scholarly journals The classification problem for torsion-free abelian groups of finite rank

2002 ◽  
Vol 16 (1) ◽  
pp. 233-258 ◽  
Author(s):  
Simon Thomas
Keyword(s):  
Finite Rank ◽  
Abelian Groups ◽  
Classification Problem ◽  
Free Abelian Groups ◽  
Torsion Free

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

2001 ◽  
Vol 7 (3) ◽  
pp. 329-344 ◽  
Author(s):  
Simon Thomas
Keyword(s):  
Finite Rank ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Classification Problem ◽  
Dimensional Vector Space ◽  
Torsion Free Abelian Group ◽  
Linearly Independent ◽  
Free Abelian Groups ◽  
Image Position ◽  
Torsion Free

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 behx(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

THE CLASSIFICATION PROBLEM FOR p-LOCAL TORSION-FREE ABELIAN GROUPS OF RANK TWO

2006 ◽  
Vol 06 (02) ◽  
pp. 233-251 ◽  
Author(s):  
GREG HJORTH ◽  
SIMON THOMAS
Keyword(s):  
Abelian Groups ◽  
Classification Problem ◽  
Classification Problems ◽  
Borel Reducibility ◽  
Free Abelian Groups ◽  
Torsion Free

We prove that if p ≠ q are distinct primes, then the classification problems for p-local and q-local torsion-free abelian groups of rank two are incomparable with respect to Borel reducibility.

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