View source for Torsion-free abelian groups of rank 1

{{context|date=May 2017}}
Infinitely generated [[Abelian group|abelian group]]s have very complex structure and are far less well understood than finitely generated abelian groups. Even [[Torsion-free abelian group|torsion-free abelian group]]s are vastly more varied in their characteristics than [[Vector space|vector space]]s. Torsion-free abelian groups of [[Rank of an abelian group|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:

:<math> n a + m b = 0 \;</math>

==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<sub>p<sup>n</sup></sub>'' such that:

:<math>p^n a_{p^n} = a\;</math>

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'':

:<math>T(a)=\{t_2,t_3,t_5,\ldots\}\;</math>.

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

:<math>n a + m b = 0\;</math>

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==
* {{cite journal | author=Reinhold Baer |  title=Abelian groups without elements of finite order | journal=Duke Mathematical Journal | volume=3 | year=1937 | issue=1 | pages=68–122 | doi=10.1215/S0012-7094-37-00308-9 | url=https://projecteuclid.org/journals/duke-mathematical-journal/volume-3/issue-1/Abelian-groups-without-elements-of-finite-order/10.1215/S0012-7094-37-00308-9.short}}
* {{cite book | author=Phillip A. Griffith | title=Infinite Abelian group theory | series=Chicago Lectures in Mathematics | publisher=University of Chicago Press | year=1970 | isbn=0-226-30870-7 }} Chapter VII.

[[Category:Abelian group theory]]

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