---
tag: JZNQTTMG58
kind: concept
name: Torsion group
msc: [20F50, 20K10]
specializes: [G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Torsion_group
same_as:
  wikidata: Q1551033
  wikipedia: Torsion group
---

## Summary

A torsion group, also called a periodic group, is a [[group|G5S1M9]] in which every element has finite [[order|0WE2NHZJ7B]].
Every finite group is a torsion group, but there are also infinite ones.
The Burnside problem asks whether a finitely generated torsion group must be finite; the answer is no in general.
For an abelian group, the elements of finite order form the [[torsion subgroup|0DBWXQRXQJ]].

## Explanations

### L1 human

In a torsion group, if you take any element and multiply it by itself again and again, you come back to the identity after finitely many steps.
A clock is a good picture: adding hours, you always return to 12.
The group $\Q/\Z$ of fractions modulo $1$ is an infinite example.
The fraction $\tfrac{3}{7}$ added to itself $7$ times gives $3$, which is $0$ modulo $1$.
The integers are not a torsion group, because adding $1$ to itself never returns to $0$.