# Torsion-free abelian groups of rank 1

Infinitely generated abelian groups have very complex structure and are far less well understood than finitely generated abelian groups. Even torsion-free abelian groups are vastly more varied in their characteristics than vector spaces. Torsion-free abelian groups of rank 1 are far more amenable than those of higher rank, and a satisfactory classification exists, even though there are an uncountable number of isomorphism classes.

## Definition

A torsion-free abelian group of rank 1 is an abelian group such that every element except the identity has infinite order, and for any two non-identity elements *a* and *b* there is a non-trivial relation between them over the integers:

## Classification of torsion-free abelian groups of rank 1

For any non-identity element *a* in such a group and any prime number *p* there may or may not be another element *a _{pn}* such that:

If such an element exists for every *n*, we say the ** p-root type of a is infinity**, otherwise, if

*n*is the largest non-negative integer that there is such an element, we say the

**.**

*p*-root type of*a*is*n*We call the sequence of *p*-root types of an element *a* for all primes the **root-type** of *a*:

- .

If *b* is another non-identity element of the group, then there is a non-trivial relation between *a* and *b*:

where we may take *n* and *m* to be coprime.

As a consequence of this the root-type of *b* differs from the root-type of *a* only by a finite difference at a finite number of indices (corresponding to those primes which divide either *n* or *m*).

We call the **co-finite equivalence class of a root-type** to be the set of root-types that differ from it by a finite difference at a finite number of indices.

The co-finite equivalence class of the type of a non-identity element is a well-defined invariant of a torsion-free abelian group of rank 1. We call this invariant the **type** of a torsion-free abelian group of rank 1.

If two torsion-free abelian groups of rank 1 have the same type they may be shown to be isomorphic. Hence there is a bijection between types of torsion-free abelian groups of rank 1 and their isomorphism classes, providing a complete classification.

## References

- Reinhold Baer (1937). "Abelian groups without elements of finite order".
*Duke Mathematical Journal*.**3**(1): 68–122. doi:10.1215/S0012-7094-37-00308-9. - Phillip A. Griffith (1970).
*Infinite Abelian group theory*. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 0-226-30870-7. Chapter VIII.