Length function

Let $G$ be a group. A length function on $G$ is a function $L\colon G\to \mathbb {R} ^{+}$ satisfying:

$L(e)$	$=$	$0,$
$L(g)$	$=$	$L(g^{-1}),\quad \forall g\in G,$
$L(g_{1}g_{2})$	$\leq$	$L(g_{1})+L(g_{2}),\quad \forall g_{1},g_{2}\in G.$

Coxeter groups (including the symmetric group) have combinatorial important length functions, for which each simple reflection has length 1. A longest element of a Coxeter group is called a Coxeter element, and is both important and unique up to conjugation.

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Length function at PlanetMath.