Word metric

In group theory, a branch of mathematics, a word metric on a group ${\displaystyle G}$ is a way to measure distance between any two elements of ${\displaystyle G}$. As the name suggests, the word metric is a metric on ${\displaystyle G}$, assigning to any two elements ${\displaystyle g}$, ${\displaystyle h}$ of ${\displaystyle G}$ a distance ${\displaystyle d(g,h)}$ that measures how efficiently their difference ${\displaystyle g^{-1}h}$ 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.

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