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.

Further definitions

In this section \le is a left-invariant order on a group G with identity element e. All that is said applies to right-invariant orders with the obvious modifications. Note that \le being left-invariant is equivalent to the order \le' defined by 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 g \not= e of an ordered group positive if e \le g. The set of positive elements in an ordered group is called the positive cone, it is often denoted with G_+; the slightly different notation G^+ is used for the positive cone together with the identity element. The positive cone G_+ characterises the order \le; indeed, by left-invariance we see that g \le h if and only if. In fact a left-ordered group can be defined as a group G together with a subset P satisfying the two conditions that: The order \le_P associated with P is defined by ; the first condition amounts to left-invariance and the second to the order being well-defined and total. The positive cone of \le_P is P. The left-invariant order \le is bi-invariant if and only if it is conjugacy invariant, that is if g \le h then for any x \in G we have as well. This is equivalent to the positive cone being stable under inner automorphisms. If a \in G, then the absolute value of a, denoted by |a|, is defined to be: If in addition the group G is abelian, then for any a, b \in G a triangle inequality is satisfied:.

Examples

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; this is still true for nilpotent groups but there exist torsion-free, finitely presented groups which are not left-orderable.

Archimedean ordered groups

Otto Hölder showed that every Archimedean group (a bi-ordered group satisfying an Archimedean property) is isomorphic to a subgroup of the additive group of real numbers,. If we write the Archimedean l.o. group multiplicatively, this may be shown by considering the Dedekind completion, \widehat{G} of the closure of a l.o. group under nth roots. We endow this space with the usual topology of a linear order, and then it can be shown that for each the exponential maps are well defined order preserving/reversing, topological group isomorphisms. Completing a l.o. group can be difficult in the non-Archimedean case. In these cases, one may classify a group by its rank: which is related to the order type of the largest sequence of convex subgroups.

Other examples

Free groups are left-orderable. More generally this is also the case for right-angled Artin groups. Braid groups are also left-orderable. The group given by the presentation is torsion-free but not left-orderable; note that it is a 3-dimensional crystallographic group (it can be realised as the group generated by two glided half-turns with orthogonal axes and the same translation length), and it is the same group that was proven to be a counterexample to the unit conjecture. More generally the topic of orderability of 3--manifold groups is interesting for its relation with various topological invariants. There exists a 3-manifold group which is left-orderable but not bi-orderable (in fact it does not satisfy the weaker property of being locally indicable). Left-orderable groups have also attracted interest from the perspective of dynamical systems as it is known that a countable group is left-orderable if and only if it acts on the real line by homeomorphisms. Non-examples related to this paradigm are lattices in higher rank Lie groups; it is known that (for example) finite-index subgroups in are not left-orderable; a wide generalisation of this has been recently announced.