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word metric
In group theory, a branch of mathematics, a word metric on a group is a way to measure distance between any two elements of . As the name suggests, the word metric is a metric on , assigning to any two elements , of a distance that measures how efficiently their difference 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 must first be chosen before a word metric on 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.
== Examples ==
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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