In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group G equipped with a total order "≤" that is translation-invariant. This may have different meanings. We say that (G, ≤) is a:

A group G is said to be left-orderable (or right-orderable, or bi-orderable) if there exists a left- (or right-, or bi-) invariant order on G. A simple necessary condition for a group to be left-orderable is to have no elements of finite order; however this is not a sufficient condition. It is equivalent for a group to be left- or right-orderable; however there exist left-orderable groups which are not bi-orderable.

In this section, ${\displaystyle \leq }$ is a left-invariant order on a group ${\displaystyle G}$ with identity element ${\displaystyle e}$. All that is said applies to right-invariant orders with the obvious modifications. Note that ${\displaystyle \leq }$ being left-invariant is equivalent to the order ${\displaystyle \leq '}$ defined by ${\displaystyle g\leq 'h}$ if and only if ${\displaystyle h^{-1}\leq g^{-1}}$ being right-invariant. In particular, a group being left-orderable is the same as it being right-orderable.

In analogy with ordinary numbers, we call an element ${\displaystyle g\not =e}$ of an ordered group positive if ${\displaystyle e\leq g}$. The set of positive elements in an ordered group is called the positive cone, it is often denoted with ${\displaystyle G_{+}}$; the slightly different notation ${\displaystyle G^{+}}$ is used for the positive cone together with the identity element.

The positive cone ${\displaystyle G_{+}}$ characterises the order ${\displaystyle \leq }$; indeed, by left-invariance we see that ${\displaystyle g\leq h}$ if and only if ${\displaystyle g^{-1}h\in G_{+}}$. In fact, a left-ordered group can be defined as a group ${\displaystyle G}$ together with a subset ${\displaystyle P}$ satisfying the two conditions that:

The order ${\displaystyle \leq _{P}}$ associated with ${\displaystyle P}$ is defined by ${\displaystyle g\leq _{P}h\Leftrightarrow g^{-1}h\in P}$; the first condition amounts to left-invariance and the second to the order being well-defined and total. The positive cone of ${\displaystyle \leq _{P}}$ is ${\displaystyle P}$.

The left-invariant order ${\displaystyle \leq }$ is bi-invariant if and only if it is conjugacy-invariant, that is if ${\displaystyle g\leq h}$ then for any ${\displaystyle x\in G}$ we have ${\displaystyle xgx^{-1}\leq xhx^{-1}}$ as well. This is equivalent to the positive cone being stable under inner automorphisms.

If ${\displaystyle a\in G}$, then the absolute value of ${\displaystyle a}$, denoted by ${\displaystyle |a|}$, is defined to be: $${\displaystyle |a|:={\begin{cases}a,&{\text{if }}a\geq 0,\\a^{-1},&{\text{otherwise}}.\end{cases}}}$$ If in addition the group ${\displaystyle G}$ is abelian, then for any ${\displaystyle a,b\in G}$ a triangle inequality is satisfied: ${\displaystyle |a+b|\leq |a|+|b|}$.