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 do not form subgroups in general.
However, the group algebra of a group with a length functions forms a filtered algebra: the axiom corresponds to the filtration axiom.
This article is copied from an article on Wikipedia® - the free encyclopedia created and edited by its online user community. The text was not checked or edited by anyone on our staff. Although the vast majority of Wikipedia® encyclopedia articles provide accurate and timely information, please do not assume the accuracy of any particular article. This article is distributed under the terms of GNU Free Documentation License.