---
tag: 9BM25E20JQ
kind: concept
name: Orderable group
msc: ["20F60", "06F15"]
specializes: [G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Partially_ordered_group
---

## Summary

An orderable group, or bi-orderable group, is a [[group|G5S1M9]] that admits a total order preserved by multiplication on both sides.
In other words, it is a group that can be made into a [[linearly ordered group|P2E8BNYC7B]].
Orderability is a property of the group, not extra structure, and a group can have many different orders.
Torsion-free abelian groups and free groups are orderable, while groups with nontrivial elements of finite order are not.

## Explanations

### L1 human

Some groups can have their elements lined up so that the line is respected by the group operation.
The integers and the rationals under addition are examples.
A group with an element $g \ne e$ and $g^n = e$ cannot be lined up like this.
If $g$ were larger than $e$, then $g^2$, $g^3$, and so on would be larger still, and $g^n = e$ could not happen.