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:[ 1] [ 2] [ 3]
L
(
e
)
=
0
,
L
(
g
− − -->
1
)
=
L
(
g
)
L
(
g
1
g
2
)
≤ ≤ -->
L
(
g
1
)
+
L
(
g
2
)
,
∀ ∀ -->
g
1
,
g
2
∈ ∈ -->
G
.
{\displaystyle {\begin{aligned}L(e)&=0,\\L(g^{-1})&=L(g)\\L(g_{1}g_{2})&\leq L(g_{1})+L(g_{2}),\quad \forall g_{1},g_{2}\in G.\end{aligned}}}
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
S
i
:=
{
g
∣ ∣ -->
L
(
g
)
≤ ≤ -->
i
}
{\displaystyle S_{i}:=\{g\mid L(g)\leq i\}}
do not form subgroups in general.
However, the group algebra of a group with a length functions forms a filtered algebra : the axiom
L
(
g
h
)
≤ ≤ -->
L
(
g
)
+
L
(
h
)
{\displaystyle L(gh)\leq L(g)+L(h)}
corresponds to the filtration axiom.
References
^ Lyndon, Roger C. (1963), "Length functions in groups", Mathematica Scandinavica , 12 : 209–234, doi :10.7146/math.scand.a-10684 , JSTOR 24489388 , MR 0163947
^ Harrison, Nancy (1972), "Real length functions in groups", Transactions of the American Mathematical Society , 174 : 77–106, doi :10.2307/1996098 , MR 0308283
^ Chiswell, I. M. (1976), "Abstract length functions in groups", Mathematical Proceedings of the Cambridge Philosophical Society , 80 (3): 451–463, doi :10.1017/S0305004100053093 , MR 0427480
This article incorporates material from Length function on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License .