In the mathematical field of geometric group theory, a length function is a function that assigns a number to each element of a group.
A length function L : G → R+ on a group G is a function satisfying:[1][2][3] Compare with the axioms for a metric and a filtered algebra.
Coxeter groups (including the symmetric group) have combinatorially important length functions, using the simple reflections as generators (thus each simple reflection has length 1).
A longest element of a Coxeter group is both important and unique up to conjugation (up to different choice of simple reflections).
This article incorporates material from Length function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.