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:

n a + m b = 0 \;

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 apn such that:

p^n a_{p^n} = a\;

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:

T(a)=\{t_2,t_3,t_5,\ldots\}\;.

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

n a + m b = 0\;

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