---
tag: PEMNN32A43
kind: concept
name: Torsion-free group
msc: [20F99, 20K15]
specializes: [G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Torsion_(algebra)
---

## Summary

A torsion-free group is a [[group|G5S1M9]] in which the identity is the only element of finite [[order|0WE2NHZJ7B]].
It is the opposite extreme to a [[torsion group|JZNQTTMG58]].
For an abelian group $A$, the quotient of $A$ by its [[torsion subgroup|0DBWXQRXQJ]] is torsion-free.
A finitely generated torsion-free abelian group is a [[free abelian group|0WJX8MGG37]] $\Z^n$.

## Explanations

### L1 human

In a torsion-free group, no element other than the identity ever comes back to the identity when you repeat it.
The integers are the basic example: starting from $3$ and adding $3$ again and again, you get $6, 9, 12, \dots$ and never $0$.
A quarter turn of a circle comes back to the start after four steps.
So a group that contains a quarter turn is not torsion-free.
Torsion-free groups are the groups with no hidden clocks inside.