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.[1]

Definitions

edit

An abelian group   is said to be torsion-free if no element other than the identity   is of finite order.[2][3][4] 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.

Groups of rank 1

edit

Rank

edit

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  .

Classification

edit

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[5][6] that   is a complete isomorphism invariant for rank-1 torsion free abelian groups.

Classification problem in general

edit

The hardness of a classification problem for a certain type of structures on a countable set can be quantified using model theory and descriptive set theory. In this sense it has been proved that the classification problem for countable torsion-free abelian groups is as hard as possible.[7]

Notes

edit
  1. ^ See for instance the introduction to Thomas, Simon (2003), "The classification problem for torsion-free abelian groups of finite rank", J. Am. Math. Soc., 16 (1): 233–258, doi:10.1090/S0894-0347-02-00409-5, Zbl 1021.03043
  2. ^ Fraleigh (1976, p. 78)
  3. ^ Lang (2002, p. 42)
  4. ^ Hungerford (1974, p. 78)
  5. ^ 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.
  6. ^ Phillip A. Griffith (1970). Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 0-226-30870-7. Chapter VII.
  7. ^ Paolini, Gianluca; Shelah, Saharon (2021). "Torsion-Free Abelian Groups are Borel Complete". arXiv:2102.12371 [math.LO].

References

edit