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.

Table of contents |

2 Properties 3 Curiosities about large rank groups |

## 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*

*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 **Z**^{n} 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

**Q**yields a short exact sequence of

**Q**-vectorspaces 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:

## 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 2*n*-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 *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 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 *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.