Linearly ordered group

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.[1]

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:

1. for ${\displaystyle g,h\in P}$ we have also ${\displaystyle gh\in P}$;
2. let ${\displaystyle P^{-1}=\{g^{-1},g\in P\}}$, then ${\displaystyle G}$ is the disjoint union of ${\displaystyle P,P^{-1}}$ and ${\displaystyle \{e\}}$.

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:

$${\displaystyle |a|:={\begin{cases}a,&{\text{if }}a\geq 0,\\-a,&{\text{otherwise}}.\end{cases}}}$$

If in addition the group ${\displaystyle G}$ is abelian, then for any ${\displaystyle a,b\in G}$ a triangle inequality is satisfied: ${\displaystyle |a+b|\leq |a|+|b|}$.

Any left- or right-orderable group is torsion-free, that is it contains no elements of finite order besides the identity. Conversely, F. W. Levi showed that a torsion-free abelian group is bi-orderable;[2] this is still true for nilpotent groups[3] but there exist torsion-free, finitely presented groups which are not left-orderable.

• Cyclically ordered group
• Hahn embedding theorem
• Partially ordered group

• Deroin, Bertrand; Navas, Andrés; Rivas, Cristóbal (2014). "Groups, orders and dynamics". arXiv:1408.5805 [math.GT].
• Levi, F.W. (1942), "Ordered groups.", Proc. Indian Acad. Sci., A16 (4): 256–263, doi:10.1007/BF03174799, S2CID 198139979
• Fuchs, László; Salce, Luigi (2001), Modules over non-Noetherian domains, Mathematical Surveys and Monographs, vol. 84, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1963-0, MR 1794715
• Ghys, É. (2001), "Groups acting on the circle.", L'Enseignement Mathématique, 47: 329–407