Editing Word metric (section)

=== Variations ===

The word metric has an equivalent definition formulated in more geometric terms using the [[Cayley graph]] of ''G'' with respect to the generating set ''S''. When each edge of the Cayley graph is assigned a metric of length 1, the distance between two group elements ''g'',''h'' in ''G'' is equal to the shortest length of a path in the Cayley graph from the vertex ''g'' to the vertex ''h''.

The word metric on ''G'' can also be defined without assuming that the generating set ''S'' is closed under inverse. To do this, first symmetrize ''S'', replacing it by a larger generating set consisting of each <math> s </math> in ''S'' as well as its inverse <math> s^{-1} </math>. Then define the word metric with respect to ''S'' to be the word metric with respect to the symmetrization of ''S''.