Pure N-high Subgroups, P-adic Topology and Direct Sums of Cyclic Groups

1974 ◽  
Vol 26 (02) ◽  
pp. 322-327 ◽  
Author(s):  
Khalid Benabdallah ◽  
John Irwin
Keyword(s):  
Pure Subgroup ◽  
Primary Group ◽  
Direct Sums ◽  
Cyclic Groups

This paper is divided into two sections. In the first, we characterize the subgroups N of a reduced abelian primary group for which all pure N-high subgroups are bounded. This condition on pure N-high subgroups occurs in several instances, for instance, all pure N-high subgroups of a primary group G are bounded if G is the smallest pure subgroup of G containing N; all N-high subgroups are bounded if N ≠ 0 and all N-high subgroups are closed in the p-adic topology.

scholarly journals Pure-complete subgroups of direct sums of Prüfer groups

1972 ◽  
Vol 13 (1) ◽  
pp. 47-48 ◽  
Author(s):  
Paul Hill
Keyword(s):  
Abelian Group ◽  
Pure Subgroup ◽  
Direct Sums ◽  
Primary Abelian Group ◽  
Cyclic Groups ◽  
Direct Sum

Suppose that G is a p-primary abelian group. The subgroup G[p] = {x∈G:px=0} is called the socle of G and any subgroup S of G[p] is called a subsocle of G. If each subsocle of G supports a pure subgroup, then G is said to be pure-complete [1]. It is well known that, if G a direct sum of cyclic groups, then G is necessarily pure-complete. Further results about pure-complete groups are contained in [1] and [3].

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

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.

Quasi-isomorphism of direct sums of cyclic groups

1965 ◽  
Vol 16 (1-2) ◽  
pp. 33-36 ◽  
Author(s):  
R. A. Beaumont ◽  
R. S. Pierce
Keyword(s):  
Direct Sums ◽  
Cyclic Groups

scholarly journals Subgroups of direct products closely approximated by direct sums

2017 ◽  
Vol 29 (5) ◽  
pp. 1125-1144 ◽  
Author(s):  
Maria Ferrer ◽  
Salvador Hernández ◽  
Dmitri Shakhmatov
Keyword(s):  
Direct Product ◽  
Coding Theory ◽  
List Type ◽  
Direct Products ◽  
Topological Groups ◽  
Product Topology ◽  
Direct Sums ◽  
Cyclic Groups ◽  
Finite Set ◽  
Infinite Set

AbstractLet I be an infinite set, let {\{G_{i}:i\in I\}} be a family of (topological) groups and let {G=\prod_{i\in I}G_{i}} be its direct product. For {J\subseteq I}, {p_{J}:G\to\prod_{j\in J}G_{j}} denotes the projection. We say that a subgroup H of G is(i)uniformly controllable in G provided that for every finite set {J\subseteq I} there exists a finite set {K\subseteq I} such that {p_{J}(H)=p_{J}(H\cap\bigoplus_{i\in K}G_{i})}, (ii)controllable in G provided that {p_{J}(H)=p_{J}(H\cap\bigoplus_{i\in I}G_{i})} for every finite set {J\subseteq I},(iii)weakly controllable in G if {H\cap\bigoplus_{i\in I}G_{i}} is dense in H, when G is equipped with the Tychonoff product topology.One easily proves that (i) {\Rightarrow} (ii) {\Rightarrow} (iii). We thoroughly investigate the question as to when these two arrows can be reversed. We prove that the first arrow can be reversed when H is compact, but the second arrow cannot be reversed even when H is compact. Both arrows can be reversed if all groups {G_{i}} are finite. When {G_{i}=A} for all {i\in I}, where A is an abelian group, we show that the first arrow can be reversed for all subgroups H of G if and only if A is finitely generated. We also describe compact groups topologically isomorphic to a direct product of countably many cyclic groups. Connections with coding theory are highlighted.

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