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In group theory, a word metric on a discrete group G {\displaystyle G} is a way to measure distance between any two elements of G {\displaystyle G}. As the name suggests, the word metric is a metric on G {\displaystyle G} , assigning to any two elements g {\displaystyle g} , h {\displaystyle h} of G {\displaystyle G} a distance d {\displaystyle d} that measures how efficiently their difference g − 1 h {\displaystyle g^{-1}h} can be expressed as a word whose letters come from a generating set for the group. The word metric on G is very closely related to the Cayley graph of G: the word metric measures the length of the shortest path in the Cayley graph between two elements of G.
A generating set for G {\displaystyle G} must first be chosen before a word metric on G {\displaystyle G} is specified. Different choices of a generating set will typically yield different word metrics. While this seems at first to be a weakness in the concept of the word metric, it can be exploited to prove theorems about geometric properties of groups, as is done in geometric group theory.