---
tag: 1P8087YGYQ
kind: definition
name: Linearly ordered group
msc: ["06F15", "20F60"]
defines: P2E8BNYC7B
uses: [A49BATWQEH, 0316AYH1SC, 0G77XFBV00]
sources:
  - url: https://en.wikipedia.org/wiki/Linearly_ordered_group
---

## Definition

Let $G$ be a group and let $\le$ be a [[total order|0316AYH1SC]] on $G$.
The pair $(G, \le)$ is a *linearly ordered group*, or *totally ordered group*, if for all $a, b, c \in G$,
$$a \le b \implies ca \le cb \text{ and } ac \le bc.$$
Equivalently, it is an [[ordered group|A49BATWQEH]] whose order is total.

## Remarks

Example: $\Z^2$ under addition with the [[lexicographic order|0G77XFBV00]] is a linearly ordered group.
If $(a, b) \le (c, d)$, then either $a < c$, or $a = c$ and $b \le d$, and both cases survive adding $(x, y)$ to both sides.
Non-example: $\Z^2$ with the componentwise order is an ordered group, but $(1, 0)$ and $(0, 1)$ are not comparable, so the order is not total.
In terms of the positive cone $P = \set{g : e \le g}$, the order is total if and only if $G = P \cup P^{-1}$.
A linearly ordered group has no elements of finite order other than $e$: if $g > e$, then $g^n > e$ for all $n \ge 1$.