Rank of an abelian group

The rank, or torsion-free rank, of an abelian group measures how large a group is in terms of how large a vector space one would need to "contain" it, or alternatively how large of a free abelian group it can contain.

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 set of 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 Q-vector space containing the torsion-free 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 Z-linearly independent subset of A

An abelian group is often thought of as composed of its torsion part T, and its torsion-free part A/T. The t.f. rank describes how complicated the torsion-free part can be. There is a complete classification of t.f. rank 1 torsion-free groups. Larger ranks are more difficult to classify, and no current system of classifying rank 2 torsion-free 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 torsion-free 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 torsion-free 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 2n-2 torsion-free 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 well-defined.

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 r₁ + ... + r_k = n, each r_i being a positive integer, A is the direct sum of k indecomposable groups, the first with rank r₁, the second r₂, ..., 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 two torsion-free groups, one would not be sure that they were not isomorphic.

Other silly examples include torsion-free rank 2 groups A_n,m and B_n,m such that Aⁿ is isomorphic to Bⁿ 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.