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| Infinitely generated [[abelian group]]s have very complex structure and are far less well understood than [[finitely generated abelian groups]]. Even [[torsion-free abelian group]]s are vastly more varied in their characteristics than [[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.
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| ==Definition==
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| 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:
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| :<math> n a + m b = 0 \;</math>
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| ==Classification of torsion-free abelian groups of rank 1==
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| 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:
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| :<math>p^n a_{p^n} = a\;</math>
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| 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'' '''.
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| We call the sequence of ''p''-root types of an element ''a'' for all primes the '''root-type''' of ''a'':
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| :<math>T(a)=\{t_2,t_3,t_5,\ldots\}\;</math>.
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| If ''b'' is another non-identity element of the group, then there is a non-trivial relation between ''a'' and ''b'':
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| :<math>n a + m b = 0\;</math>
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| where we may take ''n'' and ''m'' to be [[coprime]].
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| 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'').
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| 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.
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| 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.
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| 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.
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| ==References==
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| * {{cite journal | author=Reinhold Baer | authorlink=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 }}
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| * {{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 VIII.
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| [[Category:Abelian group theory]]
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