# Torsion subgroup of nilpotent group

From Groupprops

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

View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions

## Definition

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

- It is the subset comprising all the elements of that have finite order. This subset turns out to be a subgroup.
- It is the largest subgroup of that is a periodic nilpotent group.
- It is the subgroup generated by all the elements in that have finite order.

### Equivalence of definitions

This follows from equivalence of definitions of periodic nilpotent group.