groups

1952 ◽  
Vol 3 (3) ◽  
pp. 177-195 ◽  
Author(s):  
L. Fuchs
Keyword(s):  
Cyclic Groups ◽  
Direct Sum

On the Full Transitivity of a Cotorsion Hull

2006 ◽  
Vol 13 (1) ◽  
pp. 79-84 ◽  
Author(s):  
Tariel Kemoklidze
Keyword(s):  
Complete Group ◽  
Cyclic Groups ◽  
Direct Sum ◽  
Full Transitivity

Abstract A cotorsion hull of the separable 𝑝-group 𝑇 is considered when 𝑇 is a direct sum of torsion-complete groups. It is proved that in the considered case its cotorsion hull is fully transitive if and only if 𝑇 is a direct sum of cyclic groups or is a torsion-complete group.

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.

Recursively presented Abelian groups: Effective p-Group theory. I

1981 ◽  
Vol 46 (3) ◽  
pp. 617-624 ◽  
Author(s):  
Charlotte Lin
Keyword(s):  
Abelian Group ◽  
University Of California ◽  
Structure Theory ◽  
Primary Group ◽  
Cyclic Groups ◽  
Direct Sum ◽  
Classical Mathematics ◽  
Hebrew University ◽  
Infinite Height ◽  
Complete Theories

The study of effectiveness in classical mathematics is rapidly expanding, through recent research in algebra, topology, model theory, and functional analysis. Well-known contributors are Barwise (Wisconsin), Crossley (Monash), Dekker (Rutgers), Ershoff (Novosibirsk), Feferman (Stanford), Harrington (Berkeley), Mal′cev (Novosibirsk), Morley (Cornell), Nerode (Cornell), Rabin (Hebrew University), Shore (Cornell). Further interesting work is due to Kalantari (University of California, Santa Barbara), Metakides (Rochester), Millar (Wisconsin), Remmel (University of California, San Diego), Nurtazin (Novosibirsk). Areas investigated include enumerated algebras, models of complete theories, vector spaces, fields, orderings, Hilbert spaces, and boolean algebras.We investigate the effective content of the structure theory of p-groups. Recall that a p-group is a torsion abelian group in which the (finite) order of each element is some power of a fixed prime p. (In the sequel, “group” = “additively written abelian group”.)The structure theory of p-groups is based on the two elementary notions of order and height. Recall that the order of x is the least integer n such that nx = 0. The height of x is the number of times p divides x, that is, the least n such that x = pny for some y in the group but x ≠ pn+1y for any y. If for each n ∈ ω there is a “pnth-root” yn, so that x = pnyn, then we say that x has infinite height. In 1923, Prüfer related the two notions as criteria for direct sum decomposition, provingTheorem. Every group of bounded order is a direct sum of cyclic groups, andTheorem. Every countable primary group with no (nonzero) elements of infinite height is a direct sum of cyclic groups.

On Purifiable Subsocles of a Primary Abelian Group

1971 ◽  
Vol 23 (1) ◽  
pp. 48-57 ◽  
Author(s):  
John Irwin ◽  
James Swanek
Keyword(s):  
Abelian Group ◽  
Direct Summand ◽  
Pure Subgroup ◽  
Primary Abelian Group ◽  
Cyclic Groups ◽  
Direct Sum ◽  
Interesting Connection

In this paper we shall investigate an interesting connection between the structure of G/S and G, where S is a purifiable subsocle of G. The results are interesting in the light of a counterexample by Dieudonné [3, p. 142] who exhibits a primary abelian group G, where G/S is a direct sum of cyclic groups, but G is not a direct sum of cyclic groups. Surprisingly, the assumption of the purifiability of S allows G to inherit the structure of G/S. In particular, we show that if G/S is a direct sum of cyclic groups and S supports a pure subgroup H, then G is a direct sum of cyclic groups and if is a direct summand of G which is of course a direct sum of cyclic groups. It is also shown that if G/S is a direct sum of torsion-complete groups and S supports a pure subgroup H, then G is a direct sum of torsion-complete groups and H is a direct summand of G, and is also a direct sum of torsion-complete groups.

The nice basis rank of a primary abelian group

2011 ◽  
Vol 48 (2) ◽  
pp. 247-256
Author(s):  
Peter Danchev ◽  
Patrick Keef
Keyword(s):  
Abelian Group ◽  
Divisible Group ◽  
Primary Abelian Group ◽  
Cyclic Groups ◽  
Direct Sum ◽  
Divisible Hull ◽  
Reduced Group

An abelian p-group G has a nice basis if it is the ascending union of a sequence of nice subgroups, each of which is a direct sum of cyclic groups. It is shown that if G is any group, then G ⊕ D has a nice basis, where D is the divisible hull of pωG. This leads to a consideration of the nice basis rank of G, i.e., the smallest rank of a divisible group D such that G ⊕ D has a nice basis. This concept is used to show that there exist a reduced group G and a non-reduced group H, both without a nice basis, such that G ⊕ H has a nice basis

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