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- W1971295898 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. 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 h x ( p ) = sup{ n ϵ ℕ ∣ There exists y ϵ G such that p n y = x } ϵ ℕ ∪{∞}; and the characteristic χ (x) of x is defined to be the function" @default.
- W1971295898 created "2016-06-24" @default.
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- W1971295898 date "2001-10-01" @default.
- W1971295898 modified "2023-09-26" @default.
- W1971295898 title "On the Complexity of the Classification Problem for Torsion-Free Abelian Groups of Finite Rank" @default.
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- W1971295898 doi "https://doi.org/10.2307/2687752" @default.
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