Definition:Word Metric

Definition
Let $G$ 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 $d_S (g, h)$ be the minimum length among the finite sequences $(x_1, \dots, x_n)$ with each $x_i \in S$ such that $g \cdot x_1 \cdots x_n = h$.

Informally, $d_S (g, h)$ 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