In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

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: :

\beginL(e) &= 0,\\
L(g^) &= L(g)\\
L(g_1 g_2) &\leq L(g_1) + L(g_2), \quad\forall g_1, g_2 \in G.
\end

Compare with the

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

s for a

metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...

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 In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite 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 In mathematics, the longest element of a Coxeter group is the unique element of maximal length in a finite Coxeter group with respect to the chosen generating set consisting of simple reflections. It is often denoted by ''w''0. See and . Prop ...

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 In mathematics, a filtration \mathcal is an indexed family (S_i)_ of subobjects of a given algebraic structure S, with the index i running over some totally ordered index set I, subject to the condition that ::if i\leq j in I, then S_i\subseteq S ...

, meaning that the

sublevel set In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is: : L_c(f) = \left\~, When the number of independent variables is two, a level set is calle ...

S_i := \

do not form

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

s in general. However, the group algebra of a group with a length functions forms a filtered algebra: the axiom

L(gh) \leq L(g)+L(h)

corresponds to the filtration axiom. {{PlanetMath attribution, id=4365, title=Length function Group theory Geometric group theory