More precisely, let A be an abelian group and T the torsion subgroup, T = { a in A : na = 0 for some nonzero integer n }. Let Q denote the rational numbers. The t.f. rank of A is equal to all of the following cardinal numbers:
* The vector space dimension of the tensor product of Q with A
* The vector space dimension of the smallest Qvector space containing the torsionfree group A/T
* The largest cardinal d such that A contains a copy of the direct sum of d copies of the integers
* The cardinality of a maximal Zlinearly indepedent subset of A
An Abelian group is often thought of as composed of its torsion part T, and its torsionfree part A/T. The t.f. rank describes how complicated the the torsion free part can be. There is a complete classification of t.f. rank 1 torsionfree groups. Larger ranks are more difficult to classify, and no current system of classifying rank 2 torsionfree groups is considered very effective.
Larger ranks, especially infinite ranks, are often the source of entertaining paradoxical groups. For instance for every cardinal d, there are many torsionfree abelian groups of rank d that cannot be written as a direct sum of any pair of their proper subgroups. Such groups are called indecomposable, since they are not simply built up from other smaller groups. These examples show that torsionfree rank 1 groups (which are relatively well understood) are not the building blocks of all abelian groups.
Furthermore, for every integer n ≥ 3, there is a rank 2n2 torsionfree abelian group that is simultaneously a sum of two indecomposable groups, and a sum of n indecomposable groups. Hence for ranks 4 and up, even the number of building blocks is not welldefined.
Another example, due to A.L.S. Corner, shows that the situation is as bad as one could possibly imagine: Given integers n ≥ k ≥ 1, there is a torsion free group A of rank n, such that for any partition of n into r_1 + ... + r_k = n, each r_i being a positive integer, A is the direct sum of k indecomposable groups, the first with rank r_1, the second r_2, ..., the k'th with rank r_k. This shows that a single group can have all possible combinations of a given number of building blocks, so that even if one were to know complete decompositions of 2 torsionfree groups, one would not be sure that they were not isomorphic.
Other silly examples include torsionfree rank 2 groups A_(n,m) and B_(n,m) such that A^n is isomorphic to B^n if and only if n is divisible by m.
When one allows infinite rank, one is treated to a group G contained in a group K such that K is indecomposable and is generated by G and a single element, and yet every nonzero direct summand of G has yet another nonzero direct summand.
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