Torsion subgroup of nilpotent group

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This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
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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.