length function

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

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

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