# Linearly ordered group

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.

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.

Free groups are left-orderable. More generally this is also the case for right-angled Artin groups.^{[4]} Braid groups are also left-orderable.^{[5]}