Length function

In the mathematical field of geometric group theory, a length function is a function that assigns a number to each element of a group.
Definition.
A length function "L" : "G" → R+ on a group "G" is a function satisfying:
Compare with the axioms for a metric and a filtered algebra.
Word metric.
An important example of a length is the word metric: given a presentation of a group by generators and relations, the length of an element is the length of the shortest word expressing it.
Coxeter groups (including the symmetric group) have combinatorial important length functions, using the simple reflections as generators (thus each simple reflection has length 1). See also: length of a Weyl group element.
A longest element of a Coxeter group is both important and unique up to conjugation (up to different choice of simple reflections).
Properties.
A group with a length function does "not" form a filtered group, meaning that the sublevel sets formula_2 do not form subgroups in general.
However, the group algebra of a group with a length functions forms a filtered algebra: the axiom formula_3 corresponds to the filtration axiom.