Torsion subgroup of nilpotent group

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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.