Torsion-free abelian groups of rank 1

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

*
* Chapter VIII.


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