---
tag: F1WYS82NNB
kind: concept
name: Left-orderable group
msc: ["20F60", "06F15"]
specializes: [G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Linearly_ordered_group
---

## Summary

A left-orderable group is a [[group|G5S1M9]] with a total order that is preserved by multiplication on the left.
Every [[orderable group|9BM25E20JQ]] is left-orderable, but the converse fails.
[[Braid groups|BNR4H5GCTP]] are left-orderable, a result of Dehornoy.
Left orders link group theory with dynamics, because countable left-orderable groups are the countable groups that act faithfully on the real line by order-preserving homeomorphisms.

## Explanations

### L1 human

Imagine lining up the elements of a group so that multiplying everything on the left by a fixed element keeps the line in order.
Multiplying on the right may scramble it.
This is weaker than asking both sides to respect the order.
For countable groups, this is the same as acting on the real line by moving points around while keeping their order.