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].

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.

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

The Generalized Kulikov Criterion

1969 ◽  
Vol 21 ◽  
pp. 1192-1205 ◽  
Author(s):  
Charles Megibben
Keyword(s):  
Abelian Group ◽  
Primary Abelian Group ◽  
Cyclic Groups ◽  
Direct Sum

In 1941, Kulikov (5) showed that a p-primary abelian group G is a direct sum of cyclic groups if and only if G is the union of an ascending sequence of subgroups each of which has a finite bound on the heights of its elements. An easy reformulation of the Kulikov criterion is: A p-primary abelian group G is a direct sum of cyclic groups if and only if G[p] = ⴲn<ωSn where, for each n, the non-zero elements of Sn have precisely height n. This statement suggests the consideration of reduced p-groups G such that G[p] = ⴲa<λSα where, for each α, Sα – {0} ⊆ pαG – pα+lG. We shall call such p-groups summable (the term principal p-group has been used by Honda (4)). Recall that the length of a reduced p-group G is the first ordinal λ such that pλG = 0.

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.

scholarly journals INTERCHANGE RINGS

2016 ◽  
Vol 101 (3) ◽  
pp. 310-334
Author(s):  
CHARLES C. EDMUNDS
Keyword(s):  
Abelian Group ◽  
Prime Power ◽  
Binary Operation ◽  
Finite Abelian Group ◽  
Linear Operators ◽  
Prime Power Order ◽  
Cyclic Groups ◽  
Isomorphism Classes ◽  
Direct Sum ◽  
Interchange Law

An interchange ring,$(R,+,\bullet )$, is an abelian group with a second binary operation defined so that the interchange law$(w+x)\bullet (y+z)=(w\bullet y)+(x\bullet z)$ holds. An interchange near ring is the same structure based on a group which may not be abelian. It is shown that each interchange (near) ring based on a group $G$ is formed from a pair of endomorphisms of $G$ whose images commute, and that all interchange (near) rings based on $G$ can be characterized in this manner. To obtain an associative interchange ring, the endomorphisms must be commuting idempotents in the endomorphism semigroup of $G$. For $G$ a finite abelian group, we develop a group-theoretic analogue of the simultaneous diagonalization of idempotent linear operators and show that pairs of endomorphisms which yield associative interchange rings can be diagonalized and then put into a canonical form. A best possible upper bound of $4^{r}$ can be given for the number of distinct isomorphism classes of associative interchange rings based on a finite abelian group $A$ which is a direct sum of $r$ cyclic groups of prime power order. If $A$ is a direct sum of $r$ copies of the same cyclic group of prime power order, we show that there are exactly ${\textstyle \frac{1}{6}}(r+1)(r+2)(r+3)$ distinct isomorphism classes of associative interchange rings based on $A$. Several examples are given and further comments are made about the general theory of interchange rings.

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.

A note on rank and direct decompositions of torsion-free Abelian groups. II

1969 ◽  
Vol 66 (2) ◽  
pp. 239-240 ◽  
Author(s):  
A. L. S. Corner
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Direct Sums ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Free Abelian Groups ◽  
Rank 2 ◽  
Direct Decompositions ◽  
Torsion Free

According to well-known theorems of Kaplansky and Baer–Kulikov–Kapla nsky–Fuchs (4, 2), the class of direct sums of countable Abelian groups and the class of direct sums of torsion-free Abelian groups of rank 1 are both closed under the formation of direct summands. In this note I give an example to show that the class of direct sums of torsion-free Abelian groups of finite rank does not share this closure property: more precisely, there exists a torsion-free Abelian group G which can be written both as a direct sum G = A⊕B of 2 indecomposable groups A, B of rank ℵ0 and as a direct sum G = ⊕n ε zCn of ℵ0 indecomposable groups Cn (nεZ) of rank 2, where Z is the set of all integers.

FINITE-FINITARY GROUPS OF AUTOMORPHISMS

2002 ◽  
Vol 01 (04) ◽  
pp. 375-389 ◽  
Author(s):  
B. A. F. WEHRFRITZ
Keyword(s):  
Abelian Group ◽  
Finite Subgroup ◽  
Linear Groups ◽  
Cyclic Groups ◽  
The Core ◽  
Direct Sum ◽  
Abelian Case ◽  
Finitary Linear ◽  
Group A ◽  
Groups Of Automorphisms

In this paper we attempt to describe the structure of groups G of automorphisms of an abelian group M with the property that M(g - 1) is finite for every element g of G. These groups are closely related to the finitary linear groups over finite fields. The abelian case is critical for our work and the core result of this paper is the following. An abelian group A is isomorphic to a group G as above with M torsion if and only if A is torsion and has a residually-finite subgroup B with A/B a direct sum of cyclic groups.

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