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On the Complexity of the Classification Problem for Torsion-Free Abelian Groups of Finite Rank

Published online by Cambridge University Press:  15 January 2014

Simon Thomas
Simon Thomas*
Affiliation:
Mathematics Department, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, USA. E-mail: sthomas@math.rutgers.edu
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Extract

In this paper, we shall discuss some recent contributions to the project [15, 14, 2, 18, 22, 23] of explaining why no satisfactory system of complete invariants has yet been found for the torsion-free abelian groups of finite 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 ℚn which contain n linearly independent elements. Thus the collection of torsion-free abelian groups of rank at most n can be naturally identified with the set S (ℚn) of all nontrivial additive subgroups of ℚn. In 1937, Baer [4] solved the classification problem for the class S(ℚ)of rank 1 groups as follows.

Let ℙ be the set of primes. If G is a torsion-free abelian group and 0 ≠ x ϵ G, then the p-height of x is defined to be

hx(p) = sup{n ϵ ℕ ∣ There exists y ϵ G such that pny = x} ϵ ℕ ∪{∞}; and the characteristic χ (x) of x is defined to be the function

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 7 , Issue 3 , September 2001 , pp. 329 - 344
Copyright
Copyright © Association for Symbolic Logic 2001

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