Torsion-free abelian group

Definitions

Main article: Abelian group

An abelian group \langle G, + ,0\rangle is said to be torsion-free if no element other than the identity e is of finite order. Explicitly, for any n 0, the only element x \in G for which nx = 0 is x = 0.

A natural example of a torsion-free group is \langle \mathbb Z,+,0\rangle , as only the integer 0 can be added to itself finitely many times to reach 0. More generally, the free abelian group \mathbb Z^r is torsion-free for any r \in \mathbb N. An important step in the proof of the classification of finitely generated abelian groups is that every such torsion-free group is isomorphic to a \mathbb Z^r.

A non-finitely generated countable example is given by the additive group of the polynomial ring \mathbb Z[X] (the free abelian group of countable rank).

More complicated examples are the additive group of the rational field \mathbb Q, or its subgroups such as \mathbb Z[p^{-1}] (rational numbers whose denominator is a power of p). Yet more involved examples are given by groups of higher rank.

Groups of rank 1

Rank

The rank of an abelian group A is the dimension of the \mathbb Q-vector space \mathbb Q \otimes_{\mathbb Z} A. Equivalently it is the maximal cardinality of a linearly independent (over \Z) subset of A.

If A is torsion-free then it injects into \mathbb Q \otimes_{\mathbb Z} A. Thus, torsion-free abelian groups of rank 1 are exactly subgroups of the additive group \mathbb Q.

Classification

Torsion-free abelian groups of rank 1 have been completely classified. To do so one associates to a group A a subset \tau(A) of the prime numbers, as follows: pick any x \in A \setminus {0}, for a prime p we say that p \in \tau(A) if and only if x \in p^kA for every k \in \mathbb N. This does not depend on the choice of x since for another y \in A\setminus {0} there exists n, m \in \mathbb Z\setminus{0} such that ny = mx. Baer proved that \tau(A) is a complete isomorphism invariant for rank-1 torsion free abelian groups.

Classification problem in general

The hardness of a classification problem for a certain type of structures on a countable set can be quantified using model theory and descriptive set theory. In this sense it has been proved that the classification problem for countable torsion-free abelian groups is as hard as possible.

Notes

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References

1. Thomas, Simon. (2003). "The classification problem for torsion-free abelian groups of finite rank". J. Am. Math. Soc..
2. {{harvtxt. Fraleigh. 1976
3. {{harvtxt. Lang. 2002
4. {{harvtxt. Hungerford. 1974
5. Reinhold Baer. (1937). "Abelian groups without elements of finite order". [[Duke Mathematical Journal]].
6. Phillip A. Griffith. (1970). "Infinite Abelian group theory". University of Chicago Press.
7. (2021). "Torsion-Free Abelian Groups are Borel Complete".