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Word MetricIn mathematics, the word metric is a metric defined on a group, depending on a set of generators for the group. Given a group G, and a set of generators for it, every element x of G can be written as a finite product of generators and their inverses, called a word representing x. The distance from the identity element e and x is defined to be the length of the shortest word representing x. The distance between x and y in G is then defined to be the distance between e and . Intuitively, one wants the natural left (or right) action of the group on itself to be an isometry. This is easily checked to satisfy the axioms for a metric.
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