Torsion-free abelian groups of rank 1
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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:
- [math]\displaystyle{ 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 apn such that:
- [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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
- 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. 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.
- Phillip A. Griffith (1970). Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 0-226-30870-7. Chapter VII.
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