Rank of an abelian group

In mathematics, 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 a free abelian group it can contain as a subgroup.

Definition

An abelian group is often thought of as composed of its torsion subgroup T, and its torsion-free part A/T. The t.f. rank describes how complicated the torsion-free part can be.

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 the abelian groups Q and 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 Z
• The cardinality of a maximal Z-linearly independent subset of A

Following the same pattern, we may also define t.f. ranks of all modules over any principal ideal domain R. Instead of Q we then use the field of fractions of R.

Properties

There are many abelian groups of rank 0, but the only torsion-free one is the trivial group {0}.

As one would expect, the rank of Zⁿ is n for every natural number n. More generally, the rank of any free abelian group (as explained in that article) coincides with its t.f. rank.

The following fact can often be used to compute ranks: if

<math>0\to A\to B\to C\to 0\;<math>

is a short exact sequence of abelian groups, then

<math>\operatorname{rank}(B)=\operatorname{rank}(A)+\operatorname{rank}(C)\;.<math>

(Proof: tensoring the given sequence with Q yields a short exact sequence of Q-vector spaces since Q is flat; vector space dimensions are additive on short exact sequences.)

Another useful formula, familiar from vector space dimensions, is the following about arbitrary direct sums:

<math>\operatorname{rank}\left(\bigoplus_{j\in J}A_j\right) = \sum_{j\in J}\operatorname{rank}(A_j).<math>

Curiosities about large rank groups

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.