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 Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such group (mathematics), groups and topology, topological and geometry, geometric pro ...

, a length function is a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

that assigns a number to each element of a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

Definition

A length function ''L'' : ''G'' → R⁺ on a

''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 f ...

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 mathema ...

and a

filtered algebra In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory. A filtered algebra over the field k is an a ...

Word metric

An important example of a length is the

word metric In group theory, a word metric on a discrete group G is a way to measure distance between any two elements of G . As the name suggests, the word metric is a metric on G , assigning to any two elements g , h of G a distance d(g,h) that mea ...

: given a

presentation of a group In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...

by generators and relations, the length of an element is the length of the shortest word expressing it.

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...

s (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 \m ...

) 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 In mathematics, the length of an element ''w'' in a Weyl group ''W'', denoted by ''l''(''w''), is the smallest number ''k'' so that ''w'' is a product of ''k'' reflections by simple roots. (So, the notion depends on the choice of a positive Weyl ch ...

. 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

: 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