Torsion-free abelian group
|Algebraic structure → Group theory|
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.
An abelian group is said to be torsion-free if no element other than the identity is of finite order. 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 , as only the integer 0 can be added to itself finitely many times to reach 0.
- A torsion-free abelian group has no non-trivial finite subgroups.
- A finitely generated torsion-free abelian group is free.
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