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 in S as well as its inverse . Then define the word metric with respect to S to be the word metric with respect to the symmetrization of S. Example in a free group Suppose that F is the free group on the two element set . A word w in the symmetric generating set is said to be reduced if the letters do not occur next to each other in w, nor do the letters . Every element is represented by a unique reduced word, and this reduced word is the shortest word representing g. For example, since the word is reduced and has length 2, the word norm of equals 2, so the distance in the word norm between and equals 2. This can be visualized in terms of the Cayley graph, where the shortest path between b and a has length 2. Theorems Isometry of the left action The group G acts on itself by left multiplication: the action of each takes each to . This action is an isometry of the word metric. The proof is simple: the distance between and equals , which equals the distance between and . Bilipschitz invariants of a group In general, the word metric on a group G is not unique, because different symmetric generating sets give different word metrics. However, finitely generated word metrics are unique up to bilipschitz equivalence: if , are two symmetric, finite generating sets for G with corresponding word metrics , , then there is a constant such that for any , . This constant K is just the maximum of the word norms of elements of and the word norms of elements of . This proof is also easy: any word over S can be converted by substitution into a word over T, expanding the length of the word by a factor of at most K, and similarly for converting words over T into words over S. The bilipschitz equivalence of word metrics implies in turn that the growth rate of a finitely generated group is a well-defined isomorphism invariant of the group, independent of the choice of a finite generating set. This implies in turn that various properties of growth,