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.

A finite cyclic group is a torsion abelian group.

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

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

• Betti number