Torsion subgroup of nilpotent group

Definition
Suppose $$G$$ is a nilpotent group. The torsion subgroup of $$G$$ is defined in the following equivalent ways:


 * 1) It is the subset comprising all the elements of $$G$$ that have finite order. This subset turns out to be a subgroup.
 * 2) It is the largest subgroup of $$G$$ that is a periodic nilpotent group.
 * 3) It is the subgroup generated by all the elements in $$G$$ that have finite order.

Equivalence of definitions
This follows from equivalence of definitions of periodic nilpotent group.