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

Publisher： Cambridge University Press

E-ISSN： 1943-5894|7|3|329-344

ISSN： 1079-8986

Source： Bulletin of Symbolic Logic, Vol.7, Iss.3, 2001-10, pp. : 329-344

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Abstract

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 ℚ n 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 (ℚ n ) of all nontrivial additive subgroups of ℚ n . In 1937, Baer [4] solved the classification problem for the class S(ℚ)of rank 1 groups as follows.

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