Torsion abelian group

In abstract algebra, a torsion abelian group is an abelian group in which every element has finite order. ^[1] For example, the torsion subgroup of an abelian group is a torsion abelian group.

Examples

Finite Cyclic

A finite cyclic group is a torsion abelian group.

Finitely Generated Abelian Group

A finitely generated abelian group is torsion iff it is finite, see Finitely generated abelian group#Classification

Infinite Torsion Abelian Group

A torsion abelian group is not necessarily finite. For instance, the group ⁠${\displaystyle \mathbb {Q} }$⁠/ ⁠${\displaystyle \mathbb {Z} }$⁠

See also

• Betti number

References

1. Dummit, David; Foote, Richard. Abstract Algebra, ISBN   978-0471433347, pp. 369