Torsion-free abelian group

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. That is, multiples of any element other than the identity element generate an infinite number of distinct elements of the group.

Definitions

Main article: Abelian group

An abelian group ${\displaystyle \langle G,*,e\rangle }$ is said to be torsion-free if no element other than the identity ${\displaystyle e}$ is of finite order.[1][2][3] Compare this notion to that of a torsion group where every element of the group is of finite order.

A natural example of a torsion-free group is ${\displaystyle \langle \mathbb {Z} ,+,0\rangle }$, as only the integer 0 can be added to itself finitely many times to reach 0.

Properties

• A torsion-free abelian group has no non-trivial finite subgroups.
• A finitely generated torsion-free abelian group is free.[4]

See also

• Rank of an abelian group
• Torsion-free abelian groups of rank 1

Notes

1. Fraleigh (1976, p. 78)
2. Lang (2002, p. 42)
3. Hungerford (1974, p. 78)
4. Lang (2002, p. 45)

References

• Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: Addison-Wesley, ISBN 0-201-01984-1
• Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN 978-1114541016
• Hungerford, Thomas W. (1974), Algebra, New York: Springer-Verlag, ISBN 0-387-90518-9.
• Lang, Serge (2002), Algebra (Revised 3rd ed.), New York: Springer-Verlag, ISBN 0-387-95385-X.
• McCoy, Neal H. (1968), Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68-15225