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Concept
Torsion-free abelian group
Summary
In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only element with finite order.
While finitely generated abelian groups are completely classified, not much is known about infinitely generated abelian groups, even in the torsion-free countable case.
Abelian group
An abelian group is said to be torsion-free if no element other than the identity is of finite order. Explicitly, for any , the only element for which is .
A natural example of a torsion-free group is , as only the integer 0 can be added to itself finitely many times to reach 0. More generally, the free abelian group is torsion-free for any . An important step in the proof of the classification of finitely generated abelian groups is that every such torsion-free group is isomorphic to a .
A non-finitely generated countable example is given by the additive group of the polynomial ring (the free abelian group of countable rank).
More complicated examples are the additive group of the rational field , or its subgroups such as (rational numbers whose denominator is a power of ). Yet more involved examples are given by groups of higher rank.
Rank of an abelian group
The rank of an abelian group is the dimension of the -vector space . Equivalently it is the maximal cardinality of a linearly independent (over ) subset of .
If is torsion-free then it injects into . Thus, torsion-free abelian groups of rank 1 are exactly subgroups of the additive group .
Torsion-free abelian groups of rank 1 have been completely classified. To do so one associates to a group a subset of the prime numbers, as follows: pick any , for a prime we say that if and only if for every . This does not depend on the choice of since for another there exists such that . Baer proved that is a complete isomorphism invariant for rank-1 torsion free abelian groups.
Official source
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