Definition:Word Metric

Definition
Let $\left({G, \circ}\right)$ be a group, and let $S$ be a generating set for $G$ which is closed under inverses (that is, $x^{-1} \in S \iff x\in S$).

The word metric on $G$ with respect to $S$ is the metric $d_S$ defined as follows:

For any $g, h \in G$, let $d_S \left({g, h}\right)$ be the minimum length among the finite sequences $\left({x_1, \dots, x_n}\right)$ with each $x_i \in S$ such that $g \circ x_1 \circ \cdots \circ x_n = h$.

Informally, $d_S \left({g, h}\right)$ is the smallest number of elements from $S$ that one needs to multiply by to get from $g$ to $h$.

Also See

 * Word Metric is a Metric, for proof that this indeed defines a metric.