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

An abelian group $\langle G, * \rangle$ is a set G, together with a binary operation * on G, such that the following axioms are satisfied:

Associativity

For all a, b and c in G, (a * b) *c = a * (b * c).

Identity element

There is an element e in G, such that e * x = x * e = x for all x in G. This element e is an identity element for * on G.

Inverse element

For each a in G, there is an element a′ in G with the property that a′ * a = a * a′ = e. The element a′ is an inverse of a with respect to *.

Commutativity

For all a, b in G, a * b = b * a.^[1][2][3]

Order

For this definition, note that in an abelian group, the binary operation is usually called addition, the symbol for addition is “+”^[4] and a repeated sum, $a + a + \cdots + a$ of the same element appearing n times is usually abbreviated “na”.^[5] Let G be a group and a ∈ G. If there is a positive integer n such that na = e, the least such positive integer n is the order of a. If no such n exists, then a is of infinite order.^[6][7][8]

Torsion

A group G is a torsion group if every element in G is of finite order. G is torsion free if no element other than the identity is of finite order.^[9][10][11]

Properties

• A torsion-free abelian group has no non-trivial finite subgroups.
• A finitely generated torsion-free abelian group is free.^[12]

See also

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

Notes

1. Fraleigh (1976, pp. 18−20)
2. Herstein (1964, pp. 26−27)
3. McCoy (1968, pp. 143−146)
4. Fraleigh (1976, p. 27)
5. Fraleigh (1976, p. 30)
6. Fraleigh (1976, pp. 50,72)
7. Herstein (1964, p. 37)
8. McCoy (1968, p. 166)
9. Fraleigh (1976, p. 78)
10. Lang (2002, p. 42)
11. Hungerford (1974, p. 78)
12. 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