Torsion subgroup
In the theory of abelian groups, the torsion subgroup A T {\displaystyle A_{T}} of an abelian group A {\displaystyle A} is the subgroup of A {\displaystyle A} consisting of all elements that have finite order (the torsion elements of A {\displaystyle A} ). An abelian group A {\displaystyle A} is called a torsion group (or periodic group) if every element of A {\displaystyle A} has finite order and is called torsion-free if every element of A {\displaystyle A} except the identity is of infinite order.