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# Torsion-free abelian group

About: Torsion-free abelian group is a(n) research topic. Over the lifetime, 71 publication(s) have been published within this topic receiving 3099 citation(s).

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TL;DR: In this paper, it was shown that the classification problem for the groups of rank n > 2 is intractible, and that there are 2' pairwise nonisomorphic groups up to isomorphism.

Abstract: In 1937, Baer [5] introduced the notion of the type of an element in a torsion-free abelian group and showed that this notion provided a complete invariant for the classification problem for torsion-free abelian groups of rank 1. Since then, despite the efforts of such mathematicians as Kurosh [23] and Malcev [25], no satisfactory system of complete invariants has been found for the torsion-free abelian groups of finite rank n > 2. So it is natural to ask whether the classification problem is genuinely more difficult for the groups of rank n > 2. Of course, if we wish to show that the classification problem for the groups of rank n > 2 is intractible, it is not enough merely to prove that there are 2' such groups up to isomorphism. For there are 2' pairwise nonisomorphic groups of rank 1, and we have already pointed out that Baer has given a satisfactory classification for this class of groups. In this paper, following Friedman-Stanley [11] and Hjorth-Kechris [15], we shall use the more sensitive notions of descriptive set theory to measure the complexity of the classification problem for the groups of rank n > 2. Recall that, up to isomorphism, the torsion-free abelian groups of rank n are exactly the additive subgroups of the n-dimensional vector space Qf which contain n linearly independent elements. Thus the collection of torsion-free abelian groups of rank 1 < r < n can be naturally identified with the set S(Qn) of all nontrivial additive subgroups of Qn. Notice that S(Qn) is a Borel subset of the Polish space ,p(Qn) of all subsets of Qn, and hence S(Qn) can be regarded as a standard Borel space; i.e. a Polish space equipped with its associated o-algebra of Borel subsets. (Here we are identifying p(Qn) with the space 2'Qn of all functions h: Qn {0, 1} equipped with the product topology.) Furthermore, the natural action of GLn(Q) on the vector space Qn induces a corresponding Borel action on S(Qn); and it is easily checked that if A, B C S(Qn), then A _V B iff there exists an element f E GLn(Q) such that p(A) = B. It follows that the isomorphism relation on S(Qn) is a countable Borel equivalence relation. (If X is a standard Borel space, then a Borel equivalence relation on X is an equivalence relation E C X2 which is a Borel subset of X2. The Borel equivalence relation E is said to be countable iff every E-equivalence class is countable.)

55 citations

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01 Jan 1983

TL;DR: In this article, it was shown that if G is a finite rank torsion free abelian group, then the following statements are equivalent: (i) G is pure subgroup of a finite-rank completely decomposable group C; (ii) g is a homomorphic image of D; and (iii) typeset(G) is finite and for each type τ, G(τ) = Gτ ⨁ *, for some τ-homogenous completely decompositionable group Gτ, and */G*(τ ) is finite.

Abstract: M. C. R. Butler, in 1965, proved that if G is a finite rank torsion free abelian group then the following statements are equivalent: (i) G is a pure subgroup of a finite rank completely decomposable group C; (ii) G is a homomorphic image of a finite rank completely decomposable group D; (iii) typeset(G) is finite and for each type τ, G(τ) = Gτ ⨁ *, for some τ-homogenous completely decomposable group Gτ, and */G*(τ) is finite.

48 citations

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01 Jan 1974

TL;DR: In this article, it was shown that in certain situations cancellation of direct summands is possible up to isomorphism, and that a maximal completely decomposable summand is unique up to the isomorphisms of the subgroups.

Abstract: A finite rank torsion free abelian group G is almost completely decomposable if there exists a completely decomposable subgroup C with finite index in G The minimum of [G: C] over all completely decomposable subgroups C of G is denoted by i(G) An almost completely decomposable group G has, up to isomorphism, only finitely many summands If i(G) is a prime power, then the rank 1 summands in any decomposition of G as a direct sum of indecomposable groups are uniquely determined If G and H are almost completely decomposable groups, then the following statements are equivalent: (i) G Eb L t H Eb L for some finite rank torsion free abelian group L (ii) i(G) = i(H) and H contains a subgroup G' isomorphic to G such that [H: G ] is finite and prime to i(G) (iii) G ED L % H @ L where L is isomorphic to a completely decomposable subgroup with finite index in G A finite rank torsion free abelian group G is almost completely decomposable if there is a completely decomposable subgroup C having finite index in G It is well known that direct sum decompositions of such groups need not be unique In fact, this class of groups is the source of all the most familiar examples of nonunique decompositions of finite rank torsion free abelian groups This paper will show, however, that the situation is not completely unruly We show that in certain situations cancellation of direct summands is possible We show that a maximal completely decomposable summand is unique up to isomorphism We show ttat an almost completely decomposable group G has, up to isomorphism, only finite many summands We show that there are only finitely many groups H for which there exists a finite rank torsion free abelian group L such that G @ L ; H ( L Theorem 11 characterizes such groups H All groups in this paper, unless indicated otherwise, are finite rank torsion free abelian groups In general, we will follow the notation and convenPresented to the Society, February 2, 1973; received by the editors June 18, 1973 AMS (MOS) subject classifications (1970) Primary 20K15

48 citations

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47 citations