Torsion-free abelian group

About: Torsion-free abelian group is a research topic. Over the lifetime, 71 publications have been published within this topic receiving 3099 citations.

Free subgroups with torsion quotients and profinite subgroups with torus quotients

Abstract: Here “group” means abelian group. Compact connected groups contain ı-subgroups, that is, compact totally disconnected subgroups with torus quotients, which are essential ingredients in the important Resolution Theorem, a description of compact groups. Dually, full free subgroups of discrete torsion-free groups of finite rank are studied in order to obtain a comprehensive picture of the abundance of ı-subgroups of a protorus. Associated concepts are also considered. Mathematics Subject Classification (2010). Primary: 22C05; Secondary: 20K15, 22B05.

The p-rank of Extℤ(G, ℤ) in certain models of ZFC

Abstract: We prove that if the existence of a supercompact cardinal is consistent with ZFC, then it is consistent with ZFC that the p-rank of Ext ℤ(G, ℤ) is as large as possible for every prime p and for any torsion-free Abelian group G. Moreover, given an uncountable strong limit cardinal µ of countable cofinality and a partition of Π (the set of primes) into two disjoint subsets Π0 and Π1, we show that in some model which is very close to ZFC, there is an almost free Abelian group G of size 2µ = µ+ such that the p-rank of Ext ℤ(G, ℤ) equals 2µ = µ+ for every p ∈ Π0 and 0 otherwise, that is, for p ∈ Π1.

Rings on indecomposable torsion free groups of rank two

Abstract: Let G be a torsion free abelian group of rank two. The principal purpose of the present paper is to show the structure of a ring on G. We prove that if G is indecomposable with cardinality of the typeset two or three, then all of the rings on G are associative and commutative. It will be presented two examples, one about an associative ring, and other non-associative ring on the torsion-free abelian groups of rank two. Mathematics Subject Classification: 20K15

On the topological mass lattice groups

Abstract: Let G to be a torsion free abelian group. In this paper we introduce the following concepts: Some interesting results about unital lattice groups and Riesz spaces with the chief link topology on them have been presented.

The cotypeset of a torsion free abelian group of rank two

Abstract: The cotypeset (set of types of rank one factors) of a torsion free abelian group of rank two is characterized. Let G be a torsion free abelian group of finite rank, hereafter called simply a "group". The typeset of G, {type (x) I0 ? x E G}, has been studied by several authors (e.g. [3, 5, 7, 8, 9, 10]). However, the main problem: When is a set of types the typeset of a group G?, has not yet been solved, even for groups of rank two. The cotypeset of G, introduced in [10], is the set of types of all rank one factors of G. This set also seems to be important in the study of torsion free groups (see e.g. [1, 2, 11, 12, 13]). It therefore seems of interest to characterize those sets of types which are the cotypeset of a group G. In this paper we give necessary and sufficient conditions for a set of types to be the cotypeset of a rank two group. Familiarity is assumed with the notions of type and characteristic (height vector) see [6]. In particular, the inner type, IT, and outer type OT, of Warfield are used [13]. The symbols V and A are used to denote the sup and inf, respectively, of collections of characteristics. If t is a characteristic, tP denotes the value of t at the prime p. If T is a type, Tp is called finite (infinite), if tP is finite (infinite) for some characteristic t E T. If T, and T2 are types with , < T2 , then T2-T, iS the type of t2 -t', where t, G T t2 ET2 are characteristics chosen so that tl < t2 (note: oo oo = 0). The type of Z is denoted by 0. Finally, we call two types, T, and 2, equivalent on a subset P of the primes (T, T2 on P) if there exist characteristics E sl t2 E T2 such that tP = tf for allp E P. The characterization of rank two groups by Beaumont-Wisner [4] is employed repeatedly. As usual, hp(x) denotes the p-height of an element x in a group G, and typeG(x) is the type of x in G. We use ( ) (( )*) to denote the subgroup (pure subgroup) generated by a set of elements. We begin with a simple lemma. LEMMA 1. Let S = { E=}I be a set of types and ao a type such that uO = vi V aj for all i Y j in I. Then (a)a a